Solution Approaches for the Elementary Shortest Path
Problem
Luigi Di Puglia Pugliese
∗
Francesca Guerriero
Department of Electronics, Computer Science and Systems, University of
Calabria, Rende, Italy.
Abstract
In this paper, the elementary single-source all-destinations shortest path
problem is considered. Given a directed graph, containing negative cost cycles,
the aim is to find the paths with minimum cost from a source node to each other
node, that do not contain repeated nodes. Three different solution strategies are
proposed to solve the problem under investigation and their theoretical properties are investigated. The first is a dynamic programming approach, the second
method is based on the solution of the k shortest paths problem, where k is
considered as a variable. The last solution strategy is an enumerative approach
based on a tree structure, constructed with a cycle elimination rule. Theoretical aspects related to the innovative proposed strategies to solve the problem
at hand are investigated. The practical behaviour of the defined algorithms is
also evaluated by considering a meaningful number of test problems.
Keywords: shortest path; negative cost cycles; dynamic programming; k shortest paths; enumerative method.
1
Introduction
The Shortest Path Problem (SPP) is one of the most studied problems in network
optimization [16, 23, 28]. The problem comes up in practice and arises as subproblem in many network optimization algorithms. Solution approaches for the
SPP have been studied for a long time (e.g., [7, 22, 23, 29, 46, 47]). More recently,
some improvements and computational study have been presented by Ahuja et al.
[1], Gabow and Tarjan [31] and Goldberg [35]. The reader is referred to the works
of Cherkassky et al. [11] and Gallo and Pallottino [32] for detailed surveys.
In its basic formulation, the objective is to determine the minimum cost path
through a network from a given source node to a destination node. Several polynomial time solution approaches have been developed in the scientific literature to
∗
Corresponding Author, e-mail: [email protected]
1
address the SPP in the case there are no negative cost cycles (e.g., see [17, 23, 43]).
On a network with negative arc costs, but with non-negative cost cycles, the best
currently known time bound O(nm) is achieved by the Bellman-Ford-Moore algorithm [7, 29, 46], where n and m denote the number of nodes and arcs in the network,
respectively. If the arc costs are non-negative, implementations of Dijkstra’s algorithm [23] achieve better bounds. In particular, an implementation presented by
Fredman and Tarjan [30] runs in O(m + n log n) time.
A trivial extension of the SPP is to find the shortest path tree rooted at a
source node, by considering as destination all the nodes of the network (SPT P).
This problem can be easily solved by the label-correcting methods used to address
the SPP (e.g., see [11, 32]).
An interesting extension of the SPP is the problem to find the first k shortest
paths (kSPP). A set of k shortest paths have to be determined for each node by
considering the first, the second and so on up to the k-th shortest path. This problem
arises in many real-life applications, e.g., in telecommunication and transportation
fields. In these contexts, alternative solutions should be available when the current
best is forbidden for various reasons, e.g., link interruption in telecommunication
network or street closed in transportation plant. Many solution approaches have
been defined to solve the kSPP. The scientific literature takes into account two
variants of the kSPP: 1) the pure kSPP and 2) the so-called loopless kSPP. In
the first case, a node can appear more than once in the solution paths, whereas,
in the latter we are interested in finding the first k shortest paths such that they
contain a node only once. A wide number of papers address the kSPP (e.g., see
[4, 12, 24, 25, 38, 45, 51]). Solution approaches for the loopless kSP P have been
defined by Martins et al. [44] and Yen [62]. Dreyfus [24] and Yen [62] cite several
additional papers on this subject going back as far as 1957.
When negative cost cycles are present in the network, the SPP is not well
defined, that is, a finite optimal solution does not exist. In this case, the problem
is to check whether a simple cycle, whose arc costs sum up to a negative number,
is present in the network. This problem is known as Negative Cost Cycle Detection
Problem (N CCDP). It is worth noting that the N CCDP is polynomial. Several
procedures have been defined for the N CCDP, see, e.g.,[15, 54].
The Bellman-Ford algorithm is one of the earliest and to date asymptotically
fastest algorithm for the N CCDP [11, 34]. Recently, Subramani [53] has introduced a
new approach for the N CCDP; the proposed solution strategy is based on exploiting
the connections between the N CCDP and the problem of checking whether a system
of difference constraints is feasible.
It is important to point out that the scientific literature provides several works
dealing with algorithms that list up all cycles in directed as well as undirected graphs
in which arc costs are not considered [10, 48, 56, 57, 59, 60]. Yamada and Kinoshita
[61] addressed the problem of finding all cycles in a directed graph with negative
costs and they proposed a recursive algorithm. It is worth observing that all the
2
cited references do not solve the SPP in presence of negative cost cycles, rather
they enumerate all the cycles in the network or check whether a negative cost cycle
exists.
The problem to find a shortest path in a network with negative cost cycles is still
a very challenging problem. To the best of our knowledge, no solution approaches
have been defined up to now to solve the elementary shortest path problem (ESPP).
It is, of course, well known that computing shortest paths in the presence of negative
cost cycles is a strongly N P-hard problem [33].
When constraints on the consumption of resource along the path are introduced,
the problem is referred to as the Resource Constrained Elementary Shortest Path
Problem (RCESPP). This problem has attracted the attention of many researchers
in the last years. Starting with the seminal work of Beasley and Christofides [6],
a variety of solution approaches have been developed to solve to optimality the
RCESPP (see, e.g., [19, 39, 40]).
The single-source single-destination version of the ESPP can be viewed as a
particular instance of the RCESPP, where the resource consumption constraints
are removed. In addition, it is possible to relax the elementarity constraints from
the ESPP introducing a new resource that takes into account the number of visited nodes. Thus, the relaxed ESPP can be modeled as a RCSPP. This issue is
explained in more detail in the following.
In this paper, we deal with the more general case in which an elementary shortest path must be found for all nodes. In other words, we address the single-source
all-destinations version of the ESPP. In order to solve to optimality the singlesource all-destinations ESPP, we apply the concepts proposed by Boland et al. [9]
and Righini and Salani [49]. Beside the solution approach derived from the optimal
strategies developed for the RCESPP, innovative methods are also presented. In
particular, we have designed a dynamic k shortest paths algorithm, in which k is
a variable and may assume an arbitrary value for each node of the network. An
enumerative solution approach is also presented. In addition, an in depth analysis
is carried out to evaluate the theoretical complexity of the proposed solution approaches and an extensive computational phase is conducted in order to asses the
efficiency of the defined optimal strategies.
In the context of the RCSPP, the single-source all-destinations SPP is applied
in order to perform network reduction procedure based on the cost, to determine
lower bounds on the optimal cost and to solve the dual Lagrangean problem when
the resource constraints are relaxed in a Lagrangean objective function. When the
RCESPP is considered, the aforementioned elements cannot be applied because of
the negative cost cycles. Thus, the solution approaches for the ESPP proposed in
this paper, can be used to determine lower bounds on the optimal cost and to solve
the dual Lagrangean problem associated with the RCESPP. This information can
be used to improve the performance of the state-of-the-art methods.
In addition, it is worth observing that the ESPP can be used to model the
3
currency conversion problem [50]. When the nodes represent currencies and the
arcs the transactions with costs equal to exchange rate, the problem is to find the
maximum product path, that is, the best exchange sequence. The problem can be
easily transformed in the SPP and, since the graph can contain negative cost cycles,
it can be viewed as an instance of the ESPP.
The proposed methods are able to solve also the single-source single-destination
ESPP. However, it is worth observing that well-tailored solution approaches for the
single-source single-destination version can be devised but this topic is out of the
scope of this work.
The paper is organized as follows. In Section 2 we give some notations and
definitions used in the paper. The proposed solution approaches, along with their
theoretical analysis are presented in Section 3. Section 4 is devoted to the discussion
of the computational results. Section 5 presents our conclusions. The appendix
reports the results of the computational experiments.
2
Notations and Definitions
Let G = (N , A) be a directed graph where N is the set of n nodes, whereas A
denotes the set of m ≤ |N × N | arcs. N contains also the source node s. A
cost cij is associated with each arc (i, j) ∈ A. A cycle on node i is defined as an
ordered sequence of nodes Ci = (i, j1 , . . . , jg , i), where ji 6= jk for all pairs i, k ∈
{1, . . . , g} such that i 6= k. It can also be viewed as a sequence of g + 1 arcs
{(i, j1 ), . . . , (jg−1 , jg ), (jg , i)}. It is worth observing, as shown by Allender [2], that
P
n!(ρ−1)!
the maximum number of cycles in a complete graph is equal to C̃ = nρ=1 ρ!(n−ρ)!
.
Let πsi be a path from node s to node i. It is a sequence of nodes πsi = (s =
i1 , . . . , il = i), l ≥ 2 and a corresponding sequence of l − 1 arcs such that the
h-th arc in the sequence is (ih , ih+1 ) ∈ A for h = 1, . . . , l − 1. Thus, each path
contains at least one arc. Let Mπ (j) be the multiplicity of node j in path π, that
is Mπ (j) = |{v : 0 ≤ v ≤ |π|, iv = j}|. The path π is said to be an elementary path
if Mπ (j) = 1, for all j ∈Pπ. A cycle Ci on node i ∈ N is said to be a negative cost
cycle (N CC) if c(Ci ) = (h,k)∈Ci chk < 0.
Let πsj be an elementary path from node s to node j such that i ∈ πsj for some
i ∈ N \ {s}. Let πsi = πsj ∪ {(j, i)} be the path obtained by extending path πsj to
node i through arc (j, i). Since i ∈ πsj , a cycle Ci = πij ∪ {j, i} is created.
Proposition 1. Given a path πsj and its sub-path πsi , Ci is a N CC if c(πsi ) >
c(πsj ) + cji .
Proof. The cost of the path πsj can be viewed as the sum of the cost of the sub-path
πsi and the sub-path from i to j, that is c(πsj ) = c(πsi ) + c(πij ). It follows that
c(πsi ) > c(πsj ) + cji ⇔ c(πsi ) > c(πsi ) + c(πij ) + cji ⇔ 0 > c(πij ) + cji = c(Ci ).
This concludes the proof.
4
Since the directed graph G is not assumed to be acyclic and the arc costs are not
constrained in sign, there may be N CCs in G.
The ESPP, from the source s to all other nodes, consists in finding the paths
πsi , ∀i ∈ N \ {s} such that c(πsi ) is minimized and Mπsi (j) = 1, ∀j ∈ πsi . It is
well known that the Bellman’s optimality principle is not verified for this problem,
indeed sub-paths of the optimal path for some node cannot be optimal. In what
∗ , u ∈ N , u 6= s, represents
follows, we formalize this property. Let us assume that πsu
the path with minimum cost and no repeated nodes in a graph with N CCs. The
following consideration can be formulated.
∗ from node s to node u is composed by subObservation 1. The optimal path πsu
∗ that are not guaranteed to be optimal
paths πsi from node s to each other node i ∈ πsu
for node i.
