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# Cost allocation in vertex weighted Steiner tree games

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### Cost allocation in vertex weighted Steiner tree games

1. 1. Vertex Weighted Steiner Tree Games Jeroen Kuipers∗ Stef H. Tijs† Vincent Feltkamp‡ June 2004 Abstract We introduce a variant of the minimum cost spanning tree game in which players receive a reward if they connect to a central supplier. The problem is to distribute the total proﬁt (deﬁned as the sum of the rewards minus the construction costs of the links to the supplier) among the players. We introduce an allocation rule for this type of game and we prove that the rule yields a core element if and only if the outcome is eﬃcient. We investigate under which conditions this allocation rule is indeed eﬃcient.1 IntroductionSuppose that a number of customers wish to be connected to a central supplier.The customers have to pay for the construction costs of the network yet tobuild. On the other hand, a customer receives a reward if he is connected tothe central supplier. The total proﬁt of the customers is deﬁned as the sumover the rewards of the connected customers minus the construction costs of thenetwork. The customers are interested in maximizing their total proﬁt. Thecustomers may decide that some of them will not be connected to the supplierif this increases the proﬁt.Formally, let N = {1, 2, . . . , n} denote the set of customers and let 0 denotethe central supplier. The cost of establishing a link between i ∈ N ∪ {0} andj ∈ N ∪ {0} (i = j) is denoted by dij ≥ 0. The reward of a customer i ∈ N isdenoted by ri ≥ 0. Suppose that the customers decide to establish a set E oflinks. We say that a customer i ∈ N is connected to the supplier if there existsa path from i to 0 in the graph GE = (N ∪ {0}, E). If GE contains a cycle, thenan arbitrary link in this cycle can be deleted, while the connected customers ∗ Department of Mathematics, University of Maastricht, PO Box 616, 6200 MD Maastricht,The Netherlands. † Retired. ‡ School of Management, PO Box 1203, 6201 BE Maastricht, The Netherlands. 1
2. 2. remain connected. This can only increase the total proﬁt. Moreover, if G Ehas a component, which contains more than one customer but which does notcontain the supplier, then all links within this component can be deleted, whileleaving the connected customers connected. Also this can only increase thetotal proﬁt. Hence, in search for an optimal network, we may restrict ourselvesto minimal trees with vertex set S ∪ {0}, where S ⊆ N . Let us denote the costof a minimal tree with vertex set S ∪ {0} by c(S) and denote i∈S ri by r(S).Then the maximal proﬁt that the customers can achieve equals v(N ) = max{r(T ) − c(T ) | T ⊆ N }.The problem of ﬁnding an optimal tree is known as the vertex weighted Steinertree problem, which was ﬁrst treated by Segev (1987).If the customers involved in a vertex weighted Steiner tree problem decideto build the optimal network, they will obtain a proﬁt v(N ). In this paperwe address the problem of allocating the total proﬁt among the customers.In order to deal with this allocation problem, we deﬁne a transferable utilitygame, or simply a game, associated with the problem. Formally, a game is anordered pair (N, v), where N is a ﬁnite set and v is a real valued function onthe subsets of N , assigning 0 to the empty set. The elements of N are calledplayers, and the subsets of N are called coalitions. The value v(S) is called theworth of coalition S. It is interpreted as the proﬁt that coalition S can achievewithout the cooperation of players outside S. Suppose all