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The k -centrum Chinese Postman delivery problem and a related cost allocation game
Discrete Applied Mathematics 179 (2014) 100–108
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The k-centrum Chinese Postman delivery problem and a related cost allocation game Daniel Granot a,∗ , Frieda Granot a , Harshavardhan Ravichandran b a
Sauder School of Business, University of British Columbia, Vancouver, B.C., Canada
b
Air New Zealand, Auckland, New Zealand
article
info
Article history: Received 28 October 2013 Received in revised form 18 July 2014 Accepted 20 July 2014 Available online 8 August 2014 Keywords: Chinese Postman problem Chinese Postman game k-centrum Chinese Postman problem Core Submodular game
abstract We analyze the cost allocation cooperative game, denoted by (N , c k ), induced by the kcentrum Chinese Postman (k-centrum CP) delivery problem in connected undirected and strongly connected directed graphs. For the undirected case, we show, for example, that for k = 1, 2, (N , c k ) is submodular for all graphs having non-negative edge-weights, for all locations of the post-office. For k ≥ 3, we prove that (N , c k ) is submodular for all nonnegative edge-weights and for all locations of the post office if and only if the graph is either the cyclic graph on three edges or it is a graph wherein each edge is contained in at most one cycle and these cycles, if any, have only two edges. For the directed graph case we show, for example, that the k-centrum CP game induced by a strongly connected graph G is submodular for all k ≥ 2 if and only if every arc in G is contained in precisely one directed cycle. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Cooperative game theory methodology was extensively applied to analyze cost and revenue allocation problems derived from classical optimization problems. Specifically, under the plausible assumption that the resources in these problems belong to different individuals (players), cooperative game theory was used to suggest fair allocations of the overall cost or revenue that the players can achieve by pooling together their resources. Thus, for example, a Linear Programming (LP) game, e.g., Owen [16], is derived from the classical LP problem by assuming that the right-hand-side vector belongs to different players; minimum cost spanning tree (mcst) games (e.g., Bird [1] and Granot and Huberman [10]) arise from the mcst problem by assuming that different players (e.g., communities) reside in the vertices of the graph, and Chinese Postman (CP) games (e.g., Hamers et al. [12]) arise from the classical CP problem where the cost of a complete tour by the postman has to be allocated among the different players who reside in the edges of the network. A cooperative game formulation of a cost allocation problem is given by the pair (N , c ), where N is the set of players, which, in the CP game, are the edges of the network, and c : 2N → R+ is the (coalitional) cost function, so that for each S ⊆ N, c (S ) is the cost incurred by coalition S if it operates alone. In the CP game, c (S ) is the cost of a minimal or cheapest tour that begins and ends at the post office (v0 ) and visits all the players (edges) in S at least once. The core of a game is the set of feasible allocations for which no coalition (i.e. is better off by seceding subset) of players from the rest of the group. That is, a vector x is in the core of the game (N , c ) if j∈N xj = c (N ) and j∈S xj ≤ c (S ), ∀S ⊂ N. A game with a non-empty core is said to be balanced. It follows from Shapley [18] that a submodular game has a non-empty
∗
Correspondence to: 2053 Main Mall, Vancouver, BC, Canada, V6T 1Z2. Tel.: +1 604 822 8432. E-mail addresses: [email protected] (D. Granot), [email protected] (F. Granot), [email protected] (H. Ravichandran). http://dx.doi.org/10.1016/j.dam.2014.07.021 0166-218X/© 2014 Elsevier B.V. All rights reserved.
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core, in addition to having other nice properties. A game is said to be submodular if and only if its characteristic function c is submodular. A function c is said to be submodular if and only if c (S ∪ {i}) − c (S ) ≤ c (T ∪ {i}) − c (T ), ∀i ∈ N , with T ⊆ S ⊆ N \ {i}, or, equivalently, if and only if c (S ∪ T ) + c (S ∩ T ) ≤ c (S ) + c (T ), ∀S , T ⊆ N. Finally, a game (N , c ) is said to be totally balanced if the core of every subgame (S , cS ) of (N , c ) is not empty, where cS is restricted to subsets of S. It is well known that submodular games are totally balanced. In general, a CP game can have an empty core. However, Hamers et al. [12] have shown that a CP game induced by a weakly Eulerian graph has a non-empty core, where a graph G = (V (G), E (G)) is called weakly Eulerian if its bi-connected components1 are Eulerian subgraphs. Further, Hamers [11] has shown that a CP game is submodular if it is induced by a weakly cyclic connected graph, where a graph is weakly cyclic if every edge therein is contained in at most one cycle. A graph G is called CP balanced (resp., totally balanced, submodular) if the CP game (N , c ) induced by G is balanced (resp., totally balanced, submodular) for