On the concavity of delivery games

E L S E V I E R European Journal of Operational Research 99 (1997) 445-458

EUROPEAN JOURNAL

OF OPERATIONAL RESEARCH

T h e o r y a n d M e t h o d o l o g y

On the concavity of delivery games

H e r b e r t H a m e r s *

CentER and Department of Econometrics, Tilburg University, P.O. Box 90153, 5000 LE Tilburg, Netherlands

Received March 1995

A b s t r a c t

Delivery games, introduced by Hamers, Bonn, van de Leensel and Tijs in 1994, are combinatorial optimization games that arise from delivery problems closely related to the Chinese postman problem (CPP). They showed that delivery games are not necessarily balanced. For delivery problems corresponding to the class of bridge-connected Euler graphs they showed that the related games are balanced. This paper focuses on the concavity property for delivery games. A delivery game arising from a delivery model corresponding to a bridge-connected Euler graph need not be concave. The main result will be that for delivery problems corresponding to the class of bridge-connected cyclic graphs, which is a subclass of the class of bridge-connected Euler graphs, the related delivery games are concave. Further, we discuss some extreme points of the core and the ~--value for this class of concave delivery games. © 1997 Elsevier Science B.V.

Keywords: Chinese postman problem; Cooperative games

1. I n t r o d u c t i o n

The delivery situations introduced in Hamers et al. (1994) are defined on a connected graph in which a cost function is defined on the edges. For these situations, which are closely related to the Chinese postman problem (cf. Kwan, 1962), they defined a new class of combinatorial opt imizat ion games called delivery games. They showed that these games need not be balanced. For delivery situations corresponding to br idge-connected Euler graphs they showed that the related delivery games are balanced.

This paper focuses on the concavity property of delivery games. A game satisfies the concavity property if the characteristic function of a del ivery game is a sub-modular function. In general a del ivery game is not concave, since concavity would imply that the game is balanced. But even for the class of balanced games corresponding to the delivery problems that arise from br idge-connected Euler graphs concavity is not necessari ly satisfied. We will show that delivery games corresponding to the class of br idge-connected cyclic graphs, which is a subclass of the class of bridge-connected Euler graphs, are concave.

There are several arguments to ask for concavity. Concave games are balanced, which means that one can find core-elements prescribing an allocation of the worth of the grand coalit ion among the players in such a way that no subgroup has an incentive to split off. Moreover, Shapley (1971) (cf. Edmonds, 1970) and Ishiichi

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446 H. Hamers / European Journal o f Operational Research 99 (1997) 445-458

(1981) showed that the extreme points of the core are the marginal vectors of the game if and only if the game is concave. Here, a marginal vector allocates to each player the marginal contribution this player constitutes according to a given way (a permutation) to form the grand coalition. With respect to one-point game theoretical solution concepts, concave games are nice since the Shapley value (Shapley, 1953), which is by definition the average of all marginal vectors, is in the barycentre of the core. Further, the z-value (Tijs, 1981), which is an efficient compromise between an utopia vector and a minimal right vector, can be easily calculated.

Concavity, or its counterpart convexity (super-modular), has already been a subject of study for several classes of games. For instance, airport games (cf. Littlechild and Owen, 1973) are concave and bankruptcy games (cf. Aumann and Maschler, 1985) are convex. The concavity (convexity) property is a rare property for combinatorial games. Only two classes of sequencing games, considered in Curiel, Pederzoli and Tijs (1989) and Hamers, Borm and Tijs (1995), are contained in the class of convex games. Other combinatorial games, e.g. minimum spanning tree games (cf. Granot and Huberman, 1981), travelling sales man games (cf. Potters, Curiel and Tijs, 1992), flow games (cf. Kalai and Zemel, 1982a) and LP-games (cf. Owen, 1975), do not posses the concavity or convexity property.

The paper is organized as follows. In Section 2 the delivery game is formulated and several examples describe some properties of delivery games. Subsequently the concavity result is formulated. For these concave delivery games we study some extreme points of its core by looking at marginal vectors. Finally, the z-value is evaluated for these delivery games. Section 3 contains the formal proof for the concavity result.

2. Delivery problems and delivery games

This section describes the class of delivery models and the corresponding delivery games. Before the formal description is provided we need some definitions.

Let G = (V, E, i) be an undirected connected graph, where V is the set of vertices, E the set of edges and i the incidence mapping that assigns to each edge of E an unordered pair of not necessarily different vertices of V. A walk in G = (V, E, i) is a finite sequence of edges and vertices of the form v~, e~, v 2 . . . . . ek, vk+ l with k>J0, v I . . . . . v k + ~ V , e I . . . . . e k ~ E such that i ( e j ) = { v j , vi+l} for all j ~ { 1 . . . . . k}. Such a walk is a closed walk if vj = vk+ ~. A closed walk in which all edges are distinct is called a closed trail. A path in G is a walk in which all vertices (except, possibly v I = vk+ l) and edges are distinct. A closed path, i.e. v I = vk+ 1, containing at least one edge is called a circuit.

Let G = (V, E, i) be a connected graph. We assume that each edge corresponds to one player. Formally, there is a one-to-one map g : E --> N, where N = { 1 . . . . . I E I } is the set of players. Further, we pick a vertex v 0 ~ V which is called the post office of G. We now define an S-tour of a coalition S as a closed walk that starts in the post office v 0 and visits each edge that corresponds to a player of S at least once. Formally, we have the following definition.

Definition 1. Let G = (V, E, i, v 0) be a connected undirected graph in which v 0 ~ V is the post office. Then an S-tour is a closed walk v o, e l, v I . . . . . v k - l , ek, Vo in G such that S c {g(ej) l j ~ {1 . . . . . k}}.