∗ =
Proof. We prove the result by contradiction. Let us suppose that a sub-path πsj
∗
∗
∗
πsi ∪ πij is part of the optimal solution πsu , and, for the purpose of contradiction, we
∗ is the optimal path from s to i. It follows that c(π ∗ ) = c(π ∗ )+c(π ∗ ).
assume that πsi
sj
si
ij
Let us suppose that another path π̄sj : i 6∈ π̄sj exists to reach node j. Since π̄sj is
∗ , we have that c(π̄ ) ≥ c(π ∗ ).
different from πsj
sj
sj
Let us suppose that a path π̄ji exists. If the path π̄sj is extended to node i through
∗ ∪ π̄ = C
path π̄ji , then we have that c(π̄si ) = c(π̄sj ) + c(π̄ji ). We assume that πij
ji
i
∗
is a N CC. Since we assume that πsi is the optimal path from node s to node i, it
∗ ). This means that c(π̄ ) + c(π̄ ) ≥ c(π ∗ ) − c(π ∗ ), thus
follows that c(π̄si ) ≥ c(πsi
sj
ji
sj
ij
∗
c(π̄sj ) − c(πsj ) ≥ −c(Ci ). Since c(Ci ) < 0, for a sufficiently low cost c(π̄sj ), that
∗ ) − c(C ), the relation c(π̄ ) − c(π ∗ ) ≥ −c(C ) is not verified,
is, for c(π̄sj ) < c(πsj
i
sj
i
sj
∗.
contradicting the optimality hypothesis on path πsi
In order to better explain Observation 1, let us consider the directed graph of
Figure 1.
Figure 1: Directed graph example.
∗ = {s, 1, 2, 3} of cost 0. The cost
The optimal path from node s to node 3 is πs3
∗
∗
of sub-path πs1 is equal to 1 but c(πs1 ) is not the least cost from s to 1. Indeed, the
path π̄s1 = {s, 2, 1} is the optimal path of cost 0 to node 1.
5
Observation 1 suggests that it should be necessary to keep more than one path
for some node. It follows that the optimal solution of the ESPP is not guaranteed
to be a spanning tree.
Under this respect, it is possible to devise a modified label-correcting scheme
for the SPP in order to obtain valid upper bounds for the ESPP. Let yi be the
label that stores the value of the dual price for each node i ∈ N and L denote the
list of nodes, that have to be processed. We indicate with pred ∈ <|N | the vector
of predecessors, that is predi = j is the predecessor of node i in the path πsi . A
set F S(i) of successor nodes is associated with each node i ∈ N , that is defined as
follows: F S(i) = {j : (i, j) ∈ A}.
A general scheme of a label-correcting method to obtain upper bounds for the
ESPP is reported in Algorithm 1.
Algorithm 1 : Label Correcting Scheme
1: Step 1 (Initialization phase)
2: ys = 0; yi = +∞, ∀i ∈ N \ {s}; predi = i, ∀i ∈ N ; L = {s}.
3:
4: Step 2 (Node selection)
5: Select a node i from the list L and remove it from L.
6:
7: Step 3 (Label extension)
8: for all j ∈ F S(i) do
9:
if yj > yi + cij then
10:
if j 6∈ πsi then
11:
yj = yi + cij ;
12:
predj = i;
13:
add node j to L if it does not already belong to it.
14:
end if
15:
end if
16: end for
17:
18: Step 4 (Termination check)
19: if L = ∅ then
20:
STOP. yi , ∀i ∈ N , is an upper bound on the optimal cost of the path from node s to node i.
21: else
22:
Go to Step 2.
23: end if
Algorithm 1 is a classical label-correcting method where the elementarity constraints are taken into account. The combination of conditions introduced in line 10
and line 19 ensures that the built paths do not contain repeated nodes and that the
algorithm terminates in a finite number of iterations. Since the procedure is based
on Bellman’s optimality conditions, at the end of Algorithm 1, the label yi , ∀i ∈ N ,
represents an upper bound on the elementary shortest path cost from node s to node
i.
6
3
Proposed Solution Approaches
The solution approaches proposed in this paper are based on the idea to compute
for each node the minimum set of paths such that the elementary shortest path is
found for all nodes.
Three strategies have been developed. The first one can be viewed as an application of the methods proposed to solve the RCESPP. A multi-dimensional labeling
procedure is devised and the dimension of the label is updated dynamically. The
second strategy is based on a labeling approach to solve the kSPP. In our case,
k is considered as a variable that can take different values for each node. In the
last solution approach, a tree structure is defined and an enumerative procedure is
devised in order to exploit the defined tree. It is worth observing that, while the
first solution approach can be viewed as an extension of the algorithms for solving
the RCESPP well-tailored to the ESPP, the last two strategies are proposed for
the first time to solve instances of the SPP in presence of N CCs.
3.1
Dynamic Multi-Dimensional Labeling Approach
The approaches, presented in this section, use the concept of critical node introduced
by Kohl [42]. In particular, a node i is said to be critical if there exists a cycle Ci
that is a N CC. A binary variable is introduced for each critical node in order to
keep track of whether the node is visited. This binary variable can be viewed as a
resource associated with node i, bounded to be less than or equal to one (see [6]).
Let S = {i1 , i2 , . . . , i|S| }, be the set of critical nodes and r[p], p = 1, . . . , |S| be
the binary variable associated with node ip ∈ S.
We denote as yi = (c(πsi ), ri ) the label that contains the cost and the resource consumptions vector of the path πsi from node s to node i. The resource
consumptions vector ri has the following form: ri [p] = 1, ∀p : jp ∈ πsi and
ri [p] = 0, ∀p : jp 6∈ πsi . In other words, the vector ri keeps trace of the nodes
j ∈ S that are visited along the path πsi .
1 ), r 1 ) and y 2 = (c(π 2 ), r 2 ) be two labels associated
Definition 1. Let yi1 = (c(πsi
i
i
si
i
2
1 ) ≤ c(π 2 ), r 1 [p] ≤ r 2 [p]
with node i. The label yi is dominated by the label yi1 if c(πsi
si
i
i
for p = 1, . . . , |S| and at least one inequality is strict.
Definition 2. A label yi = (c(πsi ), ri ) associated with node i is said to be efficient
or Pareto-optimal if there does not exist a label ȳi that dominates it.
Definition 3. A label yi = (c(πsi ), ri ) associated with node i is said to be feasible if
ri [p] ≤ 1, ∀p = 1, . . . , |S|.
A set D(i), containing all the efficient labels, is associated with each node i ∈ N .
The procedure defined by Desrochers [18] can be applied to solve the ESPP. It
is worth observing that the Desrochers’ algorithm is an extension of Algorithm 1,
7
without line 10, to the constrained case. In Algorithm 2, we report the steps of the
label-correcting method extended to the constrained case.
Algorithm 2 : Label-Correcting Scheme for the ESPP
1: Step 1 (Initialization phase)
2: ys1 = (0, 0); D(s) = {ys1 }; D(j) = ∅, ∀j ∈ N \ {s}; L = {s}.
3:
4: Step 2 (Node selection)
5: Select a node i from the list L and remove it from L.
6:
7: Step 3 (Label extention)
8: for all yiξ ∈ D(i) do
9:
for all j ∈ F S(i), j 6= s do
ξ
10:
Set: c(π̄sj ) = c(πsi
) + cij ; r̄j [p] = riξ [p], p = 1, . . . , |S|;
11:
if j ∈ S then
12:
r̄j [p̂] + 1 with ip̂ = j.
13:
end if
14:
if r̄i [p] ≤ 1, p = 1, . . . , |S| then
15:
if ȳj = (c(π̄sj ), r̄j ) is not dominated by any label in D(j) then
16:
D(j) = D(j) ∪ {ȳj };
17:
remove from D(j) all the labels that are dominated by ȳj ;
18:
add node j to L if it does not already belong to it.
19:
end if
20:
end if
21:
end for
22: end for
23:
24: Step 4 (Termination check)
25: if L = ∅ then
26:
STOP. The label yi = minȳ∈D(i) {c(π̄)si } is associated with the optimal elementary path from s to
i, ∀i ∈ N .
27: else
28:
Go to Step 2.
29: end if
It is possible to devise also a label-setting algorithm by slightly modifying Algorithm 2. Indeed, the set L stores all the efficient and feasible labels and it is
initialized by the label ys . At Step 2, from the list L, the lexicographically minimal
label is selected. This label is used to generate feasible and efficient labels to each
successor node. This procedure is the same as the one proposed by Boland et al. [9]
called general label-setting algorithm.
The main drawback of the procedure is that the set S of critical nodes must be
known in advance. The enumerative procedure of Yamada and Kinoshita [61] could
be executed to determine the entire set of negative cost cycles. The nodes belonging
to each negative cost cycle can be used to initialize set S and the Algorithm 2 can
be performed. However, this strategy could be inefficient, since only a sub-set of all
nodes belonging to all negative cost cycles need to be considered. In addition, the
procedure defined by Yamada and Kinoshita [61] is not polynomial.
If the set S is not known in advance, then we can relax the elementarity constraints and introduce a surrogate resource γ that counts the number of nodes in the
path to get the relaxed ESPP (ESPP R ). This idea was proposed by Feillet et al.
8
[26] and applied by Righini and Salani [49] for solving the RCESPP. The ESPP R
is a RCSPP where the resource γ is bounded to be less than or equal to the number
of nodes n. The optimal solution to the ESPP R may not be an elementary path.
The nodes that appear more than once in the optimal paths πsi , ∀i ∈ N \ {s}, are
marked as critical nodes and the RCSPP is run again until an elementary path is
found. The approach to augment the set S is the same of that proposed by Boland
et al. [9] and by Righini and Salani [49]. The difference is that in the latters only one
path has to be found. For more detail, the reader is referred to [20]. Very recently,
in [37] the authors proposed a new state-space reduction technique combined with
the strategies defined in [49]. This strategy cannot be extended to the single-source
all-destinations version of the ESPP.
The solution approaches presented in [49, 9] can be modified in order to define
well-tailored procedure for the ESPP. In particular, starting from S = ∅, the labelsetting algorithm is executed. The search is stopped when a N CC, named Ci , is
detected. The set S is incremented by adding node i and the label-setting algorithm
is run again. The procedure terminates when no N CC is detected. This approach
is similar to the general state-space augmenting algorithm proposed in [9] with the
difference that the label-setting algorithm is stopped when a negative cost cycle
is detected. In this case, the surrogate resource γ is not necessary. In Algorithm
3, we describe the steps of the aforementioned procedure, whereas the truncated
label-setting algorithm is depicted in Algorithm 4
Algorithm 3 : Dynamic Multi-Dimensional Labeling Approach
1: Step 1 (Initialization phase)
2: Set: ζ = 0; S ζ = ∅
3:
4: Step 2 (Run the truncated label-setting algorithm)
5: Run Algorithm 4 with critical node restricted to S ζ and get C.
6:
7: Step 3 (Cycle detection)
8: if C 6= ∅ then
9:
S ζ+1 = S ζ ∪ {j}, j : M (j)C = 2;
10:
ζ + +;
11:
Go to Step 2.
12: else
13:
STOP.
14: end if
The defined solution approach is quite different from those proposed by Boland et
al. [9] and Righini and Salani [49]. For this reason and for the sake of completeness,
in the next Section we prove its correctness.