players decide tocooperate, and thus receive a total proﬁt of v(N ) together. In the negotiationsabout the division of the proﬁt among the players, the members of an arbitrarycoalition S could easily claim that they should receive at least v(S) together,since otherwise S is better of by not cooperating with the other players. Thecore is deﬁned as the set of allocations that satisfy all such possible claims, i.e. Core(v) := {x ∈ RN | i∈N xi = v(N ) and i∈S xi ≥ v(S) for all S ⊆ N }.In case of the vertex weighted Steiner tree problem we assume that a subsetS ⊆ N of customers is allowed to build its own network for connecting itsmembers. In doing so, we assume that these customers are allowed to use the 2
3. 3. vertices of customers in N S. However, if the members of S decide to use sucha vertex in N S, S will not receive the reward of this customer. Hence, v(S) = max{r(S ∩ T ) − c(T ) | T ⊆ N }.We call a game deﬁned in this way a vertex weighted Steiner tree (VWST)game.Core allocations do not necessarily exist for VWST games. Consider the fol-lowing game with N = {a, b, c, A, B, C}. For players indicated with lower-caseletters we have a cost of 3 to establish a direct link to the supplier. Also, sixother links have cost 3, namely {A, b}, {A, c}, {B, a}, {B, c}, {C, a} and {C, b}.All other links have cost 5. Furthermore, the reward of upper-case players is 5and the reward of lower-case players is 0. See ﬁgure 1 in which only the linkswith cost 3 are drawn. Let us suppose that x is a core allocation. Observethat v(N ) = 1, and that also v(N {i}) = 1 for all i ∈ N . It follows thatxi = v(N ) − x(N {i}) ≤ v(N ) − v(N {i}) = 0 for all i ∈ N . Hence x ≤ 0,which contradicts the fact that x also satisﬁes x(N ) = v(N ) = 1. A similarexample was given by van Bokhoven (1994). C b a A c B Figure 1Our game model is closely related to the model of a directed Steiner tree game(see Skorin-Kapov (1990)). For a directed Steiner tree game the worth of acoalition S is determined by a minimum cost Steiner tree directed away from 3
4. 4. the root, spanning all vertices of S. The vertex weighted Steiner tree problemcan be formulated as a directed Steiner tree problem by subtracting the rewardof a vertex from the cost of all ingoing arcs of this vertex. This does not meanthat a VWST game is a directed Steiner tree game, since for a VWST game onehas to change the data of the network each time the worth of another coalitionis computed, while for a directed Steiner tree game only the set of terminalnodes changes.2 An Allocation RuleIn this section we shall describe a procedure for allocating the proﬁt of a VWSTproblem among the customers. The procedure does not necessarily allocate thetotal proﬁt v(N ), but if it does, we have found a core element. We deﬁne theallocation rule in a slightly more general setting. Let (N, c) be a game, and letr ∈ RN with r ≥ 0. We interpret the game (N, c) as a cost game and the vectorr as a reward vector. We deﬁne the savings game (N, v) by v(S) = max{r(S ∩ T ) − c(T ) | T ⊆ N }.We will refer to the game (N, v) as the reward game with respect to r. Observethat the reward game is a VWST game if (N, c) is a minimum cost spanningtree game (see Granot and Huberman (1981)).The allocation rule works as follows. We start with a reward vector r = r 0 . De-note the reward game with respect to r 0 by v 0 . Deﬁne z1 = v 0 (N ) − v 0 (N {1})and reduce the reward of player 1 by the amount z1 . Denote the resultingreward vector by r 1 and denote the reward game with respect to the new re-ward vector by v 1 . Now deﬁne z2 = v 1 (N ) − v 1 (N {2}) and reduce the rewardof player 2 by the amount z2 . Denote the resulting reward vector by r 2 andthe associated reward game by v 2 . Continue this process and ﬁnally allocatezn = v n−1 (N ) − v n−1 (N {n}). Let us denote the i-th unit vector by ei . Thenthe allocation-process can be written down as follows. 4
