Flow pack facets of the single node fixed-charge flow polytope
Alper Atamtürk∗
[email protected]
Department of Industrial Engineering and Operations Research
University of California at Berkeley
Berkeley, CA 94720–1777
June 2001
Abstract
We present a class of facet–defining inequalities for the single node fixed–charge flow polytope
and provide a comparison of valid inequalities for this polytope. We also present computational
results that show the effectiveness of these inequalities in solving fixed–charge network flow problems.
1
Introduction
The single node fixed–charge flow model is a basic structure that arises as an important relaxation
of many mixed 0–1 integer programming (MBIP) problems with fixed charges. The model consists of
a flow balance inequality for a single node with demand b and variable upper bounds on inflow and
outflow arcs, which can be formulated as
P
P
|N |
S = {x ∈ {0, 1}|N |, y ∈ IR+ :
i∈N − yi ≤ b, yi ≤ ui xi ∀i ∈ N },
i∈N + yi −
where variable yi is the flow on arc i with capacity ui , xi is a binary variable that indicates whether
arc i is open or closed, and N = N + ∪ N − .
The single node fixed–charge flow model is interesting not only because it is a relaxation of the
fixed–charge network flow problem, but also because it is possible to derive relaxations of the form
S of a general MBIP problem. Therefore, valid inequalities for S can be used as cutting planes in
branch–and–cut algorithms to solve MBIP problems. General purpose mixed–integer programming
software packages, such as CPLEX1 , MINTO [8] and bc–opt [4] use cutting planes derived from S
among others. We refer the reader to [13] for a detailed discussion on using S and related structures
as relaxations in mixed–integer programming. Valid inequalities for S have also been instrumental in
developing strong cutting planes for a variety of problems, including lot–sizing problems [3, 10] and
facility location problems [1].
The study of the polyhedral structure of the convex hull of S is initiated by Padberg et al. [9] for
the case with N − = ∅. They introduce the flow cover inequalities, which are generalized by Van Roy
and Wolsey [12]. Gu et al. [5] strengthen these inequalities through sequence independent lifting for S.
∗ Supported,
1 CPLEX
in part, by NSF grants DMI-9908705 and DMI-0070127.
is a trademark of ILOG, Inc.
1
Stallaert [11] gives a complementary class of flow cover inequalities. Marchand and Wolsey [7] derive
other inequalities for S from mixed 0–1 knapsack relaxations of S.
Here we introduce yet another class of strong inequalities for S and show the relationship between
these inequalities and the ones introduced earlier. In Section 2 we present the flow pack inequalities for
S and study their strength for a certain restriction of S. Then we strengthen and generalize them by
lifting and compare them with other inequalities for S given in the literature. In Section 3 we conclude
with a summary of computational experiments that show the effectiveness of the new inequalities in
solving capacitated fixed–charge network flow problems with a branch–and–cut algorithm.
2
Flow pack facets
Without loss of generality, we assume that (A.1) ui > 0 for all i ∈ N (otherwise yi = 0), (A.2)
P
P
b + i∈N − ui ≥ uk for all k ∈ N − (otherwise xk = 1), and (A.3) b + i∈N − ui > 0 (otherwise
yi = 0 for all i ∈ N + ). Note that if N − 6= ∅, then (A.3) follows from (A.1) and (A.2). Under these
assumptions conv(S) is full–dimensional. The linear relaxation of S is
P
P
2|N |
P = {(x, y) ∈ IR+ :
yi ≤ ui xi , xi ≤ 1 ∀i ∈ N }.