3.1.1
Correctness of the Dynamic Multi-Dimensional Labeling Approach
In this section, we prove the correctness of the proposed solution approach. In
particular, at the end of Algorithm 3, the label yi with the smallest cost among
9
Algorithm 4 : Truncated label-setting algorithm with S ζ
1: Step 1 (Initialization phase)
2: Set: ys1 = (0, 0); D(s) = {ys1 }; D(j) = ∅, ∀j ∈ N \ {s}; L = {ys1 }; C = ∅.
3:
4: Step 2 (Label selection)
5: Select the lexicografically minimal label yiξ from the list L and remove it from L.
6:
7: Step 3 (Label extension)
8: for all j ∈ F S(i), j 6= s do
ξ
9:
Set: c(π̄sj ) = c(πsi
) + cij ; r̄j [p] = riξ [p], p = 1, . . . , |S ζ |;
10:
if j ∈ S ζ then
11:
r̄j [p̂] + 1 with ip̂ = j.
12:
end if
ξ
ξ
13:
if j 6∈ S ζ and j ∈ πsi
and c(π̄sj ) < c(πsj
) then
14:
STOP. A N CC C is detected.
15:
Return C.
16:
else
17:
if r̄i [p] ≤ 1, p = 1, . . . , |S ζ | then
18:
if ȳj = (c(π̄sj ), r̄j ) is not dominated by any label in D(j) then
19:
D(j) = D(j) ∪ {ȳj }; L = L ∪ {ȳj };
20:
remove from D(j) and L all the labels that are dominated by ȳj .
21:
end if
22:
end if
23:
end if
24: end for
25:
26: Step 4 (Termination check)
27: if L = ∅ then
28:
STOP. The label yi = min{ȳ∈D(i)} {c(π̄)si } is associated with the optimal elementary path from s
to i, ∀i ∈ N .
29:
Return C.
30: else
31:
Go to Step 2.
32: end if
those belonging to the set D(i) is associated with the least cost elementary path
from s to i.
Proposition 2. At the end of Algorithm 4, two situations can occur:
1. a N CC is detected, that is, the stop conditions of line 13 are verified;
2. if the procedure is stopped by condition of line 27, then all the efficient and
feasible paths from s to each other node i ∈ N \ {s} are determined.
(a) At the end of each iteration, the following conditions hold:
i. D(s) = {ys1 } = {(0, 0)};
ii. ∀j ∈ N , if D(j) 6= ∅ [i.e.: D(j) = {yj1 , yj2 , . . . , yjl }] and j 6= s, then
yjξ , ξ = 1, . . . , l is a label related to a feasible path from node s to node
j and D(j) is an efficient set.
10
(b) Upon termination of the algorithm, if D(j) 6= ∅, j ∈ N and j 6= s, then
D(j) contains the labels of all efficient and feasible paths from node s to
node j.
Proof. Let us consider case 1. The conditions of line 13 derive from Proposition 1.
Thus, if they are verified, then a N CC is detected. If case 1 does not occur, then
all N CCs have been detected and the algorithm terminates when the list L is found
empty, that is, case 2.
In what follows, we prove case 2.
Let us consider case 2a. Condition 2(a)i holds because, initially, D(s) = {ys1 } =
{(0, 0)} and, by the rules of the algorithm, D(s) cannot change.
We prove condition 2(a)ii by induction on the iteration count. Indeed, initially,
condition 2(a)ii holds, since node s is the only node for which the set D(s) is
nonempty. Suppose that 2(a)ii holds for some node j at the beginning of some
iteration. Let yiδ be the label removed from L.
If i = s (which only occurs at the first iteration) and δ = 1, then at the end
of this iteration, we have D(j) = yj1 for all successor nodes j of s such that the
corresponding path πsj is feasible, D(j) = ∅ for all other nodes j 6= s, j ∈
/ F S(i).
Thus, the set of labels has the required property.
δ starting from s and ending
If i 6= s, then yiδ is the label of some feasible path πsi
to i that is not dominated by the other paths in D(i), by the induction hypothesis.
If the set D(j) changes, for some node j, such that j ∈ F S(i), as a result of the
iteration, a new feasible label ȳj = (c(π̄sj ), r̄j ) is obtained for node j. The created
δ followed by the arc
label is related to the feasible path πsj consisting of path πsi
(i, j). Finally, note that, by the rules of the algorithm, the newly created label is
added to D(j) only if it is an efficient label. This completes the induction proof of
2(a)ii.
Let now consider condition 2b. Using part 2(a)ii, we have that, at each iteration,
∀j ∈ N such that D(j) 6= ∅, D(j) is an efficient set. Thus, the property mentioned
is also satisfied when the algorithm terminates. In addition, the way in which
the candidate list L is updated and the termination condition (i.e., the algorithm
terminates when there are no more labels left to be scanned) guarantee that all the
labels with the potential to determine a new label for at least one node are scanned
during the execution of the algorithm.
Proposition 3. (Correctness of Algorithm 3). At the end of Algorithm 3, the label
yi with the smallest cost among those belonging to the set D(i) is associated with the
least cost elementary path from s to i, ∀i ∈ N .
Proof. From Proposition 2, it follows that all the efficient and feasible labels yi , ∀i ∈
N , are determined.
Since the set D(i) is an efficient set, the path
11
π̄si = arg min{y∈D(i)} {c(πsi )} has the cost as low as possible among all the feasible paths from s to i, that is π̄si is the optimal solution ∀i ∈ N .
3.1.2
Complexity Analysis
As far as the complexity analysis of the proposed dynamic multi-dimensional labeling
approach for the ESPP is concerned, let D be the set of efficient andP
feasible labels
with the highest cardinality. The complexity of Algorithm 3 is O(n2 nτ=1 τ 22τ ), in
the worst case.
This result is formally stated in the following lemma.
Lemma 1. In the worst case, the complexity of Algorithm 3 is O(4n n3 ).
Proof. Let |D| be the maximum number of labels at a node. The operations executed
in lines 9 - 23 of Algorithm 4 are |S||D|, that is, the test of dominance. Since the
F S can contain at most n − 1 elements, the total number of operations executed
in lines 8 - 24 is n|S||D|. Since the for-loop of line 8 is invoked n|D| times, that
is an upper bound on the total number of generated labels, thus Algorithm 4 takes
O(n2 |S||D|2 ) operations. In the worst case, the number of efficient labels stored at
each node is equal to 2|S| . It follows that Algorithm 4 takes O(n2 |S|4|S| ) operations.
Starting from S = ∅, the set of critical nodes is incremented by one at each iteration.
In the worst case, set S contains all nodes. In other words, the operations executed
by Algorithm 3 are n2 × 1 × 4 + n2 × 2 × 42 + . . . + n2 × n × 4n = 94 n2 [4n (3n − 1) + 1].
Consequently, the complexity of Algorithm 3 is O(4n n3 ), in the worst case.
In order to better explain how Algorithm 3 works, let us consider the example of
Figure 2. In Table 1, we report set S, the labels at the last iteration of Algorithm 4,
the last selected label and the detected cycle C. The labels reported in Table 1 have
the following form: < π(y) >. The superscript reported next to the labels indicates
that the related label is dominated.
Figure 2: Toy example.
At the first iteration (see column 1st iteration of Table 1), the label < s, 2, 5, 4(−8) >
is selected. When we try to extend the label to node 2 a N CC is detected. Indeed,
path πs2 = {s, 2, 5, 4} ∪ {2} has a cost equal to −5 that is less than the cost of path
πs2 = {s, 2} associated with label < s, 2(−4) >. The dummy resource is introduced
to node 2. At the 2nd iteration, the N CC {1, 2, 3, 1} is detected and node 1 is inserted
12
S
s
1
1st iteration
∅
< s(0) >
< s, 1(2) >
2
< s, 2(−4) >
3
< s, 2, 3(−2) >
4
< s, 2, 5, 4(−8) >
5
< s, 2, 5(−6) >
6
< s, 2, 5, 6(−4) >
last extracted label
C
< s, 2, 5, 4(−8) >
{2, 5, 4, 2}
2nd iteration
{2}
< s(0, 0) >
< s, 1(2, 0) >
< s, 2, 1(−1, 1) >
< s, 2(−4, 1) >D
< s, 1, 2(−5, 1) >
< s, 2, 3(−2, 1) >D
< s, 1, 2, 3(−3, 1) >
< s, 2, 3, 1, 4(4, 1) >D
< s, 2, 5, 4(−8, 1) >D
< s, 1, 4(7, 0) >
< s, 1, 2, 5, 4(−9, 1) >
< s, 2, 5(−6, 1) >
< s, 1, 2, 5(−7, 1) >
< s, 2, 5, 6(−4, 1) >D
< s, 2, 5, 4, 6(−5, 1) >D
< s, 1, 2, 5, 4, 6(−6, 1) >
< s, 1, 2, 3(−3, 1) >
{1, 2, 3, 1}
3rd iteration
{2, 1}
< s(0, 0, 0) >
< s, 1(2, 0, 1) >
< s, 2, 1(−1, 1, 1) >
< s, 2(−4, 1, 0) >
< s, 1, 2(−5, 1, 1) >
< s, 2, 3(−2, 1, 0) >
< s, 1, 2, 3(−3, 1, 1) >
< s, 2, 3, 1, 4(4, 1, 1) >D
< s, 2, 5, 4(−8, 1, 0) >
< s, 1, 4(7, 0, 1) >
< s, 1, 2, 5, 4(−9, 1, 1) >
< s, 2, 5(−6, 1, 0) >
< s, 1, 2, 5(−7, 1, 1) >
< s, 2, 5, 6(−4, 1, 0) >D
< s, 2, 5, 4, 6(−5, 1, 0) >
< s, 1, 2, 5, 4, 6(−6, 1, 1) >
< s, 1, 2, 5, 6(−5, 1, 1) >D
< s, 1, 4, 6(10, 0, 1) >
< s, 1, 4(7, 0, 1) >
∅
Table 1: Labels associated with each node at the end of the corresponding iteration
of Algorithm 4.
in the set S. The introduction of the resource at node 2 and to node 1 avoids the
generation of N CCs and the optimal solution is found at the 3rd iteration.
3.2
Labeling Approach based on the k Shortest Paths Problem
In this Section we describe a labeling approach in which the first k paths are determined for each node i ∈ N . The aim of the procedure is to determine the smallest
set of paths for each node i. Under this respect, k is a variable and it can take
different values for each node. The main idea is to start with k = 1 for each node,
whenever a N CC is detected, k is incremented for a specific node and the labeling
algorithm is run again. When no N CC is detected, from the set of paths associated
with each node, the optimal solution is chosen.
It is worth observing that when the kSPP is solved, the paths are ranked by
nondecreasing cost. In our context, it is necessary to store equivalent paths. This is
necessary because paths with the same cost can have a different structure, that is,
they can contain different nodes.
Let Ki be the number of best paths from node s to node i. The set Πi contains the
k ) ≤ c(π k+1 ), ∀k = 1, . . . , K − 1
Ki paths ordered by nondecreasing cost, that is c(πsi
i
si
k+1
k+1
k
k
k is the k-th
and if c(πsi ) = c(πsi ), then πsi 6= πsi . In other words, Πi [k] = πsi
shortest path from node s to node i.
k ∈
Definition 4. Set Πi , ∀ i ∈ N is said to be path-elementary, if each path πsi
Πi , k = 1, . . . , Ki does not contain repeated nodes.