all non-negative edge-weights and all locations of the post office node in G. Granot et al. [9] have characterized CP balanced, CP totally balanced, and CP submodular graphs. For example, it was shown therein that a connected undirected graph G is CP balanced (resp., CP submodular), if and only if G is weakly Eulerian (resp., weakly cyclic), and that the CP game induced by an arbitrary strongly connected directed graph is balanced for all non-negative arc-weights and all locations of the post office node in G. For related studies see, e.g., Granot and Hamers [7], Granot et al. [8] and Granot and Granot [6]. In this paper we study the cost allocation problem associated with the k-centrum Chinese Postman (k-centrum CP) problem, wherein we minimize the sum of the weights of the k largest edges in the tour. k-centrum optimization problems are optimization problems wherein only the largest k-terms in the objective function are optimized; see, e.g., Ogryczak and Tamir [14]. For k = 1, k-centrum optimization problems are referred to as bottleneck problems; see, e.g., Garfinkel and Rao [5] for the bottleneck transportation problem, or Gabow and Tarjan [3] for the bottleneck spanning tree problem in a directed graph and the bottleneck maximum cardinality matching problem. For the k-centrum shortest path problem for an arbitrary k, see, e.g., Garfinkel et al. [4], and for k-centrum optimization problems in location, see, e.g., Tamir [19]. In general, a k-centrum (i.e., k-sum) optimization problem can be solved in polynomial time whenever an associated minsum problem can be solved in polynomial time (Punnen and Aneja [17]). Let (N , c k ) denote the k-centrum CP game induced by a connected undirected graph G. We will refer to G as k-CP balanced (resp., totally balanced, submodular) if the k-centrum CP game induced by G is balanced (resp., totally balanced, submodular) for all non-negative edge-weights and all locations of the post office node. We prove in this paper that for k = 1, 2, (N , c k ) is submodular for all undirected connected graphs with non-negative edge-weights and for all locations of the post office, and that G is k-CP submodular for k ≥ 3 if and only if G is either the cyclic graph on three edges or G is a weakly cyclic graph wherein the cycles, if any, have two edges. Characterizations of k-CP balanced and k-CP totally balanced graphs are also provided. In the directed case we show, for example, that a strongly connected graph G is k-CP submodular for all k ≥ 2 if and only if each arc in G is contained in precisely one directed cycle. 2. Notation and definitions Let G = (V (G), E (G)) be an undirected (directed) graph where V (G) and E (G) denote the set of nodes and the set of edges (arcs) of G. A (directed) walk in G is a finite sequence of nodes and edges (arcs) of the form v1 , e1 , v2 , . . . , em , vm+1 with m ≥ 0, v1 , . . . , vm+1 ∈ V (G), e1 , . . . , em ∈ E (G), such that ej = {vj , vj+1 }(ej = (vj , vj+1 )) for all j ∈ {1, . . . , m}. Such a walk is said to be closed if v1 = vm+1 . A simple (directed) path in G is a (directed) walk in which all vertices (except, possibly v1 and vm+1 ) are distinct. A simple, closed (directed) path containing at least one edge (arc) is called a simple (directed) cycle. An undirected (directed) graph G is (strongly) connected if there is a (directed) path (from) between any node to any other node in G. A set of edges E¯ in (a connected graph) G is a (minimal) edge-cutset of G, if its removal, but not the removal of any proper subset thereof, disconnects G. A path, P, in G will be referred to as a bridge, if the degrees of all internal nodes of P in G are 2 and every edge in P is a (minimal) edge-cutset of G. We denote by v0 ∈ V (G) the post office in G. We define an S-tour w.r.t. v0 associated with S ⊆ E (G) as a closed (directed) walk that starts at v0 , traverses all the edges (arcs) in S at least once and ends in v0 . We can associate with an S-tour, an Eulerian multigraph HS which contains t copies of the edges (arcs) that are traversed t times in the S-tour, where t ∈ {0, 1, 2, . . . , |E (G)|}. The set of Eulerian multigraphs associated with the set S ⊆ E (G) is denoted by D(S ). Let w : E (G) → R+ be the travel cost function associated with the edges (arcs) of the graph. Then the k-centrum travel cost w k (HS ) of an S-tour (v0 , e1 , v1 , . . . , vr −1 , er , v0 ) is given by the sum of the nk most expensive edges (arcs) in the tour, where nk = min{r , k}. Formally, let τk = (E (G), (G, v0 ), w) denote the delivery problem, where E (G) is the set of players, (G, v0 ) is the undirected graph with v0 as the post office, and w is the edge travel cost function. Then the delivery game, denoted by (N , c k ), where N = E (G), corresponding to the delivery problem τk is defined for all S ⊆ E (G) by c k (S ) = min w k (HS ). HS ∈D(S )
We refer to the above delivery game as the k-centrum Chinese Postman (CP) game. The definition of the k-centrum CP game in the directed case is derived similarly and denoted as (N , c¯ k ). 1 A bi-connected component of a graph G is a maximal subgraph, Q , of G such that every pair of distinct edges in Q is contained in a simple cycle in Q .