5. 5. r0 := r; for k := 1 to n do zk := v k−1 (N ) − v k−1 (N {k}); rk := rk−1 − zk ek ; od.In the following we will refer to the vector z as the z-alloccation.Lemma 1 The reward games v 0 , . . . , v n satisfy the following properties. a) v k (S) ≥ v k−1 (S{k}) for all S ⊆ N with k ∈ S. b) v k (N ) = v k−1 (N {k}) for all k ∈ N . c) v k (N ) = v k (N {l}) for all k, l ∈ N with l ≤ k.Proof. Let S ⊆ N . Choose T ⊆ N such that v k−1 (S{k}) = r k−1 (S{k} ∩ T ) − c(T ).Then v k−1 (S{k}) = r k−1 (S{k} ∩ T ) − c(T ) ≤ rk (S ∩ T ) − c(T ) ≤ v k (S).This proves a).It follows directly from a) that v k (N ) ≥ v k−1 (N {k}). Choose T ⊆ N suchthat v k (N ) = r k (T ) − c(T ). If k ∈ T , then we have / v k (N ) = r k (T ) − c(T ) = r k−1 (T ) − c(T ) ≤ v k−1 (N {k})and b) follows. If k ∈ T , then b) follows fromv k (N ) = r k (T ) − c(T ) = r k−1 (T ) − c(T ) − zk ≤ v k−1 (N ) − zk = v k−1 (N {k}).We prove c) by induction on the number k. The games v k and v k−1 diﬀer onlyin the reward of player k. Hence, we have v k (S) = v k−1 (S) for all S ⊆ N with 5
6. 6. k ∈ S. It follows that v k (N ) = v k−1 (N {k}) = v k (N {k}) for all k ∈ N , where /the ﬁrst equality follows from ii). This proves c) in case l = k. Thus we havecompletely handled the case k = 1. Now assume that c) holds for all pairs l, kwith l ≤ k < p where p > 1. The case l = k = p is already handled, so assumel < k = p. We have v k (N {l}) ≥ v k−1 (N {l}) − zk = v k−1 (N ) − zk = v k (N ).Now c) follows, since it trivially holds that v k (N {l}) ≤ v k (N ).Theorem 2 The z-allocation satisﬁes z(S) ≤ v(N ) − v(N S) for all S ⊆ Nand z(N ) = v(N ) − v n (N ).Proof. We shall prove by induction that z(S) ≤ v k (N ) − v k (N S) for allS ⊆ {k + 1, . . . , n} and all k ∈ N . For k = n − 1 this is trivially true. Supposeit is true for all k > p. Let S ⊆ {p + 1, . . . , n}. If p + 1 ∈ S then according to /the induction hypothesis we have z(S) ≤ v p+1 (N ) − v p+1 (N S) ≤ v p (N ) − v p (N S),where the second inequality follows from the relations v p+1 (N ) = v p (N ) − zpand v p+1 (N S) ≥ v p (N S) − zp . If p + 1 ∈ S then z(S) = zp+1 + z(S{p + 1}) ≤ v p (N ) − v p (N {p}) + v p+1 (N ) − v p+1 (N S ∪ {p + 1}) = v p (N ) − v p+1 (N S ∪ {p + 1}) ≤ v p (N ) − v p (N S),where the last inequality follows from lemma 1. By lemma 1 we have zk =v k−1 (N ) − v k (N ) for all k ∈ N . Hence, z(N ) = k∈N (v k−1 (N ) − v k (N )) =v 0 (N ) − v n (N ).Corollary 3 The z-allocation is a core element of the reward game (N, v) ifand only if it is eﬃcient.Proof. That eﬃciency of z is a necessary condition follows from the deﬁnitionof a core element. Let us prove that it also suﬃcient. Let S ⊆ N . According 6
7. 7. to theorem 2 we have z(N S) ≤ v(N ) − v(S). Substituting v(N ) = z(N ), weﬁnd z(N S) ≤ z(N ) − v(S), and after rearranging we obtain z(S) ≥ v(S).The z-allocation is not necessarily a core element of (N, v), which follows triv-ially from the fact that VWST games with an empty core exist. For instance,if we compute the z-allocation for the example in Figure 1 of a 6-player VWSTgame with an empty core, we obtain z = 0 and v n (N ) = v(N ) = 1. Even whenthe core of a reward game is non-empty, the z-allocation is not necessarily anelement of it. Deﬁne the 4-player cost game (N, c) by   8 if |S| = 1  12 if |S| = 2  c(S) =  20 if |S| = 3  24 if |S| = 4, and let r = (7, 7, 7, 7). The reward game (N, v) is then given by   0 if |S| = 1  2 if |S| = 2  