A vector yi ∈ <Ki is associated with each node i that stores the costs of the
k ∈Π .
paths belonging to the set Πi , that is, yi [k] represents the cost of path πsi
i
13
Algorithm 5 : Truncated labeling algorithm for dynamic kSPP
1: Step 1 (Initialization phase)
1 = {s}, Π = {π 1 }; Π = ∅, ∀i ∈ N \ {s};
2: Set: ys [1] = 0; yi [k] = +∞, k = 1, . . . , Ki , ∀i ∈ N \ {s}; πss
s
i
ss
L = {s}; C = ∅.
3:
4: Step 2 (Node selection)
5: Select a node i from L and delete it from L.
6:
7: Step 3 (Label extension)
8: for all j ∈ F S(i), j 6= s do
k̃ AND ∃k̄ : j ∈ π k̄ , y [k̄] + c < y [1] then
9:
if yik 6= +∞, ∀k ∈ {1, . . . , Ki } AND 6 ∃k̃ : j 6∈ πsi
ij
j
si i
k̄ ∪ {(i, j)} is detected.
10:
STOP. A cycle C = πsi
11:
Return C.
12:
else
k do
13:
for all k = 1, . . . , Ki : yi [k] 6= +∞, j 6∈ πsi
14:
for all ξ = 1, . . . , Kj do
k ∪ {(i, j)} 6= π ξ ) then
15:
if yi [k] + cij < yj [ξ] OR (yi [k] + cij = yj [ξ] AND πsi
sj
16:
yj [δ + 1] = yj [δ], δ = ξ, . . . , Kj ;
δ+1
δ , δ = ξ, . . . , K ;
17:
πsj
= πsj
j
18:
yj [ξ] = yi [k] + cij ;
ξ
k ∪ {j};
19:
πsj
= πsi
20:
add node j to L if it does not already belong to it.
21:
BREAK
22:
end if
23:
end for
24:
end for
25:
end if
26: end for
27:
28: Step 4 (termination check)
29: if L = ∅ then
1 .
30:
STOP. yi [1] is the cost of the optimal elementary path πsi
31:
Return C.
32: else
33:
Go to Step 2.
34: end if
Every time a N CC C is detected, the number of paths that have to be found
for each node i ∈ C is increased, that is, Ki = Ki + 1, ∀i ∈ C, and the labeling
algorithm is executed again. When no N CC is detected, yi [1] is the cost of the
optimal elementary shortest path from node s to node i ∈ N .
The proposed solution approach iteratively solves the kSPP for a fixed value of
Ki , ∀i ∈ N . The steps of the labeling procedure are depicted in Algorithm 5.
In the proposed strategy, Algorithm 5 is iteratively run after that Ki , ∀i ∈ N , is
coherently updated. The dynamic labeling solution strategy is reported in Algorithm
6.
3.2.1
Correctness of the Dynamic Labeling Approach
In this section, we prove the correctness of the proposed solution approach. In
1 is the least cost elementary path from s
particular, at the end of Algorithm 5, πsi
14
Algorithm 6 : Dynamic Labeling Approach
1: Step 1 (Initialization phase)
2: Set: ζ = 0; Ks = 1; Kiζ = 1 ∀i ∈ N \ {s}.
3:
4: Step 2 (Run the truncated labeling algorithm)
5: Run Algorithm 5 with Kiζ , i ∈ N and get C.
6:
7: Step 3 (Cycle detection)
8: if C 6= ∅ then
9:
Kiζ+1 + 1, ∀i ∈ C, Kiζ+1 = Kiζ , ∀i ∈ N \ C;
10:
ζ + +;
11:
Go to Step 2.
12: else
13:
STOP.
14: end if
to i.
Proposition 4. At the end of Algorithm 5 we have that:
1. a N CC is detected, that is, conditions of line 9 are verified;
2. if condition of line 29 is verified, then all the first Ki elementary paths are
determined for each node i ∈ N , that is, sets Πi , ∀i ∈ N , are path-elementary.
(a) At the end of each iteration, the following conditions hold:
i. ys [1] = 0;
k is an elementary path.
ii. ∀i ∈ N \ {s}, if yi [k] 6= +∞, then πsi
(b) Upon termination of the algorithm, Πi , ∀i ∈ N , is path-elementary.
Proof. Let us consider case 1: The first part of conditions in line 9, that is, yik 6=
+∞, ∀k ∈ {1, . . . , Ki }, checks whether all the Ki paths to node i are determined.
The second part derives from Proposition 1, thus if it is verified for some k̄, a N CC
is found and the algorithm terminates. It follows that the presence of a N CC is
verified only if the entire set of paths from node s to the considered node i are
determined. If case 1 does not occur, then all N CCs have been detected and the
algorithm terminates when the list L is found empty, that is, case 2.
Case 2: Let us consider case 2a. Condition 2(a)i holds because, initially, ys [1] = 0
and, by the rules of the algorithm, ys [1] cannot change.
We prove condition 2(a)ii by induction on the iteration count. Indeed, initially,
condition 2(a)ii holds, since node s is the only node for which the cost is not equal to
+∞. Suppose that 2(a)ii holds for some node j at the beginning of some iteration.
Let i be the node removed from L.
If i = s (which only occurs at the first iteration), then at the end of this iteration,
we have yj [1] 6= +∞ for all successor nodes j of s and the corresponding path
15
πsj = {(s, j)} is elementary, yj [k] = +∞, k 6= 1 and yj [k] = +∞ for all other nodes
j 6= s, j ∈
/ F S(i). Thus, the set of paths has the required property.
k starting from s and ending to i
If i 6= s, then yi [k] is the cost of the k-th path πsi
that does not contain repeated nodes, by the induction hypothesis. If yj [k] changes,
for some node j ∈ F S(i) and k, then a new path is obtained for node j. The created
k consists of path π k followed by the arc (i, j). Finally, note that, by the
path πsj
si
k . This
rules of the algorithm, the newly created path is elementary because j 6∈ πsi
completes the induction proof of 2(a)ii.
Let now consider condition 2b. Using part 2(a)ii, we have that, at each iteration,
k , ∀j ∈ N such that y [k] 6= +∞ is an elementary path. Thus, the property
πsj
j
mentioned is also satisfied when the algorithm terminates. In addition, the way
in which the candidate list L is updated and the termination condition (i.e., the
algorithm terminates when there are no more nodes left to be scanned) guarantee
that all the nodes with the potential to determine a new path for at least one node
are scanned during the execution of the algorithm.
1 is the
Proposition 5. (Correctness of Algorithm 6). At the end of Algorithm 6, πsi
optimal path from node s to node i, ∀i ∈ N .
Proof. The algorithm terminates when no cycle is detected. From Proposition 4, we
know that if no cycles are detected, then Algorithm 5 provides a path-elementary
set Πi , ∀i ∈ N . Since the paths are ordered in nondecreasing order of the cost,
the first path of each set represents the least cost path. In addition, being Πi path1 is the optimal path.
elementary, πsi
3.2.2
Complexity Analysis
The complexity of Algorithm 6 is derived in the following lemma.
Lemma 2. In the worst case, the complexity of the proposed solution approach is
O(n9 C̃ 7 ).
Proof. Let K be the highest number of paths that have to be found. The operations
executed in lines 15 - 21 are 2K + 3. Since the forloops of line 13 and 14 take K 2 ,
the number of iterations in lines 12 - 22 is 2K 3 + 3K 2 . The F S contains at most
n − 1 elements. The for-loop of line 8 is invoked nK times. Consequently, Algorithm
5 takes O(n2 (2K 4 + 3K 3 )). Since each node can be part of each cycle, the iterations
executed by Algorithm 6 are nC̃ in the worst case. This means that the proposed
approach executes n2 (2×14 +3×13 )+n2 (2×24 +3×23 )+. . .+n2 (2×nC̃ 4 +3×nC̃ 3 ) =
2
8
2
6
4
2
3
2
15 (nC̃) (6nC̃ + 15) + 12 (nC̃) (nC̃ + 1) + 15 (nC̃) (10(nC̃) − 1) operations, thus
9
7
the complexity of Algorithm 6 is O(n C̃ ).
16
k (y [k]) >) at the
In Table 2, we report the paths and the related cost (< πsi
i
end of each iteration of Algorithm 6 when solving the ESPP on the network of
Figure 2. At the 1st iteration, node 3 is selected. Since all the conditions of line
9 are verified, Algorithm 5 terminates and the cycle {1, 2, 3, 1} is returned. The
value of Ki , i = 1, 2, 3 is incremented of one and Algorithm 5 is executed again.
1 = {s, 1, 2, 5, 4} is extended to
Node 4 is selected (see 2nd iteration). When path πs4
node 2, conditions of line 9 are satisfied, thus the N CC {2, 5, 4, 2} is detected and
Algorithm 5 is stopped. The number of paths that need to be found for nodes 2,
5 and 4 is increased. After other three iterations, see 3rd , 4th and 5th iteration of
Table 2, condition of line 29 is verified. Since C = ∅, Algorithm 6 terminates and
yi [1], ∀i ∈ N , is the optimal cost.
17
1st iteration
yi
< s(0) >
< s, 1(2) >
18
s
1
Ki
1
1
Ki
1
2
2
1
< s, 1, 2(−5) >
2
3
1
< s, 1, 2, 3(−3) >
2
4
1
< s, 1, 4(7) >
5
1
6
last extracted node
C
1
2nd iteration
yi
< s(0) >
< s, 2, 3, 1(−1) >
< s, 1(2) >
< s, 1, 2(−5) >
< s, 2(−4) >
Ki
1
2
3
2
1
< s, 1, 2, 3(−3) >
< s, 2, 3(−2) >
< s, 1, 2, 5, 4(−9) >
< s, 1, 2, 5(−7) >
1
< s, 1, 2, 5(−7) >
+∞
3
{1, 2, 3, 1}
1
< s, 1, 2, 5, 4, 6(−6) >
4
{2, 5, 4, 2}
3rd iteration
yi
< s(0) >
< s, 2, 3, 1(−1) >
< s, 1(2) >
< s, 1, 2(−5) >
< s, 2(−4) >
< s, 1, 4, 2(10) >
Ki
1
2
4
< s, 1, 2, 3(−3) >
< s, 2, 3(−2) >
< s, 1, 2, 5, 4(−9) >
< s, 2, 5, 4(−8) >
2
2
< s, 1, 2, 5(−7) >
< s, 2, 5(−6) >
1
< s, 1, 2, 5, 4, 6(−6) >
4
{2, 5, 4, 2}
2
4th iteration
yi
< s(0) >
< s, 2, 3, 1(−1) >
< s, 1(2) >
< s, 1, 2(−5) >
< s, 2(−4) >
< s, 1, 4, 2(10) >
+∞
Ki
1
2
5
< s, 1, 2, 3(−3) >
< s, 2, 3(−2) >
< s, 1, 2, 5, 4(−9) >
< s, 2, 5, 4(−8) >
< s, 2, 3, 1, 4(4) >
2
3
< s, 1, 2, 5(−7) >
< s, 2, 5(−6) >
+∞
4
1
< s, 1, 2, 5, 4, 6(−6) >
4
{2, 5, 4, 2}
1
3
Table 2: Paths and costs that are found for each node at the end of Algorithm 5.