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The multigraph that optimizes the above function for a certain value of k is denoted by HS∗k . It is important to note that HS contains 0, 1 or 2 copies of every edge in E (G). Clearly, c k is subadditive, i.e., c k (S ) + c k (T ) ≥ c k (S ∪ T ), ∀S , T ⊆ E (G) such that S ∩ T = ∅. Moreover, since w is non-negative, c k is also monotonic, i.e., c k (S ) ≤ c k (T ), ∀S ⊆ T ⊆ E (G). ∗k
3. Submodularity of the k-centrum CP game induced by undirected graphs We first consider in this section the cases when k = 1, 2, and prove that for these cases the k-centrum CP game, (N , c k ), k = 1, 2, is submodular for all connected undirected graphs with non-negative edge-weights and all locations of the post office. Note that when k = 1, the k-centrum CP problem is called the bottleneck CP problem. Proposition 1. Any connected graph is 1-CP submodular. 1 ′ Proof. Let i ∈ E (G) and S , T be such that T ⊆ S ⊆ E (G) \ {i}. Consider the walk H ′ defined as H ′ = HS∗1 + HT∗∪{ i} . Clearly, H
is a multigraph associated with a feasible tour for S ∪ {i}, i.e., H ′ ∈ D(S ∪ {i}). Therefore, c 1 (S ∪ {i}) ≤ w 1 (H ′ ), where w 1 (H ′ ) is the 1-centrum CP cost of H ′ . Now, as k = 1, w 1 (H ′ ) = max{c 1 (S ), c 1 (T ∪ {i})}. If w 1 (H ′ ) = c 1 (S ) then c 1 (S ∪{i})− c 1 (S ) ≤ w 1 (H ′ )− c 1 (S ) = 0. However, by monotonicity, c 1 (S ∪{i})− c 1 (S ) ≥ 0, which implies that c 1 (S ∪ {i}) − c 1 (S ) = 0. Since c 1 (T ∪ {i}) − c 1 (T ) ≥ 0, we derive that c 1 (S ∪ {i}) − c 1 (S ) ≤ c 1 (T ∪ {i}) − c 1 (T ). On the other hand, if w 1 (H ′ ) = c 1 (T ∪ {i}), then c 1 (S ∪ {i}) ≤ w 1 (H ′ ) = c 1 (T ∪ {i}), and since c 1 (S ) ≥ c 1 (T ) we conclude that c 1 (S ∪ {i}) − c 1 (S ) ≤ c 1 (T ∪ {i}) − c 1 (T ), and the proof is complete. Proposition 2. Any connected graph is 2-CP submodular. Proof. Let S , T ⊆ E (G), and let w : E (G) → R+ . If S ∩ T = ∅, then the subadditivity of c 2 implies its submodularity for S and T . Hence, we will only consider the case when S ∩ T ̸= ∅. Let HS∗2 and HT∗2 denote optimal 2-centrum S-tour and T -tour, respectively. For simplicity of presentation, we assume that all tours must contain at least 2 edges. Hence, we can express c 2 (S ) as c 2 (S ) = w(s1 ) + w(s2 ) where w(s1 ) ≥ w(s2 ) ≥ 0, i.e., s1 , s2 ∈ HS∗2 are the largest and second largest edges in HS∗2 , respectively. Similarly c 2 (T ) = w(t1 ) + w(t2 ), where w(t1 ) ≥ w(t2 ) ≥ 0. We assume, without loss of generality, that w(s1 ) ≥ w(t1 ). Now, consider H ′ = HS∗2 + HT∗2 . Since we have assumed that w(s1 ) ≥ w(t1 ), the 2-centrum CP cost, w 2 (H ′ ), of the tour associated with H ′ , satisfies w 2 (H ′ ) = w(s1 ) + max{w(t1 ), w(s2 )}. Case 1: w(s2 ) ≥ w(t1 ). Then, c 2 (S ∪ T ) ≤ w 2 (H ′ ) = w(s1 )+w(s2 ) = c 2 (S ), and since, by monotonicity, c 2 (S ∩ T ) ≤ c 2 (T ), we derive that c 2 (S ∪ T ) + c 2 (S ∩ T ) ≤ c 2 (S ) + c 2 (T ). Case 2: w(s1 ) ≥ w(t1 ) > w(s2 ). Since w(s1 ) > w(s2 ), the multiplicity of w1 in HS∗2 is exactly 1. Thus, s1 is not an edgecutset. Then, we claim that s1 ∈ S. Indeed, if s1 ̸∈ S and s1 is not an edge-cutset, then it is possible to construct an S-tour that does not pass through s1 and whose cost is at most 2w(s2 ), contradicting the optimality of HS∗2 . Subcase 2a: s1 ∈ S ∩ T . Then, s1 = t1 . If w(s2 ) ≥ w(t2 ), then w 2 (H ′ ) = w(s1 ) + w(s2 ), implying that c 2 (S ∪ T ) ≤ w(s1 ) + w(s2 ) = c 2 (S ). By monotonicity, c 2 (S ∩ T ) ≤ c 2 (T ), which implies that c 2 is submodular. On the other hand, if w(t2 ) > w(s2 ), w 2 (H ′ ) = w(t1 ) + w(t2 ), and thus, c 2 (S ∪ T ) ≤ w(t1 ) + w(t2 ) = c 2 (T ). Since c (S ∩ T ) ≤ c 2 (S ), we conclude that c 2 is submodular. Subcase 2b: s1 ̸∈ S ∩ T . Then, since w(t1 ) > w(s2 ), t1 ̸∈ S ∩ T and t1 ̸∈ HS∗2 . If w(t1 ) = w(t2 ), then t2 ̸∈ S ∩ T , t2 ̸∈ HS∗2 , and we have c 2 (S ∩ T ) ≤ c 2 (S \ {s1 }) ≤ 2w(s2 ) ≤ w(s2 ) + w(t2 ), c 2 (S ∪ T ) ≤ w(s1 ) + w(t1 ), which implies that c 2 (S ∪ T ) + c 2 (S ∩ T ) ≤ c 2 (S ) + c 2 (T ). If, on the other hand, w(t1 ) > w(t2 ), then we conclude that t1 is also not an edge-cutset, and therefore t1 ∈ T . We have, c 2 (S ∩ T ) ≤ min{c 2 (S \ {s1 }), c 2 (T \ {t1 })} ≤ min{2w(s2 ), 2w(t2 )} = 2 min{w(s2 ), w(t2 )} ≤ w(s2 ) + w(t2 ), c 2 (S ∪ T ) ≤ w(s1 ) + w(t1 ), which implies that c 2 (S ∪ T ) + c 2 (S ∩ T ) ≤ c 2 (S ) + c 2 (T ). We conclude that (N , c 2 ) is a submodular game for all connected undirected graphs, for all non-negative edge-weights, and for all locations of the post-office, and the proof is complete. We now demonstrate, in Example 3 below, that for k ≥ 3 the k-centrum CP game (N , c k ) may not be submodular. Example 3. Consider the graph drawn in Fig. 1, and suppose that w(e1 ) = 5, w(e2 ) = 6 and w(e3 ) = 7, and that player i resides in ei , i = 1, 2, 3. For the 3-centrum CP game (N , c 3 ) associated with Fig. 1 we have: c 3 ({1, 2, 3}) = 18, c 3 ({1, 3}) = 12, c 3 ({1, 2}) = 11 and c 3 ({1}) = 10, which can be easily shown to imply that c 3 is not submodular. In fact, since c k ({1, 2, 3}) = 23 for k > 3, it follows that (N , c k ) is not submodular for all k ≥ 3. Moreover, it can be easily shown that (N , c k ) is not even balanced for all k ≥ 4.
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Fig. 1. (N , c k ) is not submodular.
Fig. 2. A C3 with an edge attached.
4. k-CP submodular graphs for k ≥ 3 Recall that by Granot et al. [9], the ordinary CP game (N , c ) is CP-submodular if and only if (N , c ) is induced by a connected weakly cyclic graph, where a graph is weakly cyclic if each edge therein is contained in at most one cycle. This result should be contrasted with Propositions 1 and 2, in the previous section, and Proposition 4 below. Proposition 4. The k-centrum CP game induced by a connected graph G is submodular for all k ≥ 3 if and only if G is either the cyclic graph on three edges, denoted by C3 , or G is weakly cyclic and every cycle therein consists of two edges. Proof. It can be easily verified by inspection that C3 is k-CP submodular for all k ≥ 3. Next, let G be a weakly cyclic graph whose cyclic subgraphs, if any, are C2 , i.e., consist of two edges, and consider i, S , T such that T ⊆ S ⊆ N \ {i}. We first show that HT∗k ⊆ HS∗k , i.e., all the edges in HT∗k , the multigraph corresponding to c k (T ), can be assumed to be also present in HS∗k . Clearly, all edges in HT∗k which are not contained in any cycle in G are also contained in HS∗k . Consider any subgraph, C2 , of G, with edges a and b and assume, without loss of generality, that HS∗k contains a with a multiplicity of 2 (as the case when a and b both are present in HS∗k with a multiplicity of 1 is trivial). If a does not contribute towards w k (HS∗k ), and w(a) ≤ w(b), then HT∗k can be assumed to contain a with multiplicity 2, and if w(a) > w(b) then there exists an alternate optimum where HS∗k traverses both a and b once, and then HT∗k ⊆ HS∗k . If, on the other hand, a does contribute to the cost w k (HS∗k ), then clearly w(a) ≤ w(b). This implies that HT∗k must also contain a with multiplicity 2, and thus, we conclude that HT∗k ⊆ HS∗k . Therefore, c k (S ∪ {i}) − c k (S ) ≤ c k (T ∪ {i}) − c k (T ), and thus, G is k-CP submodular for all k. To prove the other direction, consider first the graph shown in Fig. 2. Let S = {PA, AB, BC }, T = {BC } and i = PB. Clearly, T ⊆ S ⊆ N \ {i}. For k = 3, HS∗3 takes the route P–A–B–C–B–A–P with c 3 (S ) = 12 and HT∗3 takes the route P–B–C–B–P with c 3 (T ) = 11. We see that edge PB lies in HT∗k but not in HS∗k . Therefore, the marginal cost of adding PB to S is greater than that of adding it to T . Hence, C3 with even one edge attached to one of its nodes is not submodular for k = 3. One could similarly show that the 3-centrum CP game induced by a 1-sum of two C3 cycles is not necessarily submodular, where 1-sum of two graphs G1 and G2 is obtained by coalescing one vertex in G1 with another in G2 . Finally, consider a graph G which is a cycle of length m. Let the cycle be defined by v1 , e1 , v2 , e2 , . . . , vm , em , v1 and let v1 be the post office. Set w(e1 ) = w(e2 ) = · · · = w(em−2 ) = 10, w(em−1 ) = 1 and w(em ) = 11. Also, let S = {e1 , e2 , . . . , em−1 }, T = {em−1 } and i = em . Clearly, T ⊆ S ⊆ N \ {i}. Consider the case when k = m − 1. Clearly, HS∗k takes the path e1 , e2 , . . . , em−2 , em−1 , em−1 , em−2 , . . . , e2 , e1 with a cost of c k (S ) = 10k. However, HT∗k takes the path em , em−1 , em−1 , em with a cost of c k (T ) = 23 for k = 3, and c k (T ) = 24 for all k ≥ 4. Again we see that edge i lies in HT∗k but not in HS∗k , and therefore, from the marginal cost definition of submodularity, it follows that the k-centrum CP game induced by G is not submodular. We can conclude that the k-centrum CP game induced by a weakly cyclic graph G that is not a tree, or C3 , or a weakly cyclic graph with only cycles of size 2, is not submodular for some k, and the proof is complete. 