Int J Game Theory (2001) 29:533±542
2001
9
99
9
On the optimality of a simple strategy for
searching graphs
Shmuel Gal
The University of Haifa, Haifa 31905, Israel (email: [email protected])
Received: December 1999/Revised version: September 2000
Abstract. Consider a search game with an immobile hider in a graph. A Chinese
postman tour is a closed trajectory which visits all the points of the graph and
has minimal length. We show that encircling the Chinese postman tour in a
random direction is an optimal search strategy if and only if the graph is
weakly Eulerian (i.e it consists of several Eulerian curves connected in a treelike structure).
Key words: Search game, weakly Eulerian graph, weakly cyclic graph, Chinese
postman tour
1. Introduction
Search games were ®rst described in Isaacs (1965) and analyzed in detail in
Gal (1980) and Alpern and Gal (to appear). In this game there is a search
space Q with a starting point O (the ``origin'') of the searcher. The searcher
moves with maximal velocity 1 and would like to minimize the time he captures the hider. This is a zero-sum game with the hider, who can choose any
hiding point inside Q (at any vertex or on any arc) having the opposite goal.
In Gal (1980) it is proven that if Q is compact then this game always has a
value. In this paper we assume that Q is a graph and the hider is immobile.
We will present an exact de®nition of this game in the next chapter. Gal
(1980) has solved this game for Eulerian graphs and for trees. Reijnierse and
Potters (1993) presented a signi®cant extension of the results by solving the
game for weakly-cyclic graphs (de®ned in Ch. 4). They also made two interesting conjectures concerning the solution for weakly Eulerian graphs (De®ned in Ch. 5). In this paper we prove, in Ch. 5, that their conjectures are
correct. In order to present the new results we brie¯y present some necessary
534
S. Gal
background material: the framework is presented in Ch. 2 and searching in a
tree is brie¯y presented in Ch. 3. Searching in weakly cyclic graphs is presented in Ch. 4 and ®nally our new results on searching weakly Eulerian
graphs are presented in Ch. 5.
2. Search in a graph
In this section we present the framework and some preliminary known results
on the search in a graph. The presentation is rather brief. Full details can be
found in [4] or [1].
In our discussion, the notion a ``graph'', Q, will mean any ®nite connected
set of arcs which intersect only at vertices of Q. Thus, Q will be represented by
a set in a three dimensional Euclidean space with vertices (nodes) consisting of
all points of Q with degree 0 2 plus, possibly, a ®nite number of points with
degree 2. (Note that we allow more than one arc to meet the same pair of
vertices.). The sum of the lengths of the arcs in Q will be called the ``measure
of Q'' and denoted by m. For any A; B A Q, d…A; B† denotes the shortest distance in Q.
A search trajectory S ˆ S…t† is a continuous trajectory in Q starting from
the origin O with maximum velocity 1 (i.e., a mapping S from ‰0; y† into Q
with S…0† ˆ O, such that for any t1 < t2 , d…s…t1 †; s…t2 †† U t2 t1 ).
A search strategy, s, is a probability mixture of search trajectories. (An
exact de®nition is given in [4] and [1] but here we use only a ®nite number of
trajectories).
A hiding strategy, h, is a probability measure in Q.
For any search trajectory, S, and hiding point, H, the capture time is the
minimum t with S…t† ˆ H.
The game is zero-sum. For any s and h the loss of the searcher (the gain of
the hider) is the expected capture time, t…s; h†, if the searcher uses s and the
hider uses h.
The value, v, of the search game with an immobile hider in Q always exists.
In studying search trajectories in Q, we shall often use the following de®nitions.
De®nition 1. A trajectory S…t† de®ned for 0 U t U T is called ``closed'' if s…0† ˆ
S…T†. A closed trajectory which visits all points of Q is called a ``tour''.
De®nition 2. A graph Q is called ``Eulerian'' if there exists a tour L with length
m. The tour L will be called an ``Eulerian tour.''
It is well known that Q is Eulerian if and only if the degree (i.e., the number of arcs attached) of all the vertices of Q is even.
De®nition 3. A closed trajectory which visits all the points of Q and has minimal
length will be called a ``Chinese postman tour''. Its length will be denoted by m.