The set of S-tours of a coalition is denoted by D(S) . To the edges of the graph G we assign deliver costs d: E ~ [0, oc) and travel costs t : E--~ [0, ~). A

postman who has to deliver the mail to a coalition S has to pick an S-tour v 0, el, o l . . . . . ok- l, ek, Vo ~ D ( S ) .

Each time the postman visits an edge the travel costs of this edge is charged. In case the postman makes a delivery in a street the deliver costs of this street is also charged. Formally, the costs of an S-tour v 0, e~, v~ , . . . ,v k_~, e k, v 0 are equal to

k

Cs(vo , el . . . . . vk -1 , ek, V o ) = E t ( e j ) + E d ( e ) . (1) j= 1 e g g I(S)

H. Hamers / European Journal of.Operational Research 99 (1997) 445-458 447

We assume that in any S-tour each street of S pays its own specific deliver costs and also at least once his specific travel costs. Hence, each player of S has individual fixed costs. The sum of these fixed costs of the members of S is equal to Ee~ g-l(s)(t(e) + d(e)) and are called the separable costs of an S-tour. Note that the separable costs are independent of the chosen S-tour. We will call the remaining costs of an S-tour the non-separable costs of an S-tour. Consequently, we can rewrite expression (1) as the sum of the separable costs and the non-separable costs, i.e.

1[;#, ] C s ( U o ' el . . . . . Vk 1' ek, Vo) = £ ( t ( e ) + d ( e ) ) + t (ej) - Y'. t (e) . e ~ g - l ( s ) e eg - l (S )

(2)

Since each coalition S wants to be delivered as cheap as possible, it will choose an S-tour in D(S) that minimizes (2). Since the separable costs are independent of the chosen S-tour coalition S will choose an S-tour in D(S) that minimizes

k

TS( VO, e I . . . . . Vk_l, e k, Vo) = ~_, t (ej) - ~_, t (e ) . (3) j= 1 e~g 1(S)

We call such an S-tour a minimal S-tour. In the following a delivery problem is denoted by F = (N, (V, E, i, v0), t, g). Here, N is the set of players,

(V, E, i, v 0) is a connected graph in which v 0 represents the post office, t : E ~ [0, ~) assigns the travel costs to the edges and g gives the one-to-one correspondence between the edges and the players. Note that the function d : E -~ [0, ~) which assigns the delivery costs to the edges is omitted in the description of F since it does not affect the choice of a minimal tour (cf. (3)).

Before delivery games are formally introduced we recall some well known facts concerning cooperative games.

A cooperative cost game is a pair (N, c) where N is a finite set of players and c is a mapping c: 2 N ~ with c(0) = 0 and 2 N is the collection of all subsets of N.

A game (N, c) is called concave if for all coalitions S, T ~ 2 N and all j ~ N with S c T c N \ { j } it holds that

c ( T U {j}) - c(T) <~ c(S U {j}) - c(S) .

Equivalently, a game (N, v) is called concave if and only if for all coalitions S, T ~ 2 N it holds that

(4)

c(SU r) + c(S• r) + c(r). (5)

Hence, a game (N, c) is concave if and only if the map c : 2 N o • is sub-modular. A core element x = (xi)i~ i ~ AN is such that no coalition has an incentive to split of, i.e.

~ _ , x i = c ( N ) and Y'~xi<~c(S ) f o r a l l S ~ 2 u. i~N i~S

The core Core(c) consists of all core elements. A game is called balanced if its core is non-empty. The result of Shapley (1971) yields that concave games are balanced.

By defining the cost value of a coalition S as the non-separable costs of a minimal S-tour, we obtain a cooperative game corresponding to a delivery problem. This cooperative game is called the delivery game and is formally defined in the following way.

448 H. Hamers / European Journal o f Operational Research 99 (1997) 4 4 5 - 4 5 8

Vl e ~ l e~, 2 J

Vo v2 e 4 ~ 6 ~

v5 I/ 7'a V3 e 3 %

V4 Fig. 1.

VO

:i es, 1

V3

Fig. 2.

Definition 2. The delivery game (N, c) corresponding to a delivery problem F = (N, (V, E, i, v0), t, g) is defined for all S c N by

c ( S ) = min t ( e i ) - ~_, t ( e ) . (6) vo,e I . . . . . ek,vo ~ D( S) e E g - l( S)

The first example shows that in general delivery games are not balanced.

Example 1. Consider the delivery problem described in Fig. 1 in which player j is assigned to the edge ej, for all j ~ { 1 . . . . . 5}. The notation e j, k in Fig. 1 means that edge ej has travel costs k.

Then c ( N ) = 1 since v o, e,, v l, e 2, v 2, e3, 03, e4, Vo, es, v2, es, v 0 is a minimal N-tour with non-separable costs equal to t (e 5)= 1. For all S ~ A : = { { 1 , 2 , 5 } , { 3 , 4 , 5 } , { 1 , 2 , 3 , 4 } } we have that c ( S ) = O . Suppose x ~ Core(c). Then

0 < 2 = 2 c ( N ) = E ~,,xi<~ Y'~ c ( S ) = 0 . S E A i E S S E A

Contradiction, so no core elementS exists.

Before we can give the following example we recall the notion of a bridge in a graph and provide the definitions of an Euler graph and a bridge-connected Euler graph. An edge b E E is called a bridge of a connected graph G = (V, E, i) if the graph G---(V, E - { b } , ile-{b)) is a disconnected graph. A connected graph G is called an Euler graph if there exists a closed trail which includes every edge of G. Then a connected graph G = (V, E, i) is called a bridge-connected Euler graph if all the components of the graph G = (V, E - B(G) , i I e-8(c)) are Euler graphs. Here, B(G) is the set of bridges of G.

Hamers et al. (1994) showed that delivery games corresponding to bridge-connected Euler graphs are balanced. The next example shows that such delivery games need not be concave.