v(S) =  2 if |S| = 3  4 if |S| = 4. Observe that the game (N, v) has exactly one core element, namely (1, 1, 1, 1).For the z-allocation we have z1 = v(N ) − v(234) = 2. Hence, the z-allocationis not in the core of the reward game.Let us formalize the situation of this example. Since the game (N, c) is inter-preted as a cost game it makes little sense to apply the usual core concept tothis game. The anti-core of the game (N, c) is deﬁned by ACore(c) := {x ∈ RN | x(N ) = c(N ) and x(S) ≤ c(S) for all S ⊆ N }.In the example the vector x := (6, 6, 6, 6) lies in the anti-core of (N, c), andthe vector r − x = (1, 1, 1, 1) lies in the core of the reward game. This is nocoincidence as is shown in the following theorem.Theorem 4 Let (N, c) be a (cost) game and let r ≥ 0 be a reward vector. If avector x ∈ ACore(c) exists with 0 ≤ x ≤ r, then r − x ∈ Core(v), where (N, v)is the reward game with respect to r. 7
8. 8. Proof. Let S ⊆ N . We have v(S) = max{r(S ∩ T ) − c(T ) | T ⊆ N } ≤ max{r(S ∩ T ) − x(T ) | T ⊆ N } = r(S) − x(S).Eﬃciency follows from v(N ) ≥ r(N ) − c(N ) = r(N ) − x(N ).Clearly, the existence of a core element for the reward game may depend on thereward vector. Let us call a cost game (N, c) strongly balanced if the rewardgame (N, v) has a non-empty core for every non-negative reward vector r.Theorem 5 Let (N, c) be a strongly balanced cost game, let r ≥ 0 be a rewardvector, and let (N, v) be the corresponding reward game. Then the z-allocationlies in Core(v).Proof. It is suﬃcient to prove that v n (N ) = 0. It follows from lemma 1 thatv n (N ) = v n (N {i}) for all i ∈ N . Hence, the only possible core allocation for(N, v n ) is the 0-allocation. Since (N, c) is strongly balanced this must indeedbe a core allocation. It follows that v n (N ) = 0.A game (N, v) is called convex if v(S) + v(T ) ≤ v(S ∩ T ) + v(S ∪ T ) for allS, T ⊆ N . It is called concave if v(S) + v(T ) ≥ v(S ∩ T ) + v(S ∪ T ) for allS, T ⊆ N .Theorem 6 Concave games are strongly balanced.Proof. Let (N, c) be a concave game, let r ≥ 0 be a reward vector, and letthe game (N, v) be the corresponding reward game. We will prove that (N, v)is convex. Let S, T ⊆ N . Choose U, V ⊆ N such that v(S) = r(S ∩ U ) − c(U )and v(T ) = r(T ∩ V ) − c(V ). Convexity of the game (N, v) follows from v(S) + v(T ) = r(S ∩ U ) + r(T ∩ V ) − c(U ) − c(V ) ≤ r(S ∩ U ) + r(T ∩ V ) − c(U ∩ V ) − c(U ∪ V ) = r(S ∩ T ∩ U ∩ V ) − c(U ∩ V ) + r((S ∩ U ) ∪ (T ∩ V )) − c(U ∪ V ) ≤ r(S ∩ T ∩ U ∩ V ) − c(U ∩ V ) + r((S ∪ T ) ∩ (U ∪ V )) − c(U ∪ V ) ≤ v(S ∩ T ) + v(S ∪ T ).Since convex games have a non-empty core, the theorem follows. 8
9. 9. 3 VWST games deﬁned on graphsUp to here we assumed that it was possible to establish any link between twocustomers or between a customer and the supplier. Let G = (N ∪ {0}, E) be aconnected undirected graph. In this section we assume that only the edges inE can be established and that the cost of establishing such a link is given by anon-negative cost function d on the edges in E. Let us call the graph G Steinerbalanced if every VWST game deﬁned on G has a non-empty core, regardless ofthe cost function d, the rewards of the players and the location of the supplier.It follows from Theorem 5 that the z-allocation is always a core element for aVWST game is deﬁned on a graph which is Steiner balanced.Some simple reasoning can give us an idea about which graphs are Steinerbalanced and which are not: suppose we are given a graph G which is notSteiner balanced. Then, by deﬁnition, it is possible to deﬁne a VWST game(N, v) on G such that it has an empty core. Hence, the z-allocation cannot beeﬃcient for this game, and it follows that v n (N ) > 0. According to lemma 1 thegame (N, v n ) also satisﬁes v n (N ) = v n (N {i}) for all i ∈ N . This shows thatthere are several optimal trees for the grand coalition, which are all embeddedin G. Furthermore, the collection of optimal trees cannot be just any collectionof trees. For instance, if two trees have only the root-vertex in common, thenthey cannot both be optimal, since the superposition of two such trees givesa strictly better tree. In the following we will follow this type of reasoning,and this will lead to necessary conditions on a graph such that it is not Steinerbalanced.Let us ﬁrst introduce some terminology for trees. Let Γ = (V, E) be a tree-graph, and let 0 ∈ V be a special vertex of this tree, called the root-vertex. Avertex v ∈ V is called a leaf of Γ if v = 0 and if the degree of v is 1. Vertexv ∈ V is called a descendant of w ∈ V if the (unique) path from 0 to v containsw.Two vertices v, w ∈ V are called adjacent if {v, w} ∈ E. Vertex v is called achild of w ∈ V if v and w are adjacent and if v is a descendant of w. Vertexw is called the father of v. Let D(v) ⊆ V denote the set of descendants of v.Note that v ∈ D(v). The subgraph of Γ with vertex-set D(v) is again a tree, 9
10. 10. and this tree is called the subtree of Γ rooted at v.Let G = (V, E) be a graph, and let T be a collection of trees embedded in G.The collection T is said to have the merger property if it satisﬁes the followingconditions.M1 There is a special vertex which is a vertex of every tree in T . This vertex is designated as the root-vertex.M2 The collection T is non-empty and the tree consisting of the root alone is not in T .M3 If v is a vertex of both Γ ∈ T and Ω ∈ T , and if c is a child of v in the tree Γ, then the subtree of Γ rooted at c contains another vertex which is also in Ω.M4 For every vertex v ∈ V that is a leaf in some tree of T , there is another tree in T that does not contain v at all.We say that the graph G has the merger property if it contains a collection oftrees with the merger property. In Figure 2 we have depicted three copies ofthe graph of Figure 1. In each copy of the graph we highlighted an embeddedtree, and one can check that the collection of these three trees have the mergerproperty. The graph therefore has the merger property. This is a necessaryconsequence of the fact that the graph is not Steiner balanced, as the followingtheorem shows. Figure 2Theorem 7 If a graph G = (V, E) does not have the merger property, then itis Steiner balanced. 10
11. 11. Proof. Suppose that G is not Steiner balanced. We will prove that G hasthe merger property. We have observed earlier that it is possible to deﬁne aVWST game (N, v) on this graph such that v(N ) > 0 and v(N ) = v(N {i})for all i ∈ N . Here the set N consists of all vertices of V except one, and theremaining vertex is designated as the root. Call an optimal tree for coalitionN minimal if there is no other optimal tree whose vertex-set is a proper subsetof the ﬁrst tree. Let T denote the collection of all optimal trees for coalition Nthat are minimal. Trivially, the conditions M1 and M2 are satisﬁed by T .Let Γ and Ω be two trees in T , and let v be a vertex of both trees. Assumethat v is not a leaf in Γ and let c be a child of v in the tree Γ. Denote thesubtree of Γ rooted at c by Γc . Note that the sum of the rewards of the verticesin Γc minus the costs of the edges in this tree is strictly greater than d(v, c),since otherwise the subtree could be deleted from Γ without reducing the valueof Γ, contradicting the minimality of Γ. Now, suppose that Ω does not containany vertex of Γc . Then the complete subtree Γc can be attached to