4
5th iteration
yi
< s(0) >
< s, 2, 3, 1(−1) >
< s, 1(2) >
< s, 1, 2(−5) >
< s, 2(−4) >
< s, 1, 4, 2(10) >
+∞
+∞
< s, 1, 2, 3(−3) >
< s, 2, 3(−2) >
< s, 1, 2, 5, 4(−9) >
< s, 2, 5, 4(−8) >
< s, 2, 3, 1, 4(4) >
< s, 1, 4(7) >
< s, 1, 2, 5(−7) >
< s, 2, 5(−6) >
< s, 1, 4, 2, 5(8) >
+∞
< s, 1, 2, 5, 4, 6(−6) >
5
∅
3.3
Enumerative Strategy
The proposed approach looks for the optimal solution through a tree structure. Each
node is associated with a sub-graph of G excluding the root node that represents the
graph G. The nodes at a certain level are associated with sub-graphs in which at
least one N CC is removed. Given a father node, all child nodes are associated with
sub-graphs in which the same N CCs are removed. Figure 3 shows the structure of
the aforementioned tree.
Figure 3: The Figure represents the tree structure of a graph with two negative cost cycles,
that is < u, v, u > and < j, u, v, j >. Each node is associated with a graph. The root node is
the graph G(N , A). Under each node is reported the detected cycle, whereas, the set of arcs
is reported over each node. From both graphs G 1 and G 2 the cycle < u, v, u > is removed.
Both cycles < u, v, u > and < j, u, v, j > are removed from graphs G 3 , G 4 and G 5 .
In what follows we formalize the search process in the tree. Let G(N , A) be a
graph associated with a generic node of the tree. The truncated labeling approach,
that is, Algorithm 5 with Ki = 1, ∀i ∈ N , is run on G(N , A). If a N CC Ci is
detected, then δ = |Ci | − 1 sub-graphs are generated starting from G(N , A) by
removing the δ-th arc of Ci . In other words, the graph G δ (N , Aδ ), δ = 1, . . . , |Ci | − 1
are generated where Aδ = A \ {(uδ , vδ )}, with (uδ , vδ ) ∈ Ci .
Since at least one arc of the detected N CC Ci is removed, the related sub-graph
does not contain this cycle. When the truncated labeling algorithm is run on this
sub-graph, if a N CC C̄i , for some node i ∈ N is detected, we are sure that Ci 6= C̄i .
This means that at a certain level σ of the tree exactly σ N CCs have been detected.
Consequently, in all the sub-graphs at level σ, at least σ N CCs are removed (the
same arc can be part of many N CCs).
19
3.3.1
Algorithm
Let y ζ ∈ <|N | be the vector that stores the best solutions for each node i ∈ N ,
determined so far, and y ∗ ∈ <|N | be the vector of the optimal solutions for each
node i ∈ N . The value ζ is the iterations counter. At the last iteration ζ̄, y ζ̄ = y ∗ ,
that is yi∗ is the cost of the optimal path from node s to node i. Let yi (G) be
the cost of the path from node s to node i obtained applying the truncated labeling
algorithm on graph G. If a N CC Cj is detected when the truncated labeling algorithm
is executed on G, then we have that yi (G) is a valid upper bound for all nodes
i ∈ N̄ , where N̄ = {u ∈ N : v 6∈ πsu , ∀v ∈ Cj }. This means that yi (G), ∀i ∈ N̄
can be used to initialize the cost of node i for the sub-graph G δ . In other words,
yi (G δ ) = yi (G), ∀i ∈ N̄ . In addition, if yi (G) < yiζ for some i ∈ N̄ , then yiζ = yi (G).
Let L be the list containing the nodes of the tree that have to be processed. The
steps of the proposed enumerative approach are reported in Algorithm 7.
Algorithm 7 : Enumerative Algorithm
1: Step 1 (Initialization phase)
2: Set: ζ = 0; y ζ [s] = 0; y ζ [i] = +∞, ∀i ∈ N \ {s}; L0 = {G}, G = (N , A); yG [s] = 0; yG [i] = +∞, ∀i ∈ N .
3:
4: Step 2 (Selection phase)
5: Select a graph Ḡ = (N , Ā) from Lζ and remove it from Lζ .
6:
7: Step 3 (Search process)
8: Apply the truncated labeling algorithm to Ḡ.
9: if a N CC Cj is detected then
10:
Generate δ = |Cj | − 1 sub-graphs G δ = (N , Ā − {(uδ , vδ )}) and add them to Lζ ;
11:
set: yG δ [i] = yḠ [i], ∀i ∈ N̄ .
12:
if yḠ [i] < y ζ [i] for some i ∈ N̄ then
13:
y ζ [i] = yḠ [i].
14:
end if
15:
y ζ+1 = y ζ ;
16:
Lζ+1 = Lζ ;
17:
ζ + +;
18:
go to Step 4.
19: else
20:
if yḠ [i] < y ζ [i] for some i ∈ N then
21:
y ζ [i] = yḠ [i].
22:
end if
23:
y ζ+1 = y ζ ;
24:
Lζ+1 = Lζ ;
25:
ζ + +;
26:
go to Step 4
27: end if
28:
29: Step 4 (Termination check)
30: if Lζ = ∅ then
31:
STOP. y ζ−1 [i] is the cost of the optimal path from node s to node i.
32: else
33:
Go to Step 2.
34: end if
20
3.3.2
Correctness of the Enumerative Algorithm
The correctness of the method follows from the search procedure. As mentioned
above, the nodes represent sub-graphs for which the detected cycle in the father
graph is excluded. Each sub-graph is defined by removing at most one arc of the
detected cycle. In the sub-graphs of the last level all the cycles are removed, thus
for each node i we have a path for each sub-graph that is an elementary path. The
path with the smallest cost among those obtained in the sub-graphs is the optimal
one.
3.3.3
Complexity Analysis
Let M be the maximum cardinality among those of the N CCs in the graph, that
is M = n. We know that the maximum number of N CCs in the graph is C̃. In
addition, for each level σ of the tree exactly σ N CCs have been removed from the
graphs at level σ. Thus the maximum number of levels in the tree is C̃.
P
τ
The complexity of the proposed approach is O(n2 C̃
τ =0 n ). This result is formally stated in the following lemma.
Lemma 3. In the worst case, the complexity of the proposed method for the ESPP
P
τ
is O(n2 C̃
τ =0 n ).
Proof. In the worst case, the original graph associated with the first node of the tree
is solved and a N CC is detected. The branching procedure is applied and δ = n − 1
sub-graphs are generated. When each of these sub-graph is solved, another N CC
is detected and n − 1 graphs of the next level are generated. This means that the
number of graphs that have to be solved at level σ is (n − 1)σ ≈ nσ . Since we
know that the levels are bounded above by C̃, thus the total number of node is
PC̃
τ
τ =0 n . In addition, the complexity to solve each graph associated with a node
with Algorithm 5 where K = 1 takes O(n2 ) (see Lemma 2). It follows that the worst
P
τ
case complexity of the proposed method is O(n2 C̃
τ =0 n ).
In Table 3, we show the tree structure explored by Algorithm 7 when solving the
network of Figure 2. We report the graph associated with each node of the tree. In
Table 4, we report for each iteration, the selected node/graph, the cost associated
with each node after Algorithm 5 with Ki = 1 is run, the label y ζ and the detected
cycle.
21
G 1 (N , A \ {(1, 2)})
G 0 (N , A)
G 2 (N , A \ {(2, 3)})
G 3 (N , A \ {(3, 1)})
G 4 (N , A \ {(1, 2); (2, 5)})
G 5 (N , A \ {(1, 2); (5, 4)})
G 6 (N , A \ {(1, 2); (4, 2)})
G 7 (N , A \ {(2, 3); (2, 5)})
G 8 (N , A \ {(2, 3); (5, 4)})
G 9 (N , A \ {(2, 3); (4, 2)})
G 10 (N , A \ {(3, 1); (2, 5)})
G 11 (N , A \ {(3, 1); (5, 4)})
G 12 (N , A \ {(3, 1); (4, 2)})
Table 3: Graph associated with the nodes of the tree structure.
22
yζ
y(G)
23
ζ
Selected node
0
1
2
3
4
5
6
7
8
9
10
11
12
G 0 (N , A)
G (N , A \ {(1, 2)})
G 2 (N , A \ {(2, 3)})
G 3 (N , A \ {(3, 1)})
4
G (N , A \ {(1, 2); (2, 5)})
G 5 (N , A \ {(1, 2); (5, 4)})
G 6 (N , A \ {(1, 2); (4, 2)})
G 7 (N , A \ {(2, 3); (2, 5)})
G 8 (N , A \ {(2, 3); (5, 4)})
G 9 (N , A \ {(2, 3); (4, 2)})
G 10 (N , A \ {(3, 1); (2, 5)})
G 11 (N , A \ {(3, 1); (5, 4)})
G 12 (N , A \ {(3, 1); (4, 2)})
1
1
2
-1
2
2
-1
-1
-1
2
2
2
2
2
2
2
-5
-4
-5
-5
-4
-4
-4
-5
-5
-5
-5
-5
-5
3
-3
-2
+∞
-3
-2
-2
-2
+∞
+∞
+∞
-3
-3
-3
4
7
-8
-9
-9
4
4
-8
7
7
-9
7
7
-9
5
-7
-6
-7
-7
+∞
-6
-6
+∞
-7
-7
+∞
-7
-7
6
+∞
-4
-5
-5
7
-4
-5
10
-5
-6
10
-5
-6
1
+∞
+∞
2
2
-1
-1
-1
-1
-1
-1
-1
-1
-1
2
+∞
+∞
+∞
+∞
-4
-4
-4
-5
-5
-5
-5
-5
-5
3
+∞
+∞
+∞
+∞
-2
-2
-2
-2
-2
-2
-3
-3
-3
4
+∞
+∞
+∞
+∞
4
4
-8
-8
-8
-9
-9
-9
-9
5
+∞
+∞
+∞
+∞
+∞
-6
-6
-6
-7
-7
-7
-7
-7
6
+∞
+∞
+∞
+∞
7
-4
-5
-5
-5
-6
-6
-6
-6
C
< 1, 2, 3, 1 >
< 2, 5, 4, 2 >
< 2, 5, 4, 2 >
< 2, 5, 4, 2 >
∅
∅
∅
∅
∅
∅
∅
∅
∅
Table 4: Information available at the end of each iteration. Under column y(G) we report, for each node i ∈ N \ {s},
the cost as output of Algorithm 5 with Ki = 1. Column y ζ shows the potential optimal cost for each node i ∈ N \ {s}.
C indicates the cycle returned by Algorithm 5.
4
Computational Experiments
The aim of this Section is to evaluate the behaviour of the proposed solution approaches and to compare them in terms of computational cost. The experiments
have been carried out on instances generated by ourselves. Indeed, the ESPP is not
addressed by the scientific literature, thus no benchmark instances are available. In
the next Section, we present the considered instances and how they were generated.
4.1
Test Problems
The computational results are carried out on two groups of test problems. The first
one is composed of random networks, the instances belonging to the second group
are derived from Vehicle Routing Problem (VRP) benchmark test problems.