Example 2. Fig. 2 represents a delivery situation in which the graph is K7, the complete graph with seven vertices. In this figure the edges that have travel costs equal to 100 are omitted. Note that K 7 is an Euler graph and, consequently K 7 is a bridge-connected Euler graph.

Take the following coalitions: T = { g ( e 3 ) , g(e4) , g(e6)}, S = { g ( e 3 ) } and {j} ={g(e2)}. The minimal T-tour is equal to v o, e 6, v 4, e 3, v 5, e 4, v 6, e 5, v o. Hence, c(T) = t(e 5) = 2. The minimal TU { j}-tour is equal to v o, e6, v4, e3, v 5, e 4, v 6, e 9, v 2, e 2, v 1, e 1, v o. Hence, c ( T U {j}) = t (e l) + t(e 9) = 2. The minimal S-tour is equal to Vo, e6, v 4, e 3, v 5, e 7, v o. Hence, c ( S ) = t(e 6) + t (e 7) = 6. The minimal S U {j}-tour is equal to

H. Hamers / European Journal of Operational Research 99 (1997) 445-458 449

V0' e6, /34, e3, /35, e8, vz, e2, /31, el, /30" Hence, c(StA {j}) = t(e6) + t(e8) + t (ei) = 5. This implies that the delivery game is not concave, since c(T tA {j}) - c(T) = 0 > - 1 = c(S U {j}) - c(S).

The following two examples consider delivery games that arise from bridge-connected cyclic graphs. Before these examples are presented we give the definition of a bridge-connected cyclic graph and show that it can be determined in polynomial time whether an arbitrary connected graph is a bridge-connected cyclic graph or not. A connected graph G = (V, E, i) is called a bridge-connected cyclic graph if all the components of the graph

= (V, E - B ( G ) , ilE_8(G)) are cycles or single vertices. A cycle is a circuit or the union of edge-disjoint circuits. This implies that each edge is a bridge or contained in exactly one circuit. From this we deduce that if an edge e ~ E , with i ( e ) = {v, v*}, from G is removed that the graph G ' = (V, E - { e } , ilE_{e )) is disconnected in case e ~ B(G). In the other case, i.e. e ~ B(G) , there is a unique path in G' between the vertices v and v *. The uniqueness of this path implies that each edge that is contained in this path is a bridge in G'. This observation is used in the proof of Proposition 1 which shows that the determination whether or not a graph is a bridge-connected cyclic graph is polynomial in time.

Proposition 1. The complexity o f determining whether or not a graph with n edges is a bridge-connected cyclic graph is o f ~(n3).

Proof. Let G = (V, E, i), with I El = n, be a bridge-connected cyclic graph. The first step will determine for each edge of G whether or not it is a bridge of G. This can be done in ~ '(n) according to the reachability problem discussed in Papadimitriou (1994). The second step takes all non-bridges of G. These are determined in the first step. For each edge e ~ E - B ( G ) , we can construct a path, say v, el, Vl, e 2 . . . . . vk-l , ek, v*, in G' = (V, E - {e}, iE_{e)), where o and v * are the end points of e in G. This path construction is LY(n) for each edge in E - B(G). Now, we have to check whether or not each edge ej, j = {1 . . . . k}, in this path is a bridge in G'. Here, we again face a reachability problem for each edge in such a path that is of the Order •(n). Summarizing, the second step is of order ~(n3). Hence, we can conclude that the algorithm that combines the first and second step is also d~(n3). []

The following example illustrates a delivery problem and the corresponding delivery game in which the underlying graph is a circuit. Note that a circuit is a special bridge-connected cyclic graph.

Example 3. Let us consider the delivery situation that is described in Fig. 3 in which player j is assigned to edge e j, for all j ~ { 1, 2, 3}.

~ ) 1

v~3~3 ~ e2,4

~ 3

" " . . . . . . . . . . . . " . . . . . . i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

" /)1 " 0 5 '08

" 0 3 "07, ' 0 9

G1 G2 G3 G4 i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : . . . . . . . . . . . . . . . .

Fig. 3.

Fig. 4.

450

Table 1

H. Hamers / European Journal of Operational Research 99 (1997) 445-458

S {1} {2} {3} {I, 2} {1, 3} {2, 3} {1, 2, 3} c(S) 2 5 3 3 4 2 0

The corresponding delivery game is presented in Table 1. We will explain the outcome for coalition {1, 3}. The only two reasonable tours starting from v 0 that have to

be considered are v o, e 1, vl, e 2, v2, e3, v 0 and v o, e~, v L, e~, v o, e3, u2, e3, v o. The minimal tour will be the first one. Consequently, the worth of {1, 3} is c({1, 3}) = min{t(e2), t(e 1) + t(e3)} = min{4, 2 + 3} = 4.

Let F = (N, (G, v0), t, g) be a delivery situation in which G is the circuit given by ~ = Vo, el, vl, e2 . . . . . Om-1, e,,, v o. Then a coalition S c N is called connected w.r.t, the circuit ~ and v 0 if S--- N or S = g({e l, e 2 . . . . . e m } \ { e k , e~+l . . . . . et-1, et}) with 1 ~< k < l ~< m. For any connected coalition S we can calculate, in a similar way as we did for coalition {1, 3} in Example 3, its worth in the corresponding delivery game. More precisely, the worth of a connected coalition S c N is given by

c ( S ) = min ( ~ t ( e ) , Y'. t ( e ) } . (7) eC g - l(S) e~ g - l(S c)

The notion of connected coalitions can be extended to bridge-connected cyclic graphs. Let G = (V, E, i) be a bridge-connected cyclic graph and v 0 ~ V, then S is called connected w.r.t. G and v 0 if for each k ~ S there exists a path that contains v 0 and g - ~ ( k ) such that for each edge e in that path it holds that g(e) ~ S. Example 4 shows that connected coalitions can be calculated by decomposing a bridge-connected cyclic graph in bridges and circuits. More specifically it is shown that a minimum delivery tour of a connected coalition can be obtained by determining the minimal delivery tours of the corresponding subcoalitions in the decomposed parts. Finally, this example will illustrate that in general a minimal tour of a coalition in a bridge-connected cyclic graph cannot be constructed from the decomposed bridge-connected cyclic graph.