5. k-CP total balancedness and k-CP balancedness For k ∈ {1, 2}, any graph is k-CP submodular, and thus it is trivially k-CP totally balanced and k-CP balanced. We prove in this section that graph G is k-CP totally balanced for all k ≥ 3 if and only if G is weakly cyclic. Recall that a graph G is weakly cyclic if and only if every edge in E (G) belongs to at most one cycle. Alternatively, G is weakly cyclic iff every bi-connected component is a cycle. Hence, it can be seen that every weakly cyclic graph can be constructed by joining simple cycles together. These cycles can either have a node in common or they can have a bridge (line graph) in between them. A weakly cyclic graph is also called a cactus graph (see, e.g., Das [2] for a recent reference). Lemma 5. If G is not weakly cyclic, then there exists a k for which the k-CP game induced by it is not k-CP totally balanced.
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Proof. Consider an undirected connected graph G which is not weakly cyclic. As trees are, by definition, weakly cyclic, G cannot be a tree, and hence, G is not acyclic. Therefore, there exist bi-connected components in G, and since G is not weakly cyclic, at least one of those components is not a cycle. In that component, there must exist a pair of nodes, say u and v , with at least three edge-disjoint paths between them. Then, as in Example 3, we could choose, say, u as the post office and introduce edge-weights which would result with a k-CP game defined on a subgraph of this component whose core is empty, implying that G is not k-CP totally balanced. Lemma 6. If G is weakly cyclic, then the k-CP game induced by G is totally balanced for every k. Proof. We first show that the cyclic graph, C , is k-CP totally balanced, which will follow if we show that C is k-CP balanced for an arbitrary number of public edges and arbitrary non-negative edge-weights. Let C = (V (C ), E (C )) be a cyclic graph of arbitrary size with non-negative edge-weights in which some of the edges are public. Denote by T a public path consisting of contiguous public edges. If exists, let e1 (T ) (resp., e2 (T )) denote the non-public edge which, counter-clockwise from the post office, is just before (resp., after) the public path T . Further, for each non-public edge e, let S (e) (resp., R(e)) denote the coalition of all players including (the one residing at) e, which are counter-clockwise (resp., clockwise) between the post office and e. Let w k (S (e)) (resp., w k (R(e))) denote the k-centrum cost of these coalitions when the tours they employ consist of a counter-clockwise (resp., clockwise) walk from the post office to e and returning clockwise (resp., counter-clockwise) to the post office. Let Q be an arbitrary subset of N = E (C ). We will show that the subgame, (Q , cQk ), of the k-centrum CP game (N , c k ) induced by C is balanced. For that purpose, we regard the edges in N \ Q as public. Case 1. The tour corresponding to cQk (Q ) is the cycle tour which traverses all edges in C . Let T1 , . . . , Tγ denote the public paths in C which contribute to cQk (Q ). Assume that these paths are arranged in a counterclockwise manner from the post office. Thus, T1 is, counter-clockwise, the closest such public path to the post office. Procedure 1 below constructs a core vector for the subgame (Q , cQk ) in Case 1. Procedure 1 Step 0: For every edge e ∈ Q , if the cost w(e) enters the calculation of cQk (Q ), set x(e) = w(e). For all other e ∈ Q set x(e) = 0. Let x = (x(e)). Clearly x(Q ) = Σ (x(e) : e ∈ Q ) ≤ cQk (Q ).
Step 1: For Stages j = 1, . . . , γ , for i = 1, 2, if ei (Tj ) exists, let ∆(ei (Tj )) = min{w k (S (ei (Tj ))) − x(S (ei (Tj ))), cQk (Q ) − x(Q )}. Increment x(ei (Tj )) by
∆(ei (Tj )). If ∆(ei (Tj )) = cQk (Q ) − x(Q ), then stop. End.