The test problems of the first group (i.e., fully random networks) have been
generated randomly by using the Netgen generator of Klinglman et al. [41]. Table 5
shows the characteristics of the considered fully random networks, where the number
of nodes, the number of arcs and the density, that is the ratio between the number of
arcs and the number of nodes are highlighted. The cost cij , ∀(i, j) ∈ A is randomly
generated from the interval [0, 100].
test
R1
R2
R3
R4
R5
R6
R7
R8
R9
R10
R11
R12
R13
R14
R15
nodes
300
300
300
350
350
350
400
400
400
450
450
450
500
500
500
arcs
3000
9000
15000
3500
10500
17500
4000
12000
20000
4500
13500
22500
5000
15000
25000
density
10
30
50
10
30
50
10
30
50
10
30
50
10
30
50
Table 5: Characteristic of the fully random networks.
For each network, a given number of instances has been generated, in such a way
that at least a fixed number of negative cost cycles is present. The procedure used to
build the test problems is detailed in what follows. Let #c be the number of negative
cost cycles and let leng be the number of arcs belonging to a given cycle. For each
24
set
F
A-A
A-B
A-P
CE
S
B
CM T
T
GW KC
paper
|set|
[27]
[3]
[3]
[3]
[13]
[52]
[58]
[14]
[55]
[36]
3
27
23
24
13
6
60
14
13
20
nodes
min max
45
135
32
80
31
78
16
101
13
101
50
100
100 100
50
199
75
385
200 483
Table 6: Characteristic of the CVRP benchmark instances.
fully random network, we have generated 30 instances by letting #c = 1, . . . , 30. In
other words, each instance has a number of cycles equal to #c. The value leng has
been randomly chosen in the interval [2, 5].
The procedure used to build the test instances relies on the solution of the
shortest path problem for each fully random network, obtaining a tree T . Let πij
be the path from node i to node j in T . Exactly #c cycles are chosen by selecting
#c arcs (j, i) ∈ A \ T . Each cycle Ci#c , #c = 1, . . . , 30 is composed by the arcs
belonging to the path πij and by the arc (j, i). Starting from each leaf node and
exploring the related branch, the first path πij , from which it is possible to generate
a cycle with leng nodes is selected. In other words, the number of arcs of each
selected path πij is equal to leng-1. The cost of the arcs belonging to each obtained
cycle Ci#c , #c = 1, . . . , 30 is modified in order to have a negative cost cycle that is
c(Ci#c ) < 0.
It is worth observing that the procedure described above does not ensure that
exactly #c negative cost cycles are introduced, but the value #c indicates the minimum number of negative cost cycles present in the network.
The second group of instances are derived from Capacited VRP (CVRP) test
problems taken from the scientific literature. We have considered ten sets of test
problems. Each set is associated with the scientific contribution in which the corresponding test problems have been introduced for the first time.
Table 6 shows the name of each set, the paper in which the related test problems have been introduced, the number of problems and the characteristics of the
instances in terms of number of nodes (i.e., the maximum and the minimum number
of nodes over the networks belonging to the considered set are reported).
In order to test the algorithm for the ESPP on the CVRP test problems, we
have considered the related RCESPP, that is the pricing problem obtained when
25
a column-generation approach is used to solve the CVRP. First, we have modified
the CVRP test problems by adding a prize at each node. As described in [49], the
value of each prize can be chosen in the range [1, 20]. The cost of a path is the
sum of the cost associated with the arcs minus the prizes collected at the nodes.
Second, we get the ESPP instances by considering the Lagrangean relaxation of the
RCESPP. Indeed, given a Lagrangean multiplier, the problem is an instance of the
ESPP. For each RCESPP test problem we have considered 20 ESPP instances by
chosing several values for the Lagrangean multiplier. These values are selected in a
such way that instances with different degree of complexity are obtained:the lower
the value of the Lagrangean multiplier, the more difficult the ESPP instances. As
suggested in [6], the Lagrangean multiplier is set equal to 0.1, 0.2, . . . , 2 for the 1st ,
the 2nd , . . . , the 20th ESPP instances. Thus we have tested 4060 ESPP instances.
4.2
Test Codes
The codes considered in this work are named DM LA, implementing Algorithm 3
described in Section 3.1; the code DLA is related to Algorithm 6, defined in Section
3.2; the enumerative approach, that is Algorithm 7, described in Section 3.3 is
implemented with the code EN .
Considering DLA, at each iteration, the costs for each node are inizialized by
considering the partial solutions obtained in the previous iteration. In particular,
the set L contains all nodes such that at least one path has been determined in
the previous iteration and the labels are those available at the end of the previous
iteration. Of course, for the nodes belonging to the detected cycle, the dimension of
the associated label is incremented by one. This type of initialization speeds-up the
search process at each iteration.
The same type of initializzation is not possible for DM LA. Indeed, at each
iteration, an additional resource is introduced, augmenting the dimension of each
label. Since the efficiency of a solution is affected by the dimension of the associated
label, a solution that in the previous iteration was dominated, with the new definition
of the label could represent an efficiet sub-path. For this reason, in DM LA the
initialization is that reported in Algorithm 4.
4.3
Computational Results
In this Section, we analyze the behaviour of the proposed solution approaches. In
the next Section we focus on the results collected when solving the first group of
instances, that is, the fully random networks. In Section 4.3.2, the resulting best
performing algorithm on the first group of instances is tested on the second one.
26
4.3.1
Computational Results on the Fully Random Networks
In what follows, we show the computational results obtained with DM LA, DLA
and EN when solving the first group of instances. The collected results are grouped
in three classes. The first one is related to the instances with #c ∈ [1, 10], the second
class to the networks with #c ∈ [11, 20] and the results obtained on the instances
with #c ∈ [21, 30] belong to the third class. They are collected in Tables A.1 A.3 of Appendix. Each table reports, for each fully random network, the average
computational cost in ms and the average number of iterations over 10 instances,
that is those with #c ∈ [1, 10], #c ∈ [11, 20] and #c ∈ [21, 30], respectively.
Results for DM LA. As expected, the computational results underline that the
higher the number of negative cost cycles, the higher the execution time. Table
7 shows that the computational cost for solving the second and the third class of
instances is 15.77 and 197.54 times higher than the execution time required to solve
the instances of the first class.
#c
1
2
3
4
5
6
7
8
9
10
AVG
time
68.64
127.92
217.36
434.72
582.40
885.05
1215.77
1886.57
2338.98
3063.86
1082.13
iter
1.07
2.27
3.67
5.80
7.40
9.33
11.00
13.20
14.60
17.00
8.53
#c
11
12
13
14
15
16
17
18
19
20
DM LA
time
3411.22
4501.15
6140.20
7985.17
11858.16
14540.33
20057.57
26119.77
33535.01
42470.75
17061.93
iter
18.00
20.07
22.40
24.67
27.53
29.93
32.53
34.93
36.93
39.47
28.65
#c
21
22
23
24
25
26
27
28
29
30
time
49217.28
57689.17
73947.59
93819.00
117422.99
148047.03
208258.21
351173.85
438542.73
599554.48
213767.23
iter
39.33
41.53
43.60
45.13
46.93
48.47
50.73
53.93
55.47
57.87
48.30
Table 7: Execution time in ms and number of iterations obtained by DM LA when
solving the first, the second and the third class of test problems.
This behaviour can be justified by considering the number of iterations. Indeed,
DM LA performs, on average, 8.53, 28.65 and 48.30 iterations, for the instances
belonging to the first, the second and the third class, respectively. Figure 4 shows
the trend of the computational cost and the number of iterations with respect to
the number of negative cost cycles.
From Figure 4, it is clear that the computational cost grows exponentially with
respect to the increase of the number of negative cost cycles, whereas, a linear trend
is observed for the number of iterations. This behaviour is due to the fact that
the higher the number of negative cost cycles, the higher the execution time per
27
Figure 4: Execution time and number of iterations of DM LA as a function of the
number of negative cost cycles.
iteration. In particular, the time per iteration is exponential with respect to the
number of negative cost cycles. This trend is shown in Figure 5.
Figure 5: Execution time per iteration of DM LA as a function of the number of
negative cost cycles.
A possible explanation of this trend can be found by considering the dimension
of the state-space induced by the definition of the labels. Indeed, the state-space is
composed by states associated with efficient partial solutions. The number of Paretooptimal solutions is strongly related to the dimension of the labels. Indeed, the
higher the number of negative cost cycles, the higher the dimension of the labels due
to the fact that an higher number of additional node resources must be introduced.
In addition, as shown in [5, 8, 21], the number of non-dominated solutions increases
in an exponential way with the size of the labels. As a conseguence, the state-space
grows exponentially with the increase of the number of the negative cost cycles.
Results for DLA. In Table 8, we report the average computational cost for each
value of #c derived from Table A.1 - A.3 of Appendix. As expected, the higher the
number of negative cost cycles the higher the execution time. Indeed, the computational cost for solving the instances belonging to the second and the third class is
3.63 and 7.05 times higher than that required for the instances of the first class. This
28
behaviour is justified by the number of iterations executed by DLA. In particular,
DLA performs 3.81 and 8.59 times higher number of iterations for the second and
the third class of instances, respectively, than that executed by the algorithm for
solving the instances belonging to the first class.
#c
1
2
3
4
5
6
7
8
9
10
AVG
time
98.80
133.12
151.84
166.40
197.60
210.08
234.00
276.64
372.32
392.08
223.29
iter
3.27
4.87
6.73
8.93
11.13
12.47
15.87
18.67
22.47
25.00
12.94
#c
11
12
13
14
15
16
17
18
19
20
DLA
time
573.04
635.44
659.36
693.68
747.77
887.13
957.85
975.53
925.61
1054.57
811.00
iter
33.20
36.33
38.60
40.73
46.07
54.00
57.73
60.67
59.60
66.27
49.32
#c
21
22
23
24
25
26
27
28
29
30
time
1045.21
1128.41
1210.57
1290.65
1278.17
1361.37
1551.69
1982.25
2415.94
2482.50
1574.67
iter
71.13
76.87
85.73
92.07
95.00
102.87
113.07
135.87
162.07
176.40
111.11
Table 8: Execution time in ms and number of iterations obtained by DLA when
solving the first, the second and the third class of test problems.
Figure 6 shows the trend of the computational cost and the number of iterations
executed by DLA.
Figure 6: Execution time and number of iterations of DLA as a function of the
number of negative cost cycles.
From Figure 6, it is evident that both the execution time and the number of iterations present the same trend: they grow exponentially with respect to the increase
of the number of negative cost cycles.
Results for EN . The computational results underline that EN is not able to solve
all the instances considered in this paper. As shown in Table 9, only some instances
29
#c
1
2
3
4
5
6
7
8
9
10
AVG
EN
time
252.72
781.05
2119.53
6428.36(1)
19172.52(1)
58197.29(1)
181989.65(1)
532805.82(2)
1115989.03(6)
2358009.72(13)
427574.57
nodes
4.00
12.60
37.60
116.50
344.71
1047.36
3464.71
12513.77
38617.60
76373.50
13253.24
Table 9: Execution time in ms and number of nodes generated by EN when solving
the first class of test problems. The superscript near the time indicates the number
of instances that are not solved. EN is not able to solve the instances of both the
second and the third class.
belonging to the first class are solved. In particular, the 93.33%, the 86.67%, the
60% and the 13.33% of instances with a number of negative cost cycles #c ∈ [4, 7],
#c = 8, #c = 9 and #c = 10, respectively, are solved. EN runs out of memory for
all the instances belonging to the second and the third class.