Example 4. Consider the delivery situation F described in Fig. 4 in which player j is assigned to edge e j, for all j ~ {1 . . . . . 12}.

A connected coalition is S = {1, 2, 4, 5, 6, 7, 8, 10}. Then the minimal S-tour is v 0, e l, v 1, e 2, v 2, e 5, v 4, e6, v5, e7, v6, elo , v8, elo , v6, es, v7, e9, v4, e5, v2, e3, v3, e4, v o. The corresponding non-separable costs are equal to 7, which implies that for the corresponding delivery game (N, c) it holds that c({1, 2, 4, 5, 6, 7, 8, 10}) = 7.

Now, decompose G in bridges and circuits. This leads to the graphs G l, G 2, G 3 and G 4 as shown in Fig. 4. Let N 1 = {1, 2, 3, 4}, N 2 = {5}, N 3 = {6, 7, 8, 9} and N 4 = {10, 11, 12}. Consider now for j ~ {1, 2, 3, 4} the delivery problems F~ = (Nj, (Gj, vj*), t[g-,(Nj) , glg-l(N)) with v~* = v o, v 2 = v z, v3* = v 4 and v4* = V 6. Let

• , )

(Nj, cj) be the dehvery game corresponding to Fj. Then for all j ~ {1 . . . . . 4} the minimal tour of S A g(Ej ) in Fj is equivalent to the minimal S-tour restricted to Gj with Ej the set of edges of Gj. Hence, a minimal delivery tour of a connected coalition can be constructed by combining the minimal delivery tours of the corresponding subcoalitions. This implies that c(S) = E4= icj(S • g( Ej)).

Now, consider the disconnected coalition T = {t, 4, 5}, then the minimal T-tour is v 0, e 1, v l, e2, u2, es, u4, es, v2, e3, v3, e4, v o. The costs of this delivery tour in G 1 and G 2 are equal to 3 and 1, respectively. Hence, c({1, 4, 5}) = 4. Now, the minimal delivery tour of T N g ( E l) = {1, 4} is equal to v 0, e 1, v 1, e l, v 0, e 4, v 3, e4, v 0 with q({1, 4}) = 2. Obviously, c2({5}) = 1. ttence, for a disconnected coalition we have c(T) > ~ = l c(T A g(Ej)) . This implies that for a disconnected coalition the minimal delivery tour cannot be obtained by combining the minimal delivery tours of the corresponding subcoalitions.


Let (G, v 0) be a bridge-connected cyclic graph and let (G l, vl* ) . . . . . (Gk, vk* ) be the decomposition of G in circuits and bridges, with vf the vertex in Gj that has the shortest distance in G to v 0. Let F = (N, (G, v0), t, g) and Fj = (N~, (Gj, vf ), t I E, gl e ) be the corresponding delivery problems with Gj = (Vj, Ej, i I ej)- Let (N, c) be the delivery game corres'pond(ng to F and (Nj., cj) be the delivery game corresponding to F~, which we will call a component delivery game of (N, c). Then we can generalize the result of Example 4 to F and obtain the following (in)equalities:

k

c( S) = ~_, c i ( S fqg (E . i ) ) if S is connected, (8) j=l

k

c( S) >~ ~, c i ( S N g ( E j ) ) if S is connected. (9) j=l

The following proposition shows that we can construct core elements for a delivery game by taking core elements of its component delivery games.

Proposition 2. Let ( N, c) be a delivery game and ( Nj, Cj), for j = {1 . . . . . k}, all its corresponding component delivery games. I f xj ~ Core(cj) for all j ~ {1 . . . . . k}, then x = ( x l, x 2 . . . . . xk ) E Core(c).

Proof. Let S c N. Then F~p ~ s( x)p = E~= 1Ep ~ s n Nj( Xj)p ~< E~=, cj(S ~ Nj) ~< c(S). The first inequality follows from the definition of the core and the second inequality is a consequence of (8) or (9). Since N is a connected set it follows from (8) and from the definition of the core that both inequalities become equalities. Hence x ~ Core(c). []

Now, we will formulate the main result.

Theorem 1. A delivery game is concave if the underlying connected graph is a bridge-connected cyclic graph.

The formal proof is presented in Section 3; here we will provide a short description of the construction of this proof. First we show that whenever the underlying graph is a circuit the corresponding delivery game is concave (cf. Theorem 2). Second, it is proved that a delivery game is concave if the underlying graph is the union of a circuit and a bridge that is not connected to the post office (cf. Theorem 3). Subsequently bridge-connected circuit graphs are introduced. These are the graphs that after removing the bridges only have circuits or single vertices as components. Hence, here the circuits are vertex disjoint. Using Theorem 2 and Theorem 3 we are able to prove that delivery games are concave if the underlying graph is a bridge-connected circuit graph (cf. Theorem 4). Finally, an extension of a bridge-connected circuit graph to a bridge-connected cyclic graph is described which provides the proof of Theorem 1 (cf. Lemma 1).

From Theorem 1, Shapley (1971) and Ishiichi (1981) it follows that the extreme points of the core of delivery games that arise from bridge-connected cyclic graphs are the marginal vectors. Let (N, c) be a delivery game and let H ( N ) be the set of all permutations if N = {1 . . . . . n}. Then the k-th coordinate of the marginal vector m~(c), ~-E H ( N ) , is defined by

m~[(c) = c ( { j ~ N [ ~- ( j ) < ~r(k)}) - c ( { j E N [ ~- ( j ) < ~- (k)}) .