We claim that the vector x produced by Procedure 1 is a core vector for the k-CP subgame (Q , cQk ) induced by C . Indeed, as far as violations of proper core constraints, it is sufficient to show that the coalitions of the form S (e) and R(e), for all nonpublic edges e, are not violated by x. Now, by the incrementation carried out in Step 1, we ensure that coalitions of the form S (e) are not violated by x. Next, suppose, on the contrary, that for j = l, the incrementation at Step 1, wherein, say, the allocation to e1 (Tl ) has been increased, resulted in a violation for coalition R(e1 (Tl )). Thus, if x denotes the vector after x(e1 (Tl )) was incremented, then x(R(e1 (Tl ))) > cQk (R(e1 (Tl ))). However, consider stage l − 1 in Step 1 of Procedure 1, and assume,
for simplicity of exposition, that e2 (Tl−1 ) exists. Then, by Step 1, x(S (e2 (Tl−1 ))) = w k (S (e2 (Tl−1 ))). Denote by A the set of all players residing in non-public edges, located strictly between e2 (Tl−1 ) and e1 (Tl ), and whose cost has entered cQk (Q ). Then,
Σ (x(e) : e ∈ A) = Σ (w(e) : e ∈ A), and x(Q ) > cQk (R(e1 (Tl ))) + Σ (w(e) : e ∈ A) + w k (S (e2 (Tl−1 ))) > cQk (Q ), implying that cQk (Q ) − x(Q ) < 0, which, by Step 1, cannot happen. A similar conclusion is reached if we consider a violation which could possibly occur when x(e2 (Tl )) was incremented. Thus, none of the coalitions of the form R(e) is violated by x. Finally, we need to show that x(Q ) = cQk (Q ). Suppose Step 1 terminated at Stage γ . If at this stage, for some i, ∆(ei (Tγ )) = cQk (Q ) − x(Q ), we are done. Otherwise, suppose, on the contrary, that ∆(ei (Tγ )) < cQk (Q ) − x(Q ), i = 1, 2.
• If e2 (Tγ ) exists, let R∗ ≡ R(e2 (Tγ ) \ {e2 (Tγ )}). If R∗ = ∅, regard x(R∗ ) as 0. Then, at this stage we have x(Q ) = x(R∗ ) + x(S (e2 (Tγ ))) = Σ (w(e) : e ∈ R∗ ) + w k (S (e2 (Tγ ))) > cQk (Q ), contradicting the assumption that ∆(ei (Tγ )) < cQk (Q ) − x(Q ), i = 1, 2. • If e2 (Tγ ) does not exist, then wk (S (e1 (Tγ ))) is the cost of a feasible tour for all players Q , which, since we are in Case 1, is larger than cQk (Q ), contradicting the assumption that ∆(e1 (Tγ )) = w k (S (e1 (Tγ ))) − x(Q ) < cQk (Q ) − x(Q ). Case 2. The tour corresponding to cQk (Q ) is not the cycle tour which traverses all edges in C .
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In this case, the cheapest k-centrum tour for Q takes the form S (ea ) + R(eb ) for some non-public edges ea and eb , and there are no non-public edges (i.e., edges in Q ) strictly between ea and eb . In Case 2 we follow Procedure 2 outlined below. Procedure 2 Step 0: For each edge e ∈ Q , let w ∗ (e) denote the total contribution of e to the cost cQk (Q ). Thus, w ∗ (e) is either 0, or w(e), or 2w(e). Let x(e) = w ∗ (e) for each such edge e and x(e) = 0 otherwise. Step 1: For each public path T contained in the tour of S (ea ) (resp., R(eb )), which contributes a total cost c (T ) to cQk (Q ), let e(T ) = e2 (T ) (resp., e(T ) = e1 (T )) denote the non-public edge immediately following T counterclockwise in S (ea ) (resp., clockwise in R(eb )). Increment x(e(T )) by c (T ). End. It is easy to show that the vector x derived by Procedure 2 in Case 2 is a core allocation vector for the subgame (Q , cQk ). We conclude that the k-centrum CP game induced by the cyclic graph C , with non-negative edges and wherein an arbitrary number of the edges is public, is CP balanced, which, in turn, implies that C is k-CP totally balanced. The result that a weakly cyclic graph G is k-CP totally balanced follows immediately from the result that every cyclic component of G is k-CP totally balanced, and that for every edge, e, which is not contained in a cycle in G (i.e., e is contained in a bridge), the cost that e contributes to c k (N ), if any, can be allocated to e itself, without violating core constraints, and the proof of Lemma 6 is complete. Combining Lemmas 5 and 6 we derive the following corollary: Corollary 7. Graph G is k-CP totally balanced for all k ≥ 3 if and only if G is weakly cyclic. Reflecting on Proposition 4 and Corollary 7, and recalling the corresponding results derived in Granot et al. [9], we recognize the effect of the introduction of the k-centrum optimization criterion on the CP-game model. Namely, while in the ordinary case, graphs G are CP submodular if and only if they are CP totally balanced if and only if they are weakly cyclic, the introduction of the k-centrum criterion eliminates this equivalence. Namely, under this criterion, for k ≥ 3, graph G is k-CP totally balanced if and only if it is weakly cyclic, while for k ≥ 3, the class of k-CP submodular graphs is a proper subset of the class of weakly cyclic graphs. Let us turn now to the characterization of graphs which are k-CP balanced for k ≥ 3. Recall that graph G is