It can be easily seen that m U 2m. for any graph.
Finding a minimal tour in a given graph is called the Chinese postman
problem. This problem can be reformulated as follows. Find in the given
graph Q a set of arcs of minimum total length, such that when these arcs are
On the optimality of a simple strategy for searching graphs
535
duplicated (i.e., traversed twice), the degree of each vertex becomes even. This
problem was solved by Edmonds (1965) and Edmonds and Johnson (1973)
using a matching algorithm which uses O…n 3 † computational steps, n being the
number of vertices in Q.
The graph obtained from Q by duplicating the above arcs has measure m
and is denoted by Q…2†.
We shall use the following result
Lemma 1. v U m=2
Proof. If the searcher uses a strategy which encircles a Chinese postman tour
with probability 1=2 for each direction, then for any hiding point the expected
discovery time does not exceed m=2.
De®nition 4. The search strategy used for proving Lemma 1 will be called
``random Chinese postman tour''.
The following result is proven in Gal (1980) and Alpen and Gal (to
appear)
Theorem 1. For any graph, the value v of the search game with an immobile
hider satis®es
m=2 U v U m=2 U m:
The lower bound is attained if and only if Q is Eulerian. The upper bound
m is attained if and only if Q is a tree.
3. Search on a tree
We now consider the search game on a tree Q. The value satis®es: v ˆ m.
The optimal search strategy is simply a random Chinese postman tour which
guarantees m=2 (ˆ m for a tree). The optimal hiding strategy is a little more
complicated. It is positive only for the leaves of Q. The probabilities for the
leaves are constructed recursively by assigning probabilities for sub-trees in
the following way: We start from the origin O with p…Q† ˆ 1 and go towards
the leaves and in any branching we split the probability of the current sub-tree
proportionally to the measures of sub-trees corresponding to the branches.
When only one arc remains in the current sub-tree we assign the remaining
probability, p…A†, to the leaf, A, at the end of this arc. We illustrate this
method for the tree depicted in Figure 1.
From O we branch into A1 ; C; O1 and O2 with proportions 1, 3, 6 and 3
1 3
, 13,
respectively. Thus, the probabilities of the corresponding sub-trees are 13
6
3
1
,
and
respectively.
Since
A
and
C
are
leaves
we
obtain
p…A
†
ˆ
1
1
13
13
13 and
3
p…C† ˆ 13. Continuing towards O1 we split the probability of the correspond6
, with proportions 15, 15 and 35 between B1 ; B2 and C1 so that:
ing sub-tree, 13
p…B1 † ˆ
6
;
65
p…B2 † ˆ
6
65
and
p…C1 † ˆ
18
:
65
3
3
3
3
Similarly p…A2 † ˆ 13
12 ˆ 26
and p…A3 † ˆ 13
12 ˆ 26
.
536
S. Gal
Fig. 1.
Fig. 2.
4. Search in weakly cyclic graphs
In the case that the graph Q is neither Eulerian nor a tree, it follows from
Theorem 1 that m=2 < v < m. In the following and the next section, we consider such graphs which, nevertheless, have a relatively simple solution with
the property that the random Chinese postman tour is optimal (as in the cases
of Eulerian graphs and trees).
De®nition 5. A graph is called weakly cyclic if between any two points there
exist at most two disjoint paths.
An equivalent requirement, presented in [6], is that the graph has no subset
topologically homeomorphic with a graph consisting of three arcs joining two
points.
The di½culty in solving search games for the three arcs graph is presented
in [4] and [1]. Note that Eulerian graph may be weakly cyclic (e.g. if all the
nodes have degree 2) but need not be weakly cyclic (e.g., 4 arcs connecting
two points).
It follows from the de®nition that if a weakly cyclic graph has a closed
curve G with arcs b1 ; . . . ; bk incident to it, then removing G disconnects Q into
k disjoint graphs Q1 ; . . . ; Qk with bi belonging to Qi . (If Qi and Qj would have
been connected then the incidence points of G with bi and with bj could be
connected by 3 disjoint paths). Thus, any weakly cyclic graph can be constructed by starting with a tree (which is obviously weakly cyclic) and replacing some of its nodes by closed curves as, for example, G and G 1 in Fig. 2 (all
edges have length 1):
We now formally de®ne the above operation:
De®nition 6. Let a graph Q contain a connected subgraph G. If we replace the
graph Q by a graph Q 0 in which G is replaced by a point B and all arcs in QnG
On the optimality of a simple strategy for searching graphs
537
Fig. 3.
which are incident to G are incident to B in Q 0 we shall say that G was shrinked.
and B is the shrinking node.