For a delivery situation F = (N, (G, v0), t, g) that arises from a bridge-connected cyclic graph we consider connected permutations. A permutation is called connected w.r.t. G if for each k E N the set {jl ~-(j) ~< 7r(k)} is a connected set. For the marginals that correspond to connected permutations we are able to give an

452 H. Hamers / European Journal o f Operational Research 99 (1997) 445-458

expression. First, we consider a delivery game (N, c) that arises from F = (N, (G, Vo), t, g), where G is the circuit ~ = v 0, e l, vj . . . . . era, v 0. Then from (7) we have for each k ~ N and each connected ~" ~ H ( N ) that

m ~ ( c ) = min ( ~ t (e ) , ~ t ( e ) } eEg-t({j~NlTr(j)<-N 'n-(k)}) e g g - I ( { j ~ N I ~r(j)~ 7r(k)} c)

- m i n ( ~ t (e ) , ~ t ( e ) } . (10) ~' e~ g - l ( { j ~ N [ Tr(j)< "n'(k)}) e g g - l ( { j E N l T r ( j ) < "n'(k)} c)

Note that in a circuit there are 2 " - 1 different marginals that correspond to a connected permutation.

E x a m p l e 5. Take the delivery situation and delivery game of Example 3. Then the connected marginals are m~ ' ( c ) : (2, l, - 3 ) , m=2(c) = (2, - 4 , 2), m ~ ' 3 ( c ) = ( 1 , - 4 , 3), m ~ 4 ( c ) = ( - 2 , - 1 , 3) with 7r I = ( 1 , 2, 3), "/7" 2 = (1, 3, 2), "/7" 3 = (3, 1, 2) and ~'4 = (3, 2, 1), respectively.

Consider now a delivery situation F = (N, (G, v0), t, g) in which the graph is a bridge-connected cyclic graph. Let (N, c) be the corresponding delivery game. Let (G I, v I ) . . . . . (G k, vk* ) be the decomposition of G in circuits and bridges, with vj.* the vertex in Gj that has the shortest distance in G to v 0 and Gj = (Vj, Ej, i I e~) for all j ~ {1 . . . . . k}. Let Fj = (Nj, (Gj, v 7 ), t I e~, gl e ) and let (N, cj) be the corresponding delivery game. Then for each k ~ N and each connected ~ ~ H ( A r) it {ollows that

m~(c) =rn~,~'J(cj) if g - l ( k ) ~Ej . (11)

Note that Ej can only be a circuit or a bridge. If Ej. = {b} cB(G) , then m'[tu~(cj) = t(b), otherwise we can use expression (11) to calculate m*[l~j(cj) since E~ is a circuit.

Finally, we show how the r-value can be calculated for a delivery game (N, c) that arises from a bridge-connected cyclic graph. Since (N, c) is concave we have for all k ~ N that

~-k(c) = Ac({k}) + (1 - A ) ( c ( N ) - c ( N \ { k } ) .

with h such that Y"k ~ N'q(c) = c(N). Let i(g-I({k})) = {vj*, v 2 }. Then the worth of c({k}) can be determined by adding the travel costs of the shortest path from v 0 to v~* and the travel costs of the shortest path from v o to v2*. The worth of c ( N ) - c (N \ {k} ) is a special coordinate of a marginal vector that correspond to some connected permutation. This implies that expression (11) can be used to calculate c(N) - c(N\{k}) .

3. The proof of the concavity result

For the formal proof of the concavity result we introduce some new notations. We assume in this section that the graph G in the delivery problem F = (N, (V, E, i, v0), t, g) is a bridge-connected cyclic graph. Take j ~ N and let e* ~ E be such that g ( e * ) = j. A minimal delivery tour of the coalitions S, T, S U {j}, T U {j}, S n T and S U T will be denoted by ds, dr, d s u (j), dT u (j), ds n r and d s u r, respectively.

Let F = (N, (V, E, i, v0), t, g ) and let U c N. Take d = v 0, e I . . . . . ek, v o ~ D(U). Then the non-separable travel costs of the delivery tour d with respect to U and a subset of edges E ' c E is given by a function f f , E ' :2 N _~ [0, ~) that is defined by

fdr'Z' ( U ) : • t ( e j ) - E t (e ) . (12) j: ejE E' e~ g- l ( U ) n E ,

Note that in case d is a minimal delivery tour of U in F , we have for the corresponding delivery game (N, c) that c(U)=fdr'E(U) (cf. (6)).

Let G = (V, E, i) be a circuit ~ , say ~ ' = v 0, e l, v~ . . . . . v m_ ~, e,,, v 0. For notational convenience, put

L W ~ ( ~ ") = v 0 , el , p I . . . . . Up_l, ep, Up, ep, Up_ 1 . . . . . Vl, e 1, v 0 if 1 <-Np~m- 1,

R W ~ ( ~ ) = u 0 , em, urn_ 1 . . . . . Up, e p , r e _ 1, e e, vp . . . . . UI, e l , v 0 i f 2 ~ < p ~ < m .


Further, put LWo°(g ~) = 0 (no left branch in the circuit) as well as RW~ '+ ~(g~) = 0 (no right branch in the circuit). Now, we can formulate the first theorem of this section.

Theo rem 2. A delivery game is concave if the underlying connected graph is a circuit.

Proof . Let (N, c) be the delivery game corresponding to a delivery problem F = (N, (G, v0), t, g ) of which the graph G = (V, E, i) is a circuit ~ , say ~ = v0, el, v 1 . . . . . Vm-l, era, Vo" We investigate the concavity condition (5).