vertex v,thereby increasing the value of the tree Ω, a contradiction. This proves that Tsatisﬁes condition M3.Finally, suppose that i ∈ N is a leaf of Γ ∈ T . Note that the reward of i isstrictly positive, since otherwise i could be deleted from Γ without reducing thevalue of the tree, which contradicts the minimality of Γ. Let Ω be a minimaloptimal tree for coalition N {i}. The tree Ω cannot contain vertex i, sinceotherwise v(N ) ≥ v(N {i}) + ri > v(N {i} = v(N ),a contradiction. This proves that T satisﬁes condition M4.Theorem 8 Every 5-persons VWST game has a non-empty core.Proof. We show that a graph which is not Steiner balanced must have atleast 7 vertices. Let G be a graph which is not Steiner balanced. Let T be acollection of trees in G with the merger property. Denote the root by 0. ChooseΓ1 ∈ T , and let v adjacent to 0 in this tree. The subtree rooted at v containsat least two leaves. This is seen as follows. Suppose the subtree has only oneleaf, i.e. the subtree is a path, with say l as an endpoint. Let Ω be an arbitrary 11
12. 12. tree of T . By applying condition M3 repeatedly, we see that Ω contains l.Since Ω was chosen arbitrarily, we conclude that all trees in T contain l. Thiscontradicts condition M4. Hence, the subtree of Γ1 rooted at v has at leasttwo leaves, and it follows that Γ1 consists of at least 4 vertices, namely 0, v,and at least two leaves, say l1 and l2 . Denote the vertex set of Γ1 by V1 .By condition M4, there exists Γ2 ∈ T that does not contain l1 . By conditionM3, the tree Γ2 cannot contain the predecessor of l1 in Γ1 either. Denote thevertex set of Γ2 by V2 . Of course, also Γ2 contains at least 4 vertices. Since, V1contains at least two vertices which are not in V2 , it follows already that |V | ≥ |V1 ∪ V2 | ≥ |V2 | + 2 ≥ 6.We want to show however that |V | ≥ 7.If Γ2 has at least 5 vertices, we are ﬁnished. So assume that |V2 | = 4. We knowthat Γ2 has at least two vertices which are not in Γ1 . Hence, it has at most twovertices in common with Γ1 . Consequently, Γ1 contains at least two verticeswhich are not in Γ2 , and it follows that |V | ≥ |V1 ∪ V2 | ≥ |V1 | + 2 ≥ 6.Again, if Γ1 has at least 5 vertices, we are ﬁnished. So assume that also |V1 | = 4.Now we conclude that Γ1 and Γ2 have exactly two vertices in common. Oneof these vertices is the root 0. The other vertex in common must be a leafin both trees, since otherwise a contradiction with condition M3 would occur.Denote this common leaf by l. By condition M4 it follows that there must bea third tree in T , say Γ3 , that does not contain l. Let f1 denote the father ofl in the tree Γ1 , and let f2 denote the father of l in Γ2 . These fathers mustbe diﬀerent vertices, and by condition M3 it follows that Γ3 cannot contain f1and f2 either. It follows that Γ3 can have at most 3 vertices in common withV1 ∪ V2 . Since Γ3 has at least 4 vertices, we conclude that Γ3 has at least 1vertex which is neither in Γ1 nor in Γ2 . This means that the graph G has atleast 7 vertices.Theorem 9 Let G = (V, E) be a tree-graph. Then G is Steiner balanced. 12
13. 13. Proof. Suppose G is not Steiner balanced. Then let T be a collection oftrees embedded in G with the merger property. Choose Γ ∈ T with a minimalnumber of vertices, and let l be a leaf of Γ. By condition M4 there is anothertree, say Ω ∈ T , that does not contain l. We will show that there are verticesf and c such that f is the father of c in Γ, while c is also a vertex of Ω andf is not. Assume that this is not true, i.e. if a vertex v of Γ is not in Ω, thenalso none of its children is in Ω. By repeated application of this assumptionit follows that if a vertex v of Γ is not in