It is easy to see that Q is a weakly cyclic graph if and only if Q contains a set of
non-intersecting closed curves such that shrinking them transforms Q into a tree.
In order to obtain a feeling about optimal solutions for such graphs we
consider a simple example in which Q is a union of an interval of length l and
a circle of circumference m l with only one point of intersection as depicted
in Fig. 3. Assume, for the moment, that the searcher's starting point, O, is at
the intersection.
Note that the length of the Chinese postman tour is m ˆ m ‡ 2l. We now
m ‡ 2l m
ˆ and the optimal
show that the value of the game satis®es v ˆ
2
2
search strategy, s, is the random Chinese postman tour.
m
The random Chinese postman tour guarantees (at most) by Lemma 1.
2
m
We now present a hiding strategy, h, which guarantees (at least) . h is
2
2l
at the end of the interval
described as follows: hide with probability p ˆ
m ‡ 2l
(at A) and with probability 1 p uniformly on the circle. It can be easily
checked that if the searcher either goes to A, returns to O and then goes
around the circle, or encircles and later goes to A, then the expected capture
m ‡ 2l
. Also, any other search trajectory yields a larger extime is equal to
2
pected capture time.
Now assume that the starting point is di¨erent from O. In this case the
value and the optimal search strategy remain the same the same but the optimal hiding strategy remains h only if the starting point is (anywhere) on the
circle. However as the starting point moves from O to A the probability of
2l
to 0.
hiding at A decreases from
m ‡ 2l
Solving the search games for weakly cyclic graphs was presented by Reijnierse and Potters [6]. In their paper they show that v ˆ m=2 and present an
algorithm for constructing optimal hiding strategies. (The optimal search
strategy, as presented in Lemma 1, is the random Chinese postman tour.)
We now present a simpler version of their algorithm by transforming the
graph Q into an ``equivalent'' tree Q as follows: Shrink each closed curve Gi ,
g
with circumference gi , and replace it by an arc ci of length i which connects
2
538
S. Gal
a (new) leaf Ci to the shrinking node Bi . All other arcs and nodes remain the
same. Let h be the optimal hiding strategy for the tree Q. Then the optimal
hiding strategy for Q is obtained as follows:
1. For a leaf of Q (which is also a leaf of Q) hide with the probability
assigned to it by h.
2. For a curve G i (represented by leaf Ci in Q) hide uniformly along it with
overall probability assigned by h to leaf Ci .
All other arcs and nodes are never chosen as hiding places.
We now use the above construction for the example presented in [6]. The
graph Q is depicted in Figure 2 and its equivalent tree, Q, is depicted in Figure
1 of the previous chapter.
Note that the curves G and G 1 are replaced by arcs OC and O1 C1 . Let h
be the optimal hiding strategy for Q. Then, the optimal hiding probability in
Q is the same for the leaves of Q and is obtained for the curves G and G 1 by
replacing the probability of hiding at the leaves C and C1 by hiding (uni3
3
(i.e. probability density 78
) and on G 1 with
formly) on G with probability 13
18
(i.e.
probability
density
).
probability 18
65
390
5. Search in weakly Eulerian graphs
Reijnierse and Potters (1993) made two conjectures:
1. v ˆ m=2 holds for the, wider family of weakly Eulerian graphs, i.e., graphs
which are obtained from a tree with some nodes replaced by Eulerian graphs.
2. v ˆ m=2 implies that the graph is weakly Eulerian.
The ®rst conjecture was actually proved to be correct by Reijnierse (1995)
using a recursive algorithm similar to the one used in [6] for the weakly cyclic
graphs.
We now present a simple proof for the ®rst conjecture and also show that
their second conjecture is correct. We ®rst extend the shrinking operation for
any Eulerian subgraph in exactly the same way of De®nition 6.