First suppose that d s o r d r is the tour v0, e l, vl, e 2 . . . . . Vrn-1, era, v0. W.l.o.g. we assume that d s = v 0, e l, v 1, e 2 . . . . . v,,_ ~, era, v 0. Then d s is also a delivery tour of S O T, and d T is a delivery tour of S f'l T, since S (1 T c T. Then

c ( S ) + c(T) = f ~ ' E ( S ) +f~'e(T) = f ~ ' S ( S U T) +ff,E(s n T)

>~f,~sur(Sr,s U r ) + f ~ r ( s v ~ r ) = c ( S U r ) + c ( S n r ) .

Now, we suppose that d s=LWd,(~)URWd2(~) with 0~<s l + l < s 2 ~ < m + l , and d r = L W d ~ ( ~ ' ) t0 RWd2(~)wi th0~<t~+ l<t z ~ < m + l .

I f t I < s 2 and t 2 > s l, then a delivery tour of S V~ T is

d s n T = LW0min(s~'q)(~') U Rw0max(s>'2)(~)

and a delivery tour of S U T is

Then

c( S) + c(T) =fdFs'E( S) + f ~ ' E ( r )

S l l 1

= E 2 t ( e j ) + ~ 2 t ( e j ) - - Et(ej) + E2t(ej) + ~ 2 t ( e j ) - - E t(ej) j= l j=s 2 j ~ S j= 1 j=t z j ~ T

max(spq)

= E 2 t ( e j ) + ~ 2 t ( e j ) - E t(ej) j= 1 j= min(s2,t2) j~ SO T

min(sl,q)

+ ~_~ 2 t (e j )+ ~ 2t (e j ) - - ~_. t(ej) j= 1 j= max( sz,t2) j~ SO T

F,E =fj;~T(SUT ) r,e T) c ( S L J T ) + c ( S f ~ T ) .

If t 1 >~ s 2 or t2~sl , then take as delivery tour for the coalitions S U T and S N T the tour

v o, e 1, vp . . . ,v~_ l, e~, v o, i.e. d s v r = V o , e l , v 1 . . . . . U r n _ l , e,n, v 0 and d s n r = v0, el, Vl,... ,V~_l, era, v o. Then

c( S) + c( r ) = yaFs'E( S) + fdr'e( T)

SI l 1

= E 2 t ( e j ) + ~-a 2t(ej) - Y'~ t ( e j ) + E 2 t ( e i ) + Y', 2 t ( e j ) - E t ( e j ) j= 1 j=s 2 j ~ S j= l j=t 2 j ~ T

>~ ~ t(ej)-- ~., t (e j )+ ~ t (ej)-- Y'. t(ej) j = l j ~ S O T j = l jESf~T

= f j ~ o r ( S U T) c ( S U T ) + c ( S N T ) .

454 H. Hamers / European Journal of Operational Research 99 (1997) 445-458

VO el Vp V* Vo el Vp

- Vp~+l - Vfq_ 1 13"*

G

Fig. 5. The transformation from G to G.

The following theorem shows that a delivery game corresponding to a delivery problem that arises from a graph existing of one circuit and one bridge that is not connected to the post office is concave.

T h e o r e m 3. A delivery game is concave if the underlying connected graph is the union of a circuit and a bridge that is not connected to the post office.

Proof . Let (N, c) be the delivery game corresponding to a delivery problem / ' = (N, (G, v0), t, g ) of which the graph G = (V, E, i) is the union of a circuit v o, v l, e:, v 2 . . . . . vp, ep+ 1, vp+! . . . . . Vm-1, era, Vo and a bridge era+ 1 that is not connected to the post office. Write i(em+ 1) = {Vp, v *}, where Vp ~ V \ { v 0} is the vertex incident with the bridge e,,+ ~ and the circuit, and v * =~ v k for all 0 ~< k ~ m - 1. Define a new edge ~p+ 1 by means of its end points re+ 1 and v *. Let G = (V, E, i) be the circuit graph which arises from G by replacing the edge ev+ l by the edge ~p+ 1, that is E = (E - {ep+ l}) tO {~p+ 1}, i(~p+ 1) = {vp, v*} and i(e) = i(e) for all e ~ E\{ep+ l} (see Fig. 5).

Let the corresponding delivery problem P = (N, (G, v0), t, ~ ) be given by

~ ( ~ , , + ~ ) = t ( e p + l ) , t ( e m + l ) = 0 , ~ ( e ) = t ( e ) fo ra l l e ~ ; Z \ { ~ p + ~ , e ~ + l } ,

~,( '~p+l)=g(ep+l), ~ ( e ) = g ( e ) fo ra l l e ~ E X { e p + l } .

Let (N, ~) be the delivery game corresponding to the delivery problem F . By construction, we obtain that for all S ~ 2 N,

c (S ) if g(em+l) ~ S ,

e ( S ) = c ( S ) - t ( e m + x ) if g(e,,+l ) ~ S .

It follows that for all j E N and all S c N \ { j } ,

c(Sto {j} - c(S) if j4= g(e~+, ) ,

~ ( S t o { j } ) - ~ ( S ) = c ( S t O { j } ) _ c ( S ) _ t ( e m + l ) if j = g ( e m + l ) "

As a result, the game (N, c) is concave iff the game (N, ~) is concave. Since the graph G is a circuit, the corresponding game (N, ~) is concave by Theorem 2 and therefore, the game (N, c) is concave. This completes the proof of Theorem 3. []

A graph G = (V, E, i) is called a bridge-connected circuit graph if all the components of the graph (V, E - B ( G ) , i le_8(c)) are circuits or single vertices. An edge e ~ E in a graph G = (V, E, i) is called a follower of a bridge b with respect to v 0 ~ V if and only if each path v 0, e I . . . . . e k, v k that contains e also contains b. The set of followers of b will be denoted by Fb(G, Vo). Note that b ~ Fb(G, Vo).