i∈N + yi −
i∈N − yi ≤ b,
Observe that an extreme point of P has at most one fractional variable in x. For C + ⊆ N + and
P
P
C ⊆ N − , (C + , C − ) is called a flow cover if i∈C + ui − i∈C − ui = b + λ with λ > 0. A fractional
extreme point (x̄, ȳ) of P has ȳk = uk − λ, x̄k = (uk − λ)/uk for k ∈ C + , or ȳk = λ, x̄k = λ/uk for k ∈
L− ⊆ N − \ C − with λ < uk . The rest of the flow and variable upper bound variables in the cover are
at their upper bounds, the flow variables not in the cover are at their lower bounds and the remaining
variable upper bound variables are at either of their bounds. Letting K = N − \ (C − ∪ L− ), the
well–known flow cover inequality
X
X
X
X
(yi + (ui − λ)+ (1 − xi )) −
min{ui , λ}xi −
yi ≤ b +
ui ,
(1)
−
i∈C +
i∈K
i∈L−
i∈C −
where a+ = max{a, 0}, cuts off the fractional extreme points characterized by flow cover (C + , C − ). If
xk , k ∈ C + is fractional, then (1) is violated by (uk − λ)λ/uk , else if xk , k ∈ L− is fractional, then it is
violated by λ(1 − λ/uk ). Van Roy and Wolsey [12] show that flow cover inequality (1) is facet–defining
for conv(S) if maxi∈C + ui > λ, ui > λ for all i ∈ L− and C − = ∅.
An alternative characterization of the fractional extreme points of P gives rise to a different class
P
of valid inequalities for S. For C + ⊆ N + and C − ⊆ N − , (C + , C − ) is called a flow pack if i∈C + ui −
P
+
i∈C − ui + µ = b with µ > 0. A fractional extreme point of P has ȳk = µ, x̄k = µ/uk for k ∈ L ⊆
N + \ C + , or ȳk = uk − µ, x̄k = (uk − µ)/uk for k ∈ C − with µ < uk . The rest of the flow and variable
upper bound variables in the pack are at their upper bounds, the flow variables not in the pack are
at their lower bounds and the remaining variable upper bound variables are at either of their bounds.
Letting K = N − \ C − , the flow pack inequality
X
X
X
X
X
yi +
(yi − min{ui , µ}xi ) +
(ui − µ)+ (1 − xi ) −
yi ≤
ui
(2)
i∈C +
i∈L+
i∈K
i∈C −
i∈C +
cuts off the fractional extreme points characterized by flow pack (C + , C − ). If xk , k ∈ L+ is fractional,
then (2) is violated by µ(1 − µ/uk ), else if xk , k ∈ C − is fractional, then it is violated by (uk − µ)µ/uk .
2
The flow pack inequality is a special case of the inequalities given in Stallaert [11] and may be viewed
as a flow cover inequality (1) for the relaxation of S, where the balance constraint is relaxed to
P
P
i∈N + yi − s ≤ −b after introducing a slack variable s. Flow pack inequalities are facet–
i∈N − yi −
defining for the convex hull of the restriction SC + = {(x, y) ∈ S : xi = 1 for i ∈ C + } of S.
Proposition 1 The following are necessary and sufficient conditions for flow pack inequality (2) to
be facet–defining for conv(SC + ):
P
1. If |C + ∪ K| ≥ 2, then (a) ui > µ for all i ∈ L+ , and (b) either L+ 6= ∅ or b + i∈N − ui > uk > µ
for some k ∈ C − .
2. If |C + ∪ K| = 1, then (a) ui > µ for all i ∈ L+ , and (b) either L+ 6= ∅ or uk > µ for some k ∈ C − .
3. If |C + ∪ K| = 0, then |L+ | = 1 and ui ≥ µ for i ∈ L+ .
P
Proof. We show only necessity here. For sufficiency see [2]. Suppose |C + ∪K| ≥ 1. Let ζ = i∈C + ∪K ui
and δij = (uj − µ)ui /ζ. If ui ≤ µ for some i ∈ L+ , then the inequality obtained by adding yi ≤ ui xi
and the flow pack inequality with L+ \ {i} is at least as strong as the original inequality. Now
suppose L+ = ∅. If there is no k ∈ C − with uk > µ, then the flow pack inequality can be obtained
by adding yi ≤ ui , i ∈ C + and −yi ≤ 0, i ∈ K. Finally, suppose no k ∈ C − with uk > µ
P
P
satisfies b + i∈N − ui > uk . Since by assumption (A.2) b + i∈N − ui ≥ uk holds for all k ∈ N − ,
P
P
b + i∈N − ui = uk for all k ∈ C − with uk > µ. Then b + i∈N − ui = uk ⇔ ζ = uk − µ ⇔ ui = δik
for all i ∈ C + ∪ K. But whenever xk = 0 for k ∈ C − with uk > µ, yi = 0 for all i ∈ C + and yi = ui
for all i ∈ K in any feasible solution. Hence if |C + ∪ K| ≥ 2, then there are not enough points on the
face defined by (2). If |C + ∪ K| = 0, then (2) reduces to a trivial inequality yi − µxi ≤ 0, i ∈ L+ and
the conditions |L+ | = 1 and ui ≥ µ for i ∈ L+ are clearly necessary and sufficient.