In addition, the computational cost grows very fast with respect to the increase
of the number of negative cost cycles. This behaviour can be justified by considering
the generated number of nodes. This trend is clearly underlined in Figure 7.
Figure 7: Execution time and number of nodes of EN as a function of the number
of negative cost cycles.
Comparison. From the considerations reported above, it is clear that EN is very
inefficient for solving the instances considered in this paper. In what follows, we focus
30
on the other two proposed algorithms (that is DM LA and DLA) and we compare
their behaviour. The results collected in Tables A.1 - A.3 of Appendix, underline
that DLA behaves the best. Indeed, on average, DM LA is 88.89 times slower than
DLA. However, it is worth observing that the number of iterations performed by
DLA is 2.03 times higher than that of DM LA. The worst performance in terms of
computational cost of DM LA is due to the average time per iteration. Indeed, the
time per iteration obtained with DLA is 93.87 times lower than than of DM LA.
The reasons of the poor performance of DM LA in terms of time per iteration has
explained in Section 4.3.1.
From Figure 8, it is possible to observe that the higher the number of negative
cost cycles, the higher the difference, in terms of execution time, between DLA and
DM LA. More specifically, DLA is 4.85, 21.04 and 135.75 times faster than DM LA
for the instances belonging to the first, the second and the third class, respectively.
Figure 8: Execution time of DM LA and DLA as a function of the number of
negative cost cycles.
4.3.2
Computational Results on CVRP Benchmark Instances
In this section, we evaluate the behaviour of the best performing algorithm, that is,
DLA by considering the CVRP benchmark instances. An execution time limit of
one hour per run has been imposed.
The results collected on the corresponding ESPP instances are summarized in
Table A.4 of the Appendix. We report the name of the set under the first column,
the column min λ reports the minimum value of the Lagrangean multiplier such that
at least one instance is solved, the last three columns show the numerical results
obtained. In particular, for each set, we have considered average results for each
value of the Lagrangean multiplier. Under column min, we report the minimum
values related to the time (in seconds), number of iterations (iter), number of cycles
(cycles) and the percentage (%) of the instances solved, evaluated over all the average
results, obtained for each value of the multiplier. Under column max the maximum
values and under column mdm the medium values on all the instances associated
31
with each set. It is worth observing that the first six sets, that is, F , A-A, A-B,
A-P , CE, and S contain instances associated with CVRP test problems for which
the optimal solution is known. The remaining four sets are composed of instances
for which the related CVRP test problems are not solved to optimality.
The average results reported in Table A.4 (see row AVG) underline that on all
the considered test problems, in the medium case, the 60% of instances are solved
within the time limit of one hour. In the worst case, only the 21% of instances is
solved, whereas 95% of instances are solved in the best case (see row AVG, column
max of Table A.4). It is worth observing that for the first six sets, at least one
instances with the Lagrangean multiplier equal to 0.1 is solved (see column min λ
of Table A.4). For the remaining sets, DLA is able to solve at least one instance
with the Lagrangean multiplier equal to 0.7, 0.6, 0.6, and 0.3 for set B, CM T , T ,
and GW KC, respectively.
In addition, even though the best performance is observed for the first six sets
of instances, the results collected on the other sets can be considered satisfactory.
Indeed, the average percentage of solved problem is of 17%, 44% and 87% in the
worst, medium and best case, respectively.
5
Conclusions
In this paper we have investigated the shortest path problem in presence of negative
cost cycles. This study is the first attempt to provide solution methods for the singlesource all-destinations shortest path problem with negative cost cycles. The scientific
literature provides strategies that are able to determine the presence of negative
cost cycles, but no work considers the resolution of the shortest path problem with
negative cost cycles.
Three different strategies have been devised to optimally solve the problem under
investigation. The main idea behind the proposed solution approaches is to compute,
for each node, the minimum number of paths such that the shortest paths from
the source node to all others nodes are determined. In addition, all the methods
dynamically increase the number of paths that have to be found for some node. The
main difference among the proposed approaches is related to the way in which the
number of paths is incremented. In the first method, the number of paths that have
to be found is determined by considering a dummy node resource that keeps trace
about the visiting at the node. This results in a constrained multi-objective shortest
path in which the node associated with the dummy resource can be visited only once
along the paths. The second proposed approach is based on the idea behind the k
shortest path methods. In particular, the value of k is different per node and it is
incremented each time a further path that passes through such a node is needed in
order to avoid the negative cost cycle. The last strategy increases the number of
paths by considering an enumerative approach that makes use of a tree built by a
32
cycle elimination procedure. The theoretical complexity of the proposed solution
approaches in the worst case is derived.
The proposed solution approaches have been evaluated enpirically on a large set
of test problems. In particular, we have considered two groups of instances. The
first one refers to fully random networks, the second group contains the instances
derived from Vehicle Routing Problem benchmarks. Referring to the first group
of instances, we have considered instances with up to 500 nodes and 25000 arcs
with 30 negative cost cycles at least present in the networks. The computational
results show that the solution approach based on the k shortest paths problem and
the optimal dynamic programming strategy derived from well-known procedures
for addressing the resource constrained elementary shortest path problem are very
efficient. Instead the enumerative approach shows very poor performance. In particular, it solves only instances with up to 10 negative cost cycles. The experiments
underline the superiority of the innovative strategy based on the k shortest paths
problem. Indeed, it outperforms the dynamic programming based approach and the
enumerative strategy. In particular, the best performing algorithm is able to solve
the instances based on fully random networks with 500 nodes, 25000 arcs and 30
negative cost cycles in about 5 seconds.
The proposed method based on the k shortest paths problem has been tested
on instances derived from the Vehicle Routing Problem benchmarks. In particular,
we have considered two class of test problems: the first one refers to those that
the literature solve to optimality, the second class constains the test problems for
which only near optimal solutions are known. Starting from the Vehicle Routing
Problem benchmarks we have derived several instances of the elementary shortest
path problem with a different degree of difficulty. The computational results suggest
that the best performing algorithm solves to optimality near to all the more difficult
instances belonging to the first class within a reasonable amount of time. For the
instances of the second class, the solution approach based on the k shortest paths
probelm does not solve the more difficult instances. However, despite to the N P hard complexity of the problem, the numerical results are satisfactory.
From the computational results, we can conclude that the solution approach
based on the k shortest paths problem is very efficient for solving the elementary
shortest path problem on medium-large size fully random networks. In addition, the
proposed method is able to solve in a reasonable amount of time instances derived
from the Vehicle Routing Problem benchmarks.
References
[1] R.K. Ahuja, K. Mehlhorn, J.B. Orlin, and R.E. Tarjan. Faster algorithms for
the shortest path problem. Technical Report CS-TR-154-88, Departement of
Computer Science, Princeton University, 1988.
33
[2] E. W. Allender. On the number of cycles possible in digraphs with large girth.
Discrete Applied Mathematics, 10:211–225, 1985.
[3] P. Augerat, J. M. Belenguer, E. Benavent, A. Corberan, D. Naddef, and G. Rinaldi. Computational results with a branch and cut code for the capacitated
vehicle routing problem. Technical Report 949-M, Universite Joseph Fourier,
Grenoble, France.
[4] J. A. Azevedo, J. Madeira, M. Costa, E. Q. V. Martins, and F. Pires. A
computational improvement for a shortest paths ranking algorithm. European
Journal of Operational Research, 73:188–191, 1994.
[5] O. Barndorff-Nielsen and M. Sobel. On the distribution of the number of admissible points in a vector random sample. Theory of Propability and its Applications, 11(2):283–305, 1966.
[6] J. E. Beasley and N. Christofides. An algorithm for the resource constrained
shortest path problem. Networks, 19(4):379–394, 1989.
[7] R. E. Bellman. On a routing problem. Quarterly of Applied Mathematics,
16(1):87–90, 1958.
[8] J. L. Bentley, H. T. Kung, M. Schkolnick, and C. D. Thompson. On the average
number of maxima in a set of vectors and applications. Journal of ACM,
25(4):536–543, 1978.
[9] N. Boland, J. Dethridge, and I. Dumitrescu. Accelerated label setting algorithms for the elementary resource constrained shortest path problem. Operations Research Letters, 34(1):58–68, 2006.
[10] S. Chen and D.R. Ryan. A comparison of three algoritms for finding fundamental cycles in a directed graph. Networks, 11:1–12, 1981.
[11] B. V. Cherkassky, A. V. Goldberg, and T. Radzik. Shortest paths algorithms:
Theory and experimental evaluation. Mathematical Programming, 73(2):129–
174, 1996.
[12] E. I. Chong, S. R. Maddila, and S. T. Morley. On finding single-source singledestination k shortest paths. In 7th International Conference Computing and
Information, July 1995.
[13] N. Christofides and S. Eilon. An algorithm for the vehicle dispatching problems.
Operational Research Quarterly, 20(3):309–318, 1969.
[14] N. Christofides, A. Mingozzi, and P. Toth. The vehicle routing problem. In
N. Christofides, A. Mingozzi, P. Toth, and C. Sandi, editors, Combinatorial
Optimization, chapter 11. John Wiley, Chichester, 1979.
34
[15] T. H. Cormen, C. E. Leiserson, and R. L. Rivest. Introduction to Algorithms.
MIT Press/McGraw-Hill Book Company, Boston, MA, second ed. edition, 1992.
[16] G. B. Dantzig. On the shortest route through a network. Management Science,
pages 187–190, 1960.
[17] M. S. Daskin. Network and Discrete Location. John Wiley and Sons, New York,
1995.
[18] M. Desrochers. An algorithm for the shortest path problem with resource constraints. Technical Report G-88-27, Les Cahiers du GERAD, 1988.
[19] L. Di Puglia Pugliese and F. Guerriero. Survey on resource constrained shortest
path problems. Technical Report 1/11, University of Calabria, 2011.
[20] L. Di Puglia Pugliese and F. Guerriero. A computational study of solution approaches for the resource constrained elementary shortest path problem. Annals
of Operations Research, 201(1):131–157, 2012.
[21] L. Di Puglia Pugliese and F. Guerriero. Shortest path problem with forbidden
paths: the elementary version. European Journal of Operational Research, 2013.
DOI: 10.1016/j.ejor.2012.11.010.
[22] R. B. Dial. Algorithm 360: Shortest path forest with topological ordering.
Communications of the ACM, 12(11):632–633, 1969.
[23] E. W. Dijkstra. A note on two problems in connexion with graphs. Numerische
Mathematik, 1(1):269–271, 1959.
[24] S. E. Dreyfus. An appraisal of some shortest path algorithms. Operations
Research, 17:395–412, 1969.
[25] D. Eppstein. Finding the k shortest paths. SIAM Journal on Computing,
28(2):652–673, 1998.