De®nition 7. A graph Q is called ``weakly Eulerian'' if it contains a set of nonintersecting Eulerian subgraphs G1 ; . . . Gk such that shrinking them transforms
Q into a tree.
An equivalent condition presented in [8] is that removing all the (open)
arcs which disconnect the graph (the `tree part') leaves a subgraph with all
nodes having an even (possibly zero) degree.
Fig. 4.
On the optimality of a simple strategy for searching graphs
539
Obviously, any Eulerian graph is also weakly-Eulerian, and any weakly
cyclic graph is also weakly Eulerian.
A weakly-Eulerian graph has the structure depicted in Figure 4.
Theorem 2. If Q is weakly Eulerian, then:
1. The value: v ˆ m=2.
2. The random Chinese postman tour is an optimal search strategy
3. An optimal hiding strategy, h, is the optimal hiding strategy for the
equivalent tree, Q, obtained by shrinking all the Eulerian curves and replacing
each curve by an arc from the shrinking node to a (new) leaf with length equal
to half the measure of the corresponding Eulerian subgraph. Then, the probability of such a leaf is uniformly spread on the corresponding Eulerian subgraph.
(The hiding probabilities of the leaves of Q remain the same.)
Proof. Let the measure of the equivalent tree, Q, be m=2. It can easily be
veri®ed that the length of a Chinese postman tour is m because each arc in the
tree part of Q has to be traversed twice while the Eulerian subgraphs can be
traversed just once. Thus, the searcher can achieve m=2.
We now show that h guarantees at least m=2 for the hider. Assume that the
lengths of the arcs of the Eulerian subgraphs are all rational. (Note that the
value and the strategies are continuous functions of the arc lengths.) Thus, we
can ®nd an e > 0 (arbitrarily small) so that the lengths of these arcs are even
number of multiples of e, say 2ke for a speci®c arc. Consider the grid R1 ; . . . ;
Rk with distances e; 3e; . . . ; …2k 1†e from the end points of this arc. Now
construct such a grid (with the same e) for all the arcs of the Eulerian subgraphs. We will now present an optimal hiding strategy which hides either at a
grid point or at a leaf of Q. In order to do so we ®rst construct a slightly dif^ as follows: shrink all the Eulerian subgraphs and at
ferent equivalent tree, Q,
each shrinking node, B, add all the grid points, of the corresponding Eulerian
subgraph, as leaves connected to B with arcs of length e each. Denote the
^ is similar
measure of the Eulerian subgraph by g. Then, the equivalent tree Q
g
arcs of length e
to Q except that the Eulerian subgraph is represented by
2e
g
each, rather than one arc of length , but both representations have the same
2
in¯uence on the remaining part of the equivalent tree (as, e.g., replacing the
arc OC, in Figure 1, by two arcs OR1 and OR 2 , with length 32 each, where R1
and R 2 are leaves). The measure of Q is m=2.
^ does not exceed the
Note that the distance between any two leaves in Q
distance of the corresponding points in Q and the same inequality holds for
^ Thus, any search trajecthe distance between the origin O and any leaf of Q.
tory which searches the leaves and the grid points of Q, corresponds to a
^ so that the time to reach any hiding point in Q is greater
search trajectory in Q
^
^ Since by using h for Q,
than or equal to the time to reach the same point in Q.
the hider guarantees at least m=2, the same strategy if used for Q also guarantees m=2. Since e is arbitrarily small we see that uniform probability density
on the Eulerian subgraphs (with the total probability of choosing it given by
g
the probability of choosing a leaf at the end of an arc with length going out
2
of the corresponding shrinking node B) is optimal.
540
S. Gal
Fig. 5.
Fig. 6.
(The probability of hiding in the Eulerian subgraphs is obviously not
unique because hiding only at the grid points, or just in the middle of each arc,
with the appropriate probability, is also optimal.)