2.1
Lifting flow pack inequalities
In order to both strengthen and generalize flow pack inequality (2) for S, for a flow pack (C + , C − ) we
let L− ⊆ N − \ C − and K = N − \ (C − ∪ L− ), and consider the projection S o = {(x, y) ∈ S : (xj , yj ) =
P
(1, uj ) for all j ∈ C + , (xj , yj ) = (0, 0) for all j ∈ L− }. S o is full-dimensional if µ + i∈K ui ≥ uk for
all k ∈ N − \ L− . Inequality
X
X
X
(yi − min{ui , µ}xi ) +
(ui − µ)+ (1 − xi ) −
yi ≤ 0
(3)
i∈L+
i∈K
i∈C −
is facet–defining for conv(S o ) under the conditions of Proposition 1. To lift (3) for S, we need to
compute
P
P
P
+
f (z) = − max
+ (y − min{ui , µ}xi ) +
i∈C − (ui − µ) (1 − xi ) −
i∈K yi
P
Pi∈L i
o
y
≤
b
−
z
y
−
s.t. :
i∈N − \L− i
i∈N + \C + i
0 ≤ yi ≤ ui xi , xi ∈ {0, 1}, yi ∈ IR, i ∈ (N + \ C + ) ∪ (N − \ L− ),
(4)
P
for z ∈ IR− , where bo = b − i∈C + ui .
We show that optimization problem (4) can be solved easily. Let (x̄, ȳ) be an optimal solution to
(4), let Y = {i ∈ L+ : x̄i = 1} and Z = {i ∈ C − : x̄i = 0}. We may assume that Y ⊆ {i ∈ L+ : ui > µ},
Z ⊆ {i ∈ C − : ui > µ}, and ȳi = ui for all i ∈ C − \ Z, otherwise we can find a solution with the
3
same or better objective value satisfying these assumptions. For the moment, assume that ȳi = 0
P
P
P
for all i ∈ K in an optimal solution. Then i∈L+ ȳi equals either bo − z + i∈C − \Z ui or i∈Y ui ,
P
P
depending on whether the balance constraint is tight or not. If bo − z + i∈C − \Z ui ≤
i∈Y ui
P
(equivalently, − i∈Y ∪Z ui + µ ≤ z), then
X
X
X
f (z)1 = −(b −
ui +
ui + z − |Y |µ +
(ui − µ)) = (|Y ∪ Z| − 1)µ + z.
i∈C +
If bo − z +
P
i∈C − \Z
f (z)2 = −(
X
i∈Y
ui >
i∈Z
i∈C − \Z
P
i∈Y
ui − |Y |µ +
ui (equivalently, −
X
(ui − µ)) = −
i∈Z
P
i∈Y ∪Z
X
ui + µ > z), then
(ui − µ).
i∈Y ∪Z
Now we show that there exists an optimal solution such that uı = min{ui : i ∈ Y ∪ Z} ≥ uı =
max{ui : i ∈ (L+ ∪ C − ) \ (Y ∪ Z)}. For contradiction, suppose uı < uı . If the balance constraint is
tight (Case 1), by exchanging ı and ı we obtain a solution with the same objective value, since |Y ∪ Z|
remains unchanged. Otherwise (Case 2), we again exchange ı and ı. If the new solution is still in Case
2, f (z)2 decreases by uı − uı . If the new solution is in Case 1, the objective value decreases again since
P
P
f (z)2 − f (z)1 = − i∈Y ∪Z (ui − µ) − (|Y ∪ Z| − 1)µ − z = − i∈Y ∪Z ui + µ − z > 0.