T h e o r e m 4. A delivery game is concave if the underlying connected graph is a bridge-connected circuit graph.


~0 "%.

Vp_ 1 ep vp v p _ 1 ep

..-_ - .~

+1 ..."

~N'~ V p-I-1 v O 0"""

~ J t - "" 2 ",.

Vp+ 3 ep+ 3 Vp+ 2

Vp ap ~p

T ~ P + l

! e p + 2

vp.Sr3-ffp.+ 3 v*--~ a . + 2 Vp+2 p-i'-z r

m

G* G Fig. 6. ~he extension from G * to G with V * (T) = {Vp, Vp+2}.

Proof . Let (N, c) be a delivery game corresponding to a delivery problem F = (N, (G, v0), t, g) of which the graph G = (V, E, i) is a bridge-connected circuit graph. Let j ~ N and S, T ~ 2 N be such that S c T c N \ { j } . We investigate the concavity condition (4)• Put g - l ( { j } ) = e*. In case the e d g e e* is not a follower of some bridge b ~ B(G) (that is, e* ff Fb(G, %) for some b ~ B(G)), then the minimal delivery tour of coalitions S and S U {j} have to visit the same edges in Fo(G, %)C~g- l (S) and so, the determination of the difference c(S U { j } ) - c(S) is independent of any set Fo(G, v o) associated with a bridge b ~ B(G) satisfying e* Fo(G, %). Consequently, we consider, without loss of generality, the restricted graph G* = (V *, E* , i l e , ) with edge set E* = E - U o ~ 8(a),e* ~ vb(a, Vo)" Now, we have to distinguish two cases.

Case 1. Suppose that there exists no bridge b ~ B(G) such that e* E Fb(G , Vo). Under these circumstances, the restricted graph G* is a circuit, say ~" = v 0, el, v 1, e 2 . . . . . v,~_ 1, era, v0. Let V * (T) c V * be the set of vertices of G that is incident with a bridge b E B(G) that has the property that Fo(G, v o) C~ T v~ fJ. Formally,

V * ( T ) = { v j ~ V * I t he reex i s t s a b E B ( G ) s.t. v j e i ( b ) and Vb(G, Vo) Nr4=¢} .

Note that as well d r as d r u ( j } has to visit all vertices of V*(T).. Take g * ( T ) = { v j i , vj~ . . . . .,vj~} with 1 ~ Jl < J2 < " " " < Jk ~< m - 1. Define the set of new vertices V' = { v jl , vj. . . . . . vjk }, with V' N V = ~ . For each p ~ { 1,. k} we define a new edge ~ + 1 by means of the end points v.* and v ÷ 1 (with the convention

• • ~ Jp . * JP Jp v m = %) and a new edge ap by means o f the_end points Vp and Vp. The set of new edges is denoted by E' = {ai~, a2~ . . . . . aj~, ~J~2~' . . . . ~j~}. Let G = (V, E, i) be the circuit graph which arises from G * by replacing the edge e j + l by the edge ~j+~, for all p ~ { 1 . . . . . k}, and adding the edge aj for all p E { 1 . . . . . k},

- - ! - - p

more formafly, V = V U V E = E U E - { e . . . . . e,} and t ( a , ) * ~ - ' " = { v j , vj }, for all p ~ { 1 . . . . . k}, i(?j + 1) = J ] J k d p" p p

{Ui , Ui + 1} ( W l t h the convention v,, = v 0) for all p {1 . . . . . ~}, and i(e) = i(e) for all E - {eye, . . . . eye} (see F@ 6)'. ~

Let the corresponding delivery problem F = (N, (G, v0), ), ~) be given by }(aj ) = 0 and )(~, ) = t(ej ) for - - - ! - - p s p p

all p ~ {1 . . . . . k}; t(e) = t(e) for all e ~ E - E ; {~(a~ )}~= 1 = N \ N ; g(~2 ) = g(e, ) for all p ~ {1 . . . . . k}; - - t J p • p • a p _ _ and ~(e) = g(e) for all e ~ E - E . Let (N, ~) be the dehvery game corresponding to the delivery problem F .

By construction, we obtain

c(rU { j } ) - c ( r )=?( ( ru { j } U~kN) n~)-~'((TU~XN)n~). (13)

Since S c T we have that V* (S) c V * (T), where V * (S) is the set of vertices of ~" that is incident with a bridge b and F~(G, v o) C~ S 4= ¢. This implies that there exists a D c N \ N such that

c(SU { j } ) - c(S) = ?((SU { j } UD) n i l ) - E((Su D) n i l ) . (14)


In view of (13) and (14) and the concavity of the delivery game (N, 5) by Theorem 2, it follows immediately that condition (4) holds.

Case 2. Suppose there exists a bridge b E B(G) such that e* ~ Fb(G, Vo). Without loss of generality, we may choose the bridge b ~ B(G) to be closest to the edge e*, that is e* ~ Fb,(G, v o) for all b' ~ Fb(G, v o) n (B(G) - {b}). Now we distinguish two subcases.