Ω, then none of the vertices in thesubtree of Γ rooted at v is in Ω. We know that l is not in Ω. Let v denote theﬁrst vertex on the path from 0 to l in Γ which is not in Ω. Then the wholesubtree of Γ rooted at v contains no vertex of Ω. By condition M3 it followsthat the father of v cannot be in Ω either, and we have derived a contradiction.Now choose father f and child c in Γ such that c is in Ω and f is not. Thepaths from 0 to c in Γ and Ω are not the same, since the path in Γ containsvertex f and the path in Ω does not. Then G cannot be a tree-graph, sincepaths between vertices in a tree are unique.An alternative proof of theorem 9 is obtained if we use a result by Granot etal. (1994) which states that a monotone minimum cost spanning tree game isconcave if it is deﬁned on a tree-graph. According to theorem 6 such a game isstrongly balanced, and hence the corresponding VWST game has a non-emptycore. It even follows from the proof of theorem 6 that such a VWST game isconvex.We were not able to prove (nor disprove) that a VWST game has a non-emptycore in case the underlying graph is series-parallel. We tried to prove directlythat a graph having the merger property cannot be series-parallel. We alsotried to formulate a VWST game deﬁned on a series-parallel graph as a linearproduction game (Owen (1975)) or as a generalized linear production game(Granot (1986)) by using a description of the arborescence polytope for series-parallel graphs (Goemans (1994)). Such an approach works ﬁne for the directedSteiner tree game (see Skorin-Kapov (1992)), but it failed for the VWST game.Finally, we tried to derive the result by inspection of the linear time algorithm 13
14. 14. (Wald and Colbourn 1982) that solves the directed Steiner tree problem in casethe underlying graph is series-parallel. Also this approach was unsuccessful.Note that if a series-parallel graph is Steiner balanced, then a core element forthe associated VWST game can be computed in O(n2 ) elementary operations,since computation of the z-allocation requires the determination of v k (N ) fork = 0, 1, . . . , n, and v k (N ) can be computed in linear time.We conjecture that a VWST game has a non-empty core if it deﬁned on aseries-parallel graph. This conjecture is based mainly on our fruitless eﬀorts toconstruct a series-parallel graph with the merger property. In fact, these eﬀortshave led us to the stronger conjecture that a graph has the merger propertyif and only if it contains the graph of ﬁgure 1 as a minor. If this conjectureis true, then this completely settles the question of which graphs are Steinerbalanced and which are not.ReferencesVan Bokhoven, M. 1994. Voluntary connection to a source. Bachelor Thesis,University of Tilburg.Goemans, M.X. 1994. Arborescence polytopes for series-parallel graphs, Dis-crete Applied Mathematics, 51:277-289.Granot, D., G. Huberman. 1981. Minimum cost spanning tree games, Mathe-matical Programming, 21:1–18.Granot, D. 1986. A generalized linear production model: a unifying model,Mathematical Programming, 34:212-222.Granot, D., M. Maschler, G. Owen, W.R. Zhu. 1996. The kernel/nucleolus ofa standard tree game, International Journal of Game Theory, 25(2):219-244.Owen, G. 1975. On the core of linear production games, Mathematical Pro-gramming, 9:358–370.Segev, A. 1987. The node-weighted Steiner tree problem, Networks, 17:1–18.Skorin-Kapov, D. 1992. A ﬁxed-cost spanning forest problem on series-parallelgraphs. Proceedings of the second Conference on Operational Research, KOI’92, 14
15. 15. Rovinj, 85–94.Wald, J.A., C.J. Colbourn. 1983. Steiner trees, partial 2-trees, and minimumIFI networks. Networks, 13:159–167. 15