Therefore, we may assume that Y ∪ Z consists of the first |Y ∪ Z| elements of {i1 , i2 , . . . , ir }
≡ {i ∈ C − ∪ L+ : ui > µ} indexed in nonincreasing order of ui . Then the lifting function f (z) can be
expressed in a closed form as


−wk+1 + µ < z ≤ −wk , k = 0, 1, . . . , r − 1,
 kµ + z,
f (z) =
(5)
kµ − wk , −wk < z ≤ −wk + µ,
k = 1, 2, . . . , r,

 rµ − w , z ≤ −w ,
r
r
Pk
where w0 = 0, wk = t=1 uit for k = 1, 2, . . . , r.
At this time we relax the assumption that ȳi = 0 for all i ∈ K. Suppose ȳi > 0 for some i ∈ K.
Then the balance constraint must be tight, otherwise a better solution is available by decreasing ȳi .
We can assume that ȳi = 0 for all i ∈ L+ , since otherwise we can obtain a solution with the same
P
P
objective value or better by decreasing both i∈K ȳi and i∈L+ ȳi in the same amount and setting
P
P
xi = 0 i ∈ L+ if applicable. Therefore i∈K ȳi = −bo + z − i∈C − \Z ui > 0, which is true only if
P
bo = b − i∈C + ui < 0. Then for such a solution, f (z) equals
X
X
X
ui ) = (|Z| − 1)µ + z.
ui − z +
f (z)3 = −( (ui − µ) + b −
i∈Z
i∈C +
i∈C − \Z
Let Z̄ give the minimum value f¯(z)3 . Z̄ consists of the elements of C − ordered in nondecreasing
P
P
P
ui such that bo − z + i∈C − \Z̄ ui < 0. Since bo − z + i∈C − \Z̄ ui < 0 ≤
i∈C + ui , it follows
P
that z > − i∈Z̄ ui + µ, implying f (z) ≤ f¯(z)3 . It remains to check the feasibility of the solution
described by f (z). If maxi∈L+ ui ≤ mini∈Z̄ ui , then f (z) is determined by only Z̄. In this case, if
P
P
− i∈Z̄ ui + mini∈Z̄ ui ≤ z, then by definition of Z̄, bo − z + i∈C − \Z̄ ui + mini∈Z̄ ui ≥ 0 and hence
P
P
the solution is feasible (when yi = 0 for all i ∈ K). Else if − i∈Z̄ ui + µ < z < − i∈Z̄ ui + mini∈Z̄ ui ,
then f (z) = (|Z̄| − 1)µ + z = f¯(z)3 . If maxi∈L+ ui > mini∈Z̄ ui , then f (z) is determined by a subset of
P
L+ and a strict subset Z of Z̄ and since bo − z + i∈C − \Z ui ≥ 0 for Z ⊂ Z̄, the solution is feasible.
This completes the characterization of the lifting function f (z).
4
It can be shown that f is superadditive on IR− , which implies that the lifting is sequence independent, that is the lifting function f (z) remains unchanged as the projected variable pairs (xj , yj ), j ∈
C + ∪ L− are introduced to inequality (3) sequentially [6, 14].
In order to write the lifted pack inequalities explicitly and show their relation to the lifted flow
cover inequalities, we define the following function. For u ∈ IR+ , let


wk ≤ u < wk+1 − µ, k = 0, 1, . . . , r − 1,
 kµ,
φL+ ∪C − (u, µ) =
kµ − wk + u, wk − µ ≤ u < wk ,
k = 1, 2, . . . , r,

 rµ − w + u, w ≤ u.
r
r
From Theorem 10 of Gu et al. [5], it can be seen that φC + ∪L− is an approximate superadditive lifting
function for flow cover inequalities. Then, after adding a slack variable s to the balance constraint of
P
P
S and relaxing it to i∈N − yi − i∈N + yi − s ≤ −b, by writing a lifted flow cover inequality for this
relaxation one obtains the following lifted flow pack inequality.