said to be k-CP balanced if the k-centrum CP game induced by G has a non-empty core for all non-negative edge-weights and for all locations of the post office. We show here that graph G is k-CP balanced if and only if none of its blocks contains an edgecutset whose cardinality is odd and less than k. A block is either a bi-connected component, or a graph consisting of a single bi-connected component. It is easy to show that a graph is k-CP balanced if and only if all its blocks are k-CP balanced, and that a bridge is k-CP balanced. Proposition 8. A block, G = (V (G), E (G)), that does not contain an edge-cutset whose cardinality is odd and less than k is k-CP balanced. Proof. If there are no odd edge-cutsets, the graph is Eulerian, and as such, it contains a closed walk that starts at the post office, traverses every edge in E (G) exactly once, and ends at the post office. Then clearly, c k (N ) = w(e1 )+w(e2 )+· · ·+w(enk ) where e1 , . . . , enk are the nk largest edges in E (G) such that w(e1 ) ≥ w(e2 ) ≥ · · · ≥ w(enk ) ≥ 0. Consider the allocation x(ei ) = w(ei ), x(ei ) = 0,
∀ i ≤ nk ∀ i > nk .
Such an allocation satisfies the relation ei ∈E (G) x(ei ) = c k (N ) and the constraints ei ∈S x(ei ) ≤ c k (S ), ∀S ⊂ E (G), and hence lies in the core. If the graph has odd edge-cutsets, then one of the edges in every one of those cuts must be traversed twice in order to complete the tour. But, we know that these odd edge-cutsets have a cardinality of at least k. Therefore, we can always traverse only the cheapest edge among the edges in the cut more than once without increasing the cost of the k-centrum tour. Hence, the allocation described above is in the core for this case as well.
Proposition 9. A block that contains an edge-cutset whose cardinality is odd and less than k is not k-CP balanced. Proof. Let the edges in this odd edge-cutset be e1 , e2 , . . . , e2l+1 , where 2l + 1 < k and w(e1 ) > w(e2 ) > · · · > w(e2l+1 ) > 0. Let this edge-cutset divide the graph G into two components, A (with the post office v0 in it), and B. Also set w(e) = 0 for all edges e not in the edge-cut.
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Suppose that x is a core element of (N , c k ). It is easy to see that all the players in A do not pay anything (i.e. x(A) = 0), because they all have zero edge-weights and can visit every edge in A at zero cost without crossing the edge-cutset. Now consider the subset of players S = {e2l+1 }∪ E (B). HS∗k traverses e2l+1 , visits all the edges in B and traverses back using e2l+1 as it is the cheapest edge in the edge-cutset. Therefore, x(e2l+1 ) + x(B) ≤ 2w(e2l+1 ). But, x(N ) = x(A) + x(e1 ) + · · · + x(e2l )+ x(e2l+1 )+ x(B) = w(e1 )+· · ·+w(e2l )+ 2w(e2l+1 ) and x(A) = 0. Therefore, x(e1 )+· · ·+ x(e2l ) ≥ w(e1 )+· · ·+w(e2l ). Consider another subset of players T = {e1 , e2 , . . . , e2l }∪ E (B). We can see that x(e1 )+· · ·+ x(e2l )+ x(B) ≤ w(e1 )+· · ·+ w(e2l ), which is only possible when x(B) = 0. Since x(A) = x(B) = 0, x(N ) = x(e1 ) + · · · + x(e2l+1 ). But we know that, x(e1 ) + x(e2 ) ≤ w(e1 ) + w(e2 ) x(e2 ) + x(e3 ) ≤ w(e2 ) + w(e3 )
... x(e2l+1 ) + x(e1 ) ≤ w(e2l+1 ) + w(e1 ). Summing up, we get, x(e1 ) + · · · + x(e2l+1 ) = x(N ) ≤ w(e1 ) + · · · + w(e2l+1 ) < c (N ), which implies that there exists no core allocation. This completes the proof. The results can be summarized as follows: Corollary 10. Graph G is k-CP balanced for all k ≥ 3 if and only if G is weakly Eulerian. Recalling that for ordinary CP games, G is CP balanced if and only if it is weakly Eulerian (Granot et al. [9]), we conclude from Corollary 10 that for k ≥ 3, the introduction of the k-centrum optimization criterion has not altered the class of graphs which induce CP games with a non-empty core for all non-negative edge-weights and all locations of the post office. 6. Globally and locally k-CP balanced, totally balanced and submodular graphs Following the lead of Herer and Penn [13] and Granot et al. [9], our k-CP balanced, totally balanced, and submodular graphs are required to induce, respectively, CP balanced, totally balanced and submodular games for all locations of the post office and all non-negative edge-weights. However, to the extent that the location of the post office in G can be advantageously chosen, requiring balancedness, total balancedness and submodularity to hold for all nodes of G may be a bit too restrictive. Accordingly, following Granot and Hamers [7], we will briefly consider in this section graphs G for which the k-centrum CP problem induces CP balanced, totally balanced, and submodular games for all non-negative edge weights and only for some location of the post office, and to which we will refer to, respectively, as k-CP locally balanced, locally totally balanced, and locally submodular graphs. By