We now illustrate the result by an example: Let Q be the union of an
Eulerian graph G, of measure g, and two arcs, of lengths 1 and 2 respectively,
leading to leafs C1 and C2 (see Fig. 5)
If O A G then, Q would be a star with three rays of lengths 1, 2, and 0:5g
1
, at C2 with prorespectively. Thus, h hides at C1 with probability
0:5g ‡ 3
2
0:5g
and uniformly on G with overall probability
. If the
bability
0:5g ‡ 3
0:5g ‡ 3
starting point is on the arc leading to C2 with distance 1 from C2 then Q
would be the tree depicted in Figure 6. Thus, the corresponding optimal hiding probabilities for C1 ; C2 , and G would be
0:5g‡2
1
;
0:5g‡3 0:5g‡1
1
;
0:5g‡3
and
0:5g‡2
0:5g
;
0:5g‡3 0:5g‡1
respectively:
We now prove the second conjecture of Reijnierse and Potters (1993), i.e.,
that all graphs which are not weakly Eulerian have value strictly smaller than
m=2:
Theorem 3. For any graph Q. If the value satis®es v ˆ m=2 then Q is weakly
Eulerian.
Proof. Let L be a Chinese postman tour (with length m) then the random
Chinese postman tour of L, is an optimal search strategy (since it guarantees
m=2). L traverses some of the arcs once ± call the set of such arcs T1 , and some
arcs twice ± call this set T2 . Thus, the graph Q…2† in which all the arcs in T2
are duplicated is Eulerian and has measure m.
We shall show that removing each arc of T2 disconnects the graph. Since
all the nodes in the set T1 must have even degrees which implies that T1 is
either an Eulerian curve or a set of Eulerian curves, it would then follow that
Q satis®es the equivalent conditions of De®nition 7.
On the optimality of a simple strategy for searching graphs
541
Since v ˆ m=2 then, for any e > 0 there exists he , an e-optimal hiding
strategy which guarantees at least m=2 e capture time. Let b A T2 with length
l…b† > 0. We now show that assuming Qnb connected leads to a contradiction:
If Qnb is connected then removing from Q…2† both b and its duplicate leaves
the graph connected (and Eulerian). Thus, there exists a tour L 0 of Qnb which
visits all the arcs of Qnb and has length m 2l…b†. Let the end points of b be
A and C and its midpoint ± B. Since both A and C are visited by L 0 , then L 0
can be completed into a Chinese postman tour of Q in the following two ways:
LA ± by traversing b in both directions when L 0 visits A or
LC ± by traversing b in both directions when L 0 visits C.
Now, the optimal search strategies sA and sC based on random encircling
of LA and LC respectively would guarantee at most m=2 expected capture
time for all hiding points H A Q. Also, for any H in the ``half arc'' ‰A; BŠ the
expected capture time t…sA ; H† satis®es
t…sA ; H† U
m
2
l…b†:
m
e it follows that, under he , the probability of hiding
2
in ‰A; BŠ is smaller than e=l…b†. Similarly, by using sC we can show that the
probability of hiding in ‰B; C Š is smaller than e=l…b†. Thus, under he , the probability of hiding in b is smaller than 2e=l…b† but then the searcher could use
the strategy s 0 which ®rst randomly encircles L 0 and later goes in the shortest
route to A and traverses the arc b. The expected capture time for any H A Qnb
m
l…b† and for any H A b : t…s 0 ; H† U m ‡ d…O; A†. Thus,
satis®es t…s 0 ; H† U
2
if e is small enough:
m
2e
2e
m l…b†
0
l…b† 1
<
t…s ; he † U
‡ …m ‡ d…O; A†† 2
l…b†
l…b† 2
2
Since t…sA ; he † >
which would contradict the e-optimality of he . Thus, removing b disconnects
Q. Q.E.D.
Combining Theorems 1 2 and 3 we have the following result:
v ˆ m=2 if and only if Q is weakly Eulerian. v < m=2 if and only if Q is not
weakly Eulerian.
An equivalent statement is that random Chinese postman tour is an optimal search strategy if and only if the graph Q is weakly Eulerian.
Searching a graph which is not weakly Eulerian is expected to lead to rather
complicated optimal search strategies. Even the ``simple'' graph with two nodes
connected by three arcs, of unit length each, leads to an optimal search strategy
which is a mixture of in®nitely many trajectories (see [4] and [1]).
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542
S. Gal
[2] Edmonds J (1965) The Chinese postman problem. Bull. Oper. Res. Soc. Amer. 13, Suppl. 1,
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[3] Edmonds J, Johnson EL (1973) Matching Euler tours and the Chinese postman problem.
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Theory 21:385±394
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