Subcase 1. Suppose that Fb(G, v o) N g - t (T ) -~ i~, i.e. the minimal delivery tour of coalition T has to visit at least one edge in Fb(G, Vo). If it happens that e* = b, then c(TU {j}) - c(r) = - t (e*)<~ c(S U { j} ) - c(S). In the remainder of the subcase, we may assume that e* 4= b. Since G is a bridge-connected circuit graph and b is the bridge closest to the edge e* E Fb(G , Vo) , the restricted graph G* = (V *, E* , i I e* ) with edge set

E* = Fb(G, Vo) - {b} - O Fb'(G, Vo) b' ~ Fb(G , v o ) n ( B(G)- {b})

is a circuit ~ , say ~ ' = ~0, el, Vl, ~2 . . . . . Din- 1, 8,,, ~0" Let ~o be the vertex incident with the bridge b and the circuit W. Take now V* (T) \{O 0} and in the same way as in case one we let arise G from G * and construct the delivery problem F = (N, (G, 30), t, ~) with (N, 5) the corresponding delivery game. Because of the assumption Fb(G, v o) n g - l ( T ) 4: 13, the minimal delivery tours of T and T U ( j ) have to visit the bridge b and moreover, both tours have to visit the same edges in E - Fb(G, ~'0) since e* E Fb(G , Vo). From this we deduce that

c ( T U { j } ) - c ( T ) = ~ ( ( T U { j } U i ~ \ N ) A N ) - ~ ( ( T U N \ N ) n N )

as well as

c(SU (j}) - c(S) >1 ?((SU {j} UD) n~V)- 7~((SUD) nN), with D c , ~ \ N . In view of the latter two (in)equalities and the concavity of the delivery game (N, 5) by Theorem 2, it follows immediately that condition (4) holds.

Subcase 2. Suppose that Fb(G , VO) n g - I (T) = ~ , that is the minimal delivery tour of coalition T does not v i s i t Fb(G , Vo). Then there exists a unique -b~B(G) such that F3(G, v o ) n g - l ( T ) = O as well as ( E - F-b(G, Vo)) n g - l (T ) 4= ~J. If (E - F3(G, Vo)) n B(G) v~ ~, let b* ~ (E - F-b(G, Vo)) n B(G) 4= ¢ be the bridge closest to bridge b (see Fig. 7).

Define the edge set E* by

[ (Fo.(G, vo)-{b })-(F-~(G, vo)-{?}), otherwise

The graph G* with edge set E * is the union of a circuit c~ and the bridge b. Let 9 o ~ V be the vertex incident with the bridge b* and the circuit c~ (in case the bridge b* does not exist, choose 90 = Vo). Let Vp be the vertex incident with the bridge b and the circuit. Define V * (T)~{~ o} for the circuit ~" and construct a graph G' from the circuit ~' in the same way as in case one. The graph G arises from G' by connecting the bridge b to

I) o v

Fig. 7.


G' to the vertex vp. Consider now the delivery problem (_P= ( N U (g(b)}, (G, Vo), it, ~), with ( N U (~'(b)), F) the corresponding delivery game. By construction, we obtain that

c ( T U { j } ) - c (T )

= F((TU N \ N U {g(/))}) AN) - { ( (TU )V\N)AJV) + farr'2~G'"°'-{b'(T U { j } )

as well as

c ( S V {j}) - o(S)

Notice that

fdr.F~(a,o0) - {;}/T U { j} ) = f,['F~(G'~°)- {?}( S U { j} ) TUIj} ~" ~ S U { j ] \ '

since the minimal delivery tours of coalitions S U {j} and T U {j} have to visit the same edges in F-b(G, v o) (namely, the edge e * which is visited by choosing a shortest path from b to e* ). In view of the latter three (in)equalities and the concavity of the delivery game (N U (g(b)}, ~), by Theorem 3, it follows immediately that condition (4) holds.

This completes the proof of Theorem 4. []

Consider a bridge-connected cyclic graph G. Recall that a bridge-connected cyclic graph is a graph that after removing the bridges only has cycles or single vertices as components. A cycle is a circuit or the union of edge-disjoint circuits. We construct now from G a bridge-connected circuit graph in the following way. Each vertex in G that is incident with at least three circuits will be replaced by a circuit in which the number of vertices is equal to the number of circuits and bridges incident to that vertex. This step is illustrated in Fig. 8.

After this step we have in this new graph that each vertex is in the intersection of at most two circuits. Now replace each vertex of this new graph that is incident with exactly two circuits by a bridge. This step is illustrated in Fig. 9.

Let G = (V, ff~, i) be the bridge-connected circuit graph that arises from G = (V, E, i) after applying the described procedure. Let iv'= (N, (G, v0), t, g) be a delivery problem corresponding to G. Then the delivery problem F = (M, (G, v0), }, ~,) that arises from G is called the extended delivery problem of F if

t ( e ) = t ( e ) forall e ~ E , } ( e ) = 0 foral l e ~ f f , - E ,

and

~ , ( e ) = g ( e ) foral l e ~ E .

Note that N c M. lint (N, c) be the delivery game corresponding to F and let (M, 5) be the delivery game corresponding to F. Then the following lemma shows that (N, c) is a subgame of (M, ~).

"".. ..

Fig. 8. A vertex replaced by a circuit.

Fig. 9. A vertex replaced by a bridge.


L e m m a 1. Let F be the extended delivery problem o f a delivery problem F in which the underlying graph is a bridge-connected cyclic graph. Let (N , c) be the delivery game corresponding to F and let (M, -d) be the delivery game corresponding to F . Then c( S) = ~( S) f o r all S c N .

P r o o f . I f in F a ve r tex that is the intersect ion of at least two ver t ices is vis i ted in a min ima l del ivery tour o f a

coal i t ion S to vis i t another circuit , then the corresponding min imal de l ivery tour of S in P has to visi t the new

bridges and some parts o f a new cycle be tween these two circuits. S ince these new edges have travel costs equal

to zero we have that the costs o f a min imal del ivery tour of S in /" co inc ide with the costs o f the corresponding

de l ivery tour o f S in _~. []

Proof of T h e o r e m 1. Fo l lows immedia te ly f rom Theo rem 4 and L e m m a 1. []

Acknowledgements

The author wishes to thank Je roen Suijs and two anonymous referees for their va luable comments and

suggest ions.

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On the concavity of delivery games