comparison, we will refer to the classes of graphs discussed in previous sections, where balancedness, total balancedness, and submodularity were required to hold for all node locations of the post office (and all non-negative edge-weights), as k-CP globally balanced, globally totally balanced and globally submodular graphs, respectively. Proposition 11. G is a k-CP globally balanced graph for all k ≥ 3 if and only if G is a k-CP locally balanced graph for all k ≥ 3. Proof. Clearly, if G is a k-CP globally balanced graph, it is also k-CP locally balanced. So, let G be a k-CP locally balanced graph, and assume, on the contrary, that it is not k-CP globally balanced. Then, by Corollary 10, G is not weakly Eulerian, which implies that at least one of the bi-connected components of G, say, M, is not Eulerian. Now, since M is not Eulerian, it must contain an odd edge-cutset, and using the same proof technique as in Proposition 9, we can demonstrate that G is not k-CP locally balanced for any location of the post office, which contradicts our assumption that G is k-CP locally balanced. Proposition 12. G is k-CP globally totally balanced for all k ≥ 3 if and only if G is k-CP locally totally balanced for all k ≥ 3. Proof. If G is k-CP globally totally balanced for all k ≥ 3 then, by definition, it must also be k-CP locally totally balanced for all k ≥ 3. Thus, by Corollary 7, the proof will follow if we show that graph G which is not weakly cyclic is not k-CP locally totally balanced for some k ≥ 3. So, suppose G is k-CP locally totally balanced, and assume, on the contrary, that G is not weakly cyclic. Then, there must exist three node-disjoint paths between two nodes, say, w1 and w2 , in G. We can then use exactly the same proof technique as in Granot and Hamers [7, Theorem 2.2], to demonstrate that for some edge-weights, the k-centrum CP game induced by G is not totally balanced for some k ≥ 3 and for all locations of the post office, contradicting our assumption that G is k-CP locally totally balanced. We will demonstrate below that, by contrast with the previous two cases, the class of k-CP locally submodular graphs properly contains the class of k-CP globally submodular graphs. Recall, from Proposition 4, that a k-CP globally submodular graph is either graph C3 , or a weakly cyclic graph wherein every cycle consists of two edges. Since graph G which is not weakly cyclic is not even k-CP locally totally balanced, as shown in the proof of Proposition 12, a k-CP locally submodular graph must be weakly cyclic. Further, it essentially follows from the proof of Proposition 4 that any weakly cyclic graph with a cycle of at least 4 edges is not k-CP locally submodular for some k ≥ 3. However, consider Fig. 2, wherein it is shown that
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the k-centrum CP game induced by graph G consisting of C3 with one appended edge is not submodular when the post office is located at node P. Then, it can be easily shown that if the post office is moved from node P to node B then the k-centrum CP game induced by G is submodular for all k. Indeed, it can be similarly easily shown that if G consists of a 1-sum of two C3 graphs, with a common vertex v , then G, with a post office at v , is k-CP locally submodular, and, in general, we derive the following characterization: Proposition 13. Graph G is k-CP locally submodular for all k ≥ 3 if and only if G is a 1-sum of its subgraphs G1 = (V1 , E1 ), G2 = (V2 , E2 ), . . . , Gn = (Vn , En ), such that each subgraph Gi is either a C3 graph or a weakly cyclic graph in which all cycles, if any, consist of two edges, ∩i=1,...,n V (Gi ) = {v0 }, and thus v0 is a node-cutset of G, and the post office is located at v0 . 7. k-CP submodularity and k-CP balancedness of directed graphs We study in this section k-centrum CP games induced by strongly connected directed graphs. Now, it can be easily verified that the proof of Proposition 1, which establishes that any connected undirected graph is 1-CP submodular, is also valid for strongly connected directed graphs. That is, the 1-centrum CP game induced by an arbitrary strongly connected directed graph is submodular for all non-negative arc-weights and all locations of the post office. However, Proposition 2 is not valid for directed graphs. Indeed: Example 14. Consider graph G = (V (G), E (G)) consisting of the cyclic graph on four vertices, directed, say, counterclockwise, and an additional edge. That is, V (G) = {v0 , v1 , v2 , v3 }, E (G) = {(v0 , v1 ), (v1 , v2 ), (v2 , v3 ), (v3 , v0 ), (v1 , v3 )}, the edge (v1 , v3 ) is directed from v1 to v3 , and the post office is located at v0 . Let S = {(v0 , v1 ), (v1 , v2 ), (v2 , v3 ), (v3 , v0 )}, T = {(v0 , v1 )}, and i = (v1 , v3 ). Suppose the weights are as follows: w((v1 , v2 )) = 100, w