For i ∈ L− , let l = argmax0≤h≤r {ui ≥ wh − µ}. If l = 0, then let Gi = {(−1, 0)}. If l > 0, then
Gi = {(−1, 0)} ∪ G1i ∪ G2i , where
G1i = {(
wk − µ
µ
− 1, µ(k − 1 −
)) : k = 2, 3, . . . , l}
uk
uk
and


 ∅
(0, lµ − wl )
G2i =

µ
 (
ui −wl +µ − 1, µ(l −
ui
ui −wl +µ ))
if ui = wl − µ,
if wl − µ < ui ≤ wl or ui > wr ,
if ui ≤ wr and wl < ui < wl+1 − µ.
Theorem 2 Let (C + , C − ) be a flow pack. If S o is full–dimensional, (αi , βi ) ∈ Gi for i ∈ L− , and the
conditions of Proposition 1 are satisfied, then the lifted flow pack inequality
X
X
(yi + φL+ ∪C − (ui , µ))(1 − xi )) +
(yi − µxi )
i∈C +
i∈L+
+
X
(ui − min{ui , µ})+ (1 − xi ) +
i∈C −
X
i∈L−
(αi yi + βi xi ) −
X
i∈K
yi ≤
X
ui ,
(6)
i∈C +
is facet–defining for conv(S).
Inequality (6) is facet–defining for conv(S), because the exact lifting function (5) for the flow pack
inequalities can be written as f (−u) = φL+ ∪C − (u) − u.
2.2
Comparison of valid inequalities
In contrast to the lifting function of the flow pack inequality (2), the lifting function of the flow
cover inequality (1) is not superadditive and its lifting is not sequence independent [5]. An intuitive
explanation for the absence of superadditivity in the cover lifting function is that the nonincreasing
order of ui may need to be violated to achieve feasibility when lifting a flow cover inequality; the lifting
problem of flow cover inequalities is a restriction, whereas the lifting problem of flow pack inequalities
is a relaxation.
5
In order to highlight the relation among valid inequalities for S, we write a form of the lifted
flow cover inequalities, which uses φC + ∪L− as a valid superadditive lifting function, even though this
function is slightly weaker than the one given in Theorem 11 of Gu et al. [5] for flow covers. Let
(C + , C − ) be a flow cover. If (αi , βi ) ∈ Gi for i ∈ L+ , then the following lifted flow cover inequality is
valid for S
X
X
((αi + 1)yi + βi xi )
(yi + (ui − λ)+ (1 − xi )) +
i∈L+
i∈C +
+
X
φC + ∪L− (ui , λ)(1 − xi ) −
X
min{ui , λ}xi +
X
yi ≤ b +
i∈K
i∈L−
i∈C −
X
ui . (7)
i∈C −
Now we compare the lifted flow pack and the lifted flow cover inequalities with the k–cover and
k–reverse–cover inequalities [7] derived from mixed 0–1 knapsack relaxations of S. Observing that
φL+ ∪C − (u, µ) + ψL+ ∪C − (u) = u (ψ is a notation used in [7]), after algebraic manipulations, the k–
cover inequality can be restated as
X
X
(yi − (ui − λ)+ xi ) +
(yi − (ui − φC + ∪L− (ui , λ))xi )
i∈C +
i∈L+
+
X
φC + ∪L− (ui , λ)(1 − xi ) −
i∈C −
X
min{ui , λ}xi −
i∈L−
X
yi ≤ b +
i∈K
X
ui ,
X
ui . (9)
(8)
i∈C −
and the k–reverse–cover inequality can be restated as
X
X
(yi + φL+ ∪C − (ui , µ)(1 − xi )) +
(yi − min{ui , µ}xi )
i∈C +
i∈L+
+
X
(ui − µ)+ (1 − xi ) −
i∈C −
X
(ui − φL+ ∪C − (ui , µ))xi −
i∈L−
X
i∈K
yi ≤
i∈C +
Lifted flow pack inequality (6) with (αi , βi ) ∈ G2i for all i ∈ L+ is at least as strong as k–reverse–
cover inequality (9). To see this, observe that for i ∈ L+ , if wl − µ < ui ≤ wl or ui > wr , then
βi = lµ − wl = φC + ∪L− (ui , λ), and thus the inequalities are the same. Else if ui ≤ wr and wl <
µ
ui
)x̄i + ( ui −w
)ȳi and
ui < wl+1 − µ, then for (x̄, ȳ) ∈ S, let ζ1 = αi x̄i + βi ȳi = µ(l − ui −w
l +µ
l +µ
µ
ζ2 = −(ui − φL+ ∪C − (ui , µ))x̄i = (lµ − ui )x̄i . Then ζ1 − ζ2 = (1 − ui −wl +µ )(ui x̄i − ȳi ) ≥ 0. Similarly,
lifted flow cover inequality (7) is at least as strong as k–cover inequality (8) for S. Yet it is still very
interesting that one gets strong valid inequalities such as (8) and (9) for S from its much simpler mixed
0–1 knapsack relaxation.