[26] D. Feillet, P. Dejax, M. Gendreau, and C. Gueguen. An exact algorithm for
the elementary shortest path problem with resource constraints: application to
some vehicle routing problems. Networks, 44(3):216–229, 2004.
[27] M. L. Fisher. Optimal solution of vehicle routing problems using minimum
k-trees. Operations Research, 42:626–642, 1994.
[28] R. W. Floyd. Algorithm 97: shortest path. Communications of the ACM,
5(6):345, 1962.
[29] L. R. Ford and D. R. Fulkerson. Flows in Networks. Princeton University Press,
Princeton, NJ, 1962.
35
[30] M. L. Fredman and R. E. Tarjan. Fibonacci heaps and their uses in improved
network optimization algorithms. Journal of the ACM, 34(3):596–615, 1987.
[31] H. N. Gabow and R. E. Tarjan. Faster scaling algorithm for network problems.
SIAM Journal on Computing, 18(5):1013–1036, 1989.
[32] G. Gallo and S. Pallottino. Shortest path algorithms. Annals of Operations
Research, 13(1):1–79, 1988.
[33] M. R. Garey and D. S. Johnson. Computers and Intractability: A Guide to the
Theory of NP-Completeness. W. H. Freeman & Co, New York, NY, USA, 1990.
[34] A. Goldberg. Shortest path algorithms: Engineering aspects. In ISAAC:
12th International Symposium on Algorithms and Computation, pages 502–513,
2001.
[35] A. V. Goldberg. Scaling algorithms for the shortest path problem. In 4th
ACM-SIAM Symposium on Discrete Algorithms, pages 222–231, 1993.
[36] B. L. Golden, E. A. Wasil, J. P. Kelly, and I-M. Chao. Metaheuristics in vehicle
routing. In T. G. Crainic and G. Laporte, editors, Fleet Management and
Logistics, pages 33–56. Kluwer, Boston, 1998.
[37] F. Guerriero and L. Di Puglia Pugliese. Multi-dimensional labelling approaches
to solve the linear fractional elementary shortest path problem with time windows. Optimization Methods & Software, 26(2):295–340, 2011.
[38] F. Guerriero, R. Musmanno, V. Lacagnina, and A. Pecorella. A class of labelcorrecting methods for the k shortest paths problem. Operations Research,
49(3):423–429, 2001.
[39] S. Irnich. Resource extension functions: properties, inversion and generalization
to segments. OR Spectrum, 30(1):113–148, 2008.
[40] S. Irnich and G. Desaulniers. Shortest path problems with resource constraints.
In G. Desaulniers, J. Desrosiers, and M. M. Solomon, editors, Column Generation, chapter 2, pages 33–65. Springer US, 2005.
[41] D. Klingman, A. Napier, , and J. Stutz. Netgen: A program for generating
large-scale (un)capacitated assignment, transportation, and minimum cost flow
network problems. Management Science, 20, 1974.
[42] N. Kohl. Exact methods for time constrained routing and related scheduling
problems. PhD thesis, Informatics and Mathematical Modelling, Technical University of Denmark, DTU, Richard Petersens Plads, Building 321, DK-2800
Kgs. Lyngby, 1995.
36
[43] T .L. Magnanti and R. T. Wong. Network design and transportation planning:
models and algorithms. Transportation Science, 18:1–55, 1984.
[44] E. Q. Martins, M. M. Pascoal, and J. L. Santos. Deviation algorithms for ranking shortest paths. International Journal of Foundations of Computer Science,
10(3):247–261, 1999.
[45] E. Q. V. Martins. An algorithm for ranking paths that may contain cycles.
European Journal of Operational Research, 18(1):123–130, 1984.
[46] E. F. Moore. The shortest path through a maze. In Interantional Symposium
on the Theory of Switching, pages 285–292. Harvard University Press, 1959.
[47] U. Pape. Implementation and efficiency of Moore-algorithm for the shortest
path problem. Mathematical Programming, 7:212–222, 1974.
[48] R. C. Read and R. E. Tarjan. Bounds on backtrack algorithms for listing cycles,
paths, and spanning trees. Networks, 5:237–252, 1975.
[49] G. Righini and M. Salani. New dynamic programming algorithms for the resource constrained elementary shortest path problem. Networks, 51(3):155–170,
2008.
[50] R.
Sedgewick
and
K.
Wayne.
http://www.cs.princeton.edu/ rs/AlgsDS07/, 2007.
Shortest
paths.
[51] D. R. Shier. On algorithms for finding the k shortest paths in a network.
Networks, 9(3):195–214, 1979.
[52] M. M. Solomon. Vehicle routing and scheduling with time window constraints:
Models and Algorithms. PhD thesis, Department of Decision Science, University
of Pennsylvania, 1983.
[53] K. Subramani. A zero-space algorithm for negative cost cycle detection in
networks. Journal of Discrete Algorithms, 5:408–421, 2007.
[54] K. Subramani and L. Kovalchick. A greedy strategy for detecting negative cost
cycles in networks. Future Generation Computer Systems, 21(4):607–623, 2005.
[55] D. Taillard. Parallel iterative search methods for vehicle routing problems.
Networks, 23:661–673, 1993.
[56] R. E. Tarjan. Enumeration of elementary circuits of a directed graph. SIAM
Journal on Computing, 2:211–216, 1974.
[57] J. C. Tiernan. An efficient search algorithm to find the elementary circuits of
a graph. Communication of the ACM, 13:722–726, 1970.
37
[58] A. Van Bredam. An Analysis of the Behavior of Heuristics for the Vehicle
Routing Problem for a Selection of Problems with Vehicle-Related. CustomerRelated, and Time-Related Constraints. PhD thesis, University of Antwerp,
1994.
[59] H. Weinblatt. A new search algorithm for finding the simple cycles of a finite
directed graph. Journal of the ACM, 19:43–56, 1973.
[60] J. T. Welch. A mechanical analysis of teh cyclic structure of undirected linear
graph. Journal of the ACM, 13:205–210, 1966.
[61] T. Yamada and H. Kinoshita. Finding all the nagative cycles in a directed
graph. Discrete Applied Mathematics, 118:279–291, 2002.
[62] J. Y. Yen. Finding the k-shortest loopless paths in a network. Management
Science, 17:711–715, 1971.
38
Appendix: Computational Results
test
R1
R2
R3
R4
R5
R6
R7
R8
R9
R10
R11
R12
R13
R14
R15
AVG
DM LA
time
iter
0.24
8.00
0.26
6.30
1.83
13.20
0.17
7.00
1.44
8.80
1.05
8.20
0.23
7.80
1.01
8.90
1.73
9.50
0.16
7.80
1.67
9.10
1.72
7.20
0.57
9.10
2.13
8.90
2.02
8.20
1.08
8.53
DLA
time
iter
0.04
8.40
0.12 16.30
0.38 22.70
0.04
8.80
0.12 10.80
0.48 22.00
0.07 11.00
0.14
7.50
0.30 15.40
0.06
8.70
0.22 11.30
0.50 17.60
0.07
8.40
0.26 12.60
0.55 12.60
0.22 12.94
EN
time
nodes
156.20
17417.80
123.73(1)
4920.00
0.74(7)
16.00
67.09(1)
4920.00
138.39(1)
3826.67
346.94(1)
3831.00
272.98(1)
18324.67
221.90(1)
3948.00
153.41(2)
1844.50
92.15(1)
4920.00
264.18(2)
3641.13
72.52(3)
702.14
238.83(1)
10164.33
161.84(2)
1844.50
634.52
4056.70
196.36
5625.16
Table A.1: Results obtained on the first class of instances. The superscript near the
time indicate the number of instances that are not solved.
test
R1
R2
R3
R4
R5
R6
R7
R8
R9
R10
R11
R12
R13
R14
R15
AVG
DM LA
time
final S
8.75
29.30
6.66
26.40
12.90
22.70
5.89
33.10
28.32
25.60
43.87
33.50
8.42
30.10
44.44
28.10
15.45
28.30
2.42
27.60
17.23
32.50
13.35
27.60
9.37
27.90
25.79
31.70
13.07
25.30
17.06
28.65
DLA
time
iter
0.09
27.80
0.39
34.00
0.71
43.70
0.16
41.20
0.48
48.60
4.26 146.70
0.23
43.20
0.73
76.60
0.84
37.90
0.21
37.20
0.54
28.80
0.78
30.70
0.39
47.20
0.74
39.60
1.59
56.60
0.81
49.32
Table A.2: Results obtained on the second class of instances. EN is not able to
solve these instances.
39
test
R1
R2
R3
R4
R5
R6
R7
R8
R9
R10
R11
R12
R13
R14
R15
AVG
DM LA
time
final S
36.68
40.00
308.04
57.40
132.59
40.80
86.58
54.90
232.17
42.10
351.16
40.90
86.98
45.20
440.89
51.70
336.15
53.50
279.45
49.50
21.39
44.20
110.73
57.50
316.13
46.20
252.83
47.60
214.74
53.00
213.77
48.30
DLA
time
iter
0.38
88.20
1.20
103.10
2.08
122.50
0.46 108.40
1.58 140.90
2.41 167.70
0.65 106.80
0.87
95.50
1.64
76.30
0.26
62.80
0.97
60.80
4.47 146.20
1.45 179.60
1.18
72.00
4.02 135.80
1.57
111.11
Table A.3: Results obtained on the third class of instances. EN is not able to solve
these instances.
40
set
F
min
λ
0.1
A-A
0.1
A-B
0.1
A-P
0.1
CE
0.1
S
0.1
B
0.7
CM T
0.6
T
0.6
GW KC
0.3
AVG
time
iter
cycles
% solved
time
iter
cycles
% solved
time
iter
cycles
% solved
time
iter
cycles
% solved
time
iter
cycles
% solved
time
iter
cycles
% solved
time
iter
cycles
% solved
time
iter
cycles
% solved
time
iter
cycles
% solved
time
iter
cycles
% solved
time
iter
cycles
% solved
min
mdm
max
0.00
1.00
0.00
33%
0.04
11.81
2.04
7%
0.00
1.00
0.00
4%
0.00
1.00
0.00
4%
0.00
1.00
0.00
38%
0.10
11.33
0.33
50%
0.03
4.12
0.95
2%
0.19
61.25
3.75
14%
11.31
195.83
11.67
38%
0.27
1.00
0.00
15%
1.19
28.93
1.87
21%
0.61
50.35
2.73
83%
74.88
458.45
13.29
79%
146.92
948.93
14.56
35%
29.38
171.47
4.28
73%
19.85
103.89
2.64
76%
11.28
173.58
3.15
78%
168.22
997.60
18.77
35%
305.52
1092.18
11.79
29%
208.41
1248.68
25.18
49%
18.34
96.94
4.73
64%
98.34
534.20
10.11
60%
3.57
269.00
13.00
100%
828.83
2262.09
33.50
100%
1015.13
3227.60
36.60
96%
360.37
1273.20
15.83
100%
380.31
1028.00
8.80
100%
111.27
816.00
17.00
100%
941.88
3134.00
33.00
100%
906.53
4975.75
27.50
57%
892.76
2775.00
51.60
92%
221.11
926.50
17.67
100%
566.18
2068.71
25.45
95%
Table A.4: Results obtained on the instances derived from CVRP benchmark test
problems. The time is given in seconds.
41