Example Suppose S is given by
y1 + y2 + y3 + y4 − y5 − y6 − y7 ≤ 3,
y1 ≤ 6x1 , y2 ≤ 4x2 , y3 ≤ 6x3 , y4 ≤ 8x4 , y5 ≤ 5x5 , y6 ≤ 2x6 , y7 ≤ 12x7 .
Let C + = {1}, C − = {5}, L+ = {2, 3}, L− = {7}, and K = {6}. Then µ = 2 and y2 − 2x2 + y3 −
2x3 + 3(1 − x5 ) − y6 ≤ 0 is valid for S o = {(x, y) ∈ S : (x1 , y1 ) = (1, 6), (x7 , y7 ) = (0, 0)}. Note that
w0 = 0, w1 = 6, w2 = 11, w3 = 15. Lifting it with x1 , y1 , x7 , and y7 , we obtain the following lifted flow
pack inequalities
y1 + 2(1 − x1 ) + y2 − 2x2 + y3 − 2x3 − 3(1 − x5 ) − y6 − y7 − 0x7 ≤ 6,
y1 + 2(1 − x1 ) + y2 − 2x2 + y3 − 2x3 − 3(1 − x5 ) − y6 − 53 y7 − 85 x7 ≤ 6,
y1 + 2(1 − x1 ) + y2 − 2x2 + y3 − 2x3 − 3(1 − x5 ) − y6 − 13 y7 − 4x7 ≤ 6.
6
Alternatively, let C + = {1, 2}, C − = {5}, L+ = {4}, L− = ∅, and K = {6, 7}. Then λ = 2 and
y1 +4(1−x4 )+y2 +2(1−x2 )−y6 −y7 ≤ 8 is valid for S o = {(x, y) ∈ S : (x4 , y4 ) = (0, 0), (x5 , y5 ) = (1, 5)}.
Here w0 = 0, w1 = 6, w2 = 10. Lifting it with x4 , y4 , x5 , and y5 , we get the following lifted flow cover
inequalities
y1 + 4(1 − x1 ) + y2 + 2(1 − x2 ) + 0y4 − 0x4 − (1 − x5 ) − y6 − y7 ≤ 8,
y1 + 4(1 − x1 ) + y2 + 2(1 − x2 ) + 21 y4 − 2x4 − (1 − x5 ) − y6 − y7 ≤ 8.
3
Computational results
In this section we present a summary of computational experiments on using the lifted flow pack
inequalities (6) in a branch–and–cut algorithm for solving capacitated fixed–charge network flow problems. The branch–and–cut algorithm was implemented with MINTO [8] (version 3.0) using CPLEX
(version 6.0) as the LP solver. All experiments were performed on a SUN Ultra 10 workstation. In
addition to the lifted pack inequalities, we used the lifted flow cover inequalities that are automatically
generated by MINTO. The lifted flow cover inequalities that MINTO generates are similar to inequality
(7) and are described in Gu et al. [5].
In order to test the effectiveness of the lifted flow pack inequalities, we ran the branch–and–cut
algorithm on a set of randomly generated capacitated fixed–charge network flow problems available
at http://ieor.berkeley.edu/∼atamturk/data. In Tables 1 and 2 we present a comparison of
the performance of the branch–and–cut algorithm when run with only MINTO’s lifted flow cover
inequalities, with adding flow pack inequalities, and finally with adding lifted flow pack inequalities.
Table 1 is for instances with 30 nodes and 50% arc density, whereas Table 2 is for instances with
60 nodes and 20% arc density. For this experiment, we also implemented a simple primal heuristic,
which rounds fractional binary variables in LP solutions in the search tree to one, in order to construct
feasible solutions quickly for pruning. We report the objective values of the initial LP relaxation after
preprocessing (zinit), optimal integral solution (zopt), and the LP relaxation at the root node of the
search tree after all cuts are added (zroot), the number of lifted flow cover cuts (lfcovs), the number of
flow pack cuts (fpacks), the number of lifted flow pack cuts (lfpacks), the number of nodes evaluated
(nodes), and the CPU time elapsed in seconds. All fractional entries are rounded to the nearest integer.
A comparison of the entries in columns zinit and zopt shows that all of these problems have big duality
gaps. LP relaxations at the root node (zroot) improve consistently with the addition of flow pack and
lifted flow pack cuts. We also see significant reductions in the number of nodes explored and in the
CPU time elapsed. Note that more lifted flow pack cuts are generated than flow pack cuts and that
lifting flow pack cuts do improve the LP relaxations. The effect of lifting flow pack inequalities is more
pronounced for the larger instances in Table 2. These computational results indicate that the flow
pack inequalities and the lifted flow pack inequalities improve the performance of the branch–and–cut
algorithm significantly for capacitated fixed–charge network flow problems.
7
Table 1: Computations with capacitated fixed–charge network flow problems: 30 nodes.
without fpacks
with fpacks
with lfpacks
problem zinit zopt zroot lfcovs nodes time zroot lfcovs fpacks nodes time zroot lfcovs lfpacks nodes time
fc.30.1
120 307 259 458 9783 210 261
286
45
5434 122 261 309
186
6320 145
fc.30.2
152 325 273 341 4689
95 276
334
75
1807
38 280 230
171
1412
43
98 294 235 252 3872
47 251
412 125 2193
68 254 245
159
1820
53
fc.30.3
100 763 712 261 3238
40 734
305
65
2213
35 738 205
104
1847
28
fc.30.4
158 301 274 146
215
4 277
172
36
145
4 277 128
60
149
4
fc.30.5
fc.30.6
126 272 238 296 8759 105 241
276
64
3019
72 245 243
187
4357
93
97 231 204 187 1471
14 207
167
30
458
8 212 176
98
554
10
fc.30.7
176 347 300 431 13623 270 305
381
72
4257 113 305 290
182
6063 147
fc.30.8
85 741 708 334 3172
56 709
257
43
1818
37 710 313
145
1793
40
fc.30.9
93 204 171 252 2991
32 173
249
66
1217
17 179 166
118
507
11
fc.30.10
Table 2: Computations with capacitated fixed–charge network flow problems: 60 nodes.
without fpacks
with fpacks
problem zinit zopt zroot lfcovs nodes time zroot lfcovs lfpacks nodes
fc.60.1
171 487 394 612 79448 2062 413
751
225 23631
fc.60.2
230 584 445 830 82050 1018 459
975
347 33580
161 493 414 494 11124 300 420
698
194 15091
fc.60.3
120 442 337 780 141882 4175 354
999
322 30956
fc.60.4
160 414 332 451 20436 309 336
587
155
9775
fc.60.5
199 480 388 681 127285 3656 399
738
237 28929
fc.60.6
fc.60.7
177 492 422 463
7450 184 426
497
101
5350
202 500 440 486 14360 353 453
671
159
7897
fc.60.8
164 397 358 342
3620
54 365
312
47
1557
fc.60.9
759
196
8675
fc.60.10 180 913 823 820 44523 1982 827
with lfpacks
time zroot lfcovs lfpacks nodes time
1122 415 468
350 21444 1054
1812 490 471
450 15070 634
497 420 358
279
4929 150
1652 354 532
423 19386 960
238 349 259
245
1734
39
1063 421 491
373 20553 939
373 443 375
305
4090 176
425 455 438
330
5239 289
25 365 227
142
915
16
508 834 450
255 11912 534
Acknowledgment
I am grateful to Z. Gu, G. L. Nemhauser and M. W. P. Savelsbergh for their valuable comments and
to L. A. Wolsey for pointing out that flow cover inequalities can be used to show validity of flow pack
inequalities, which shortened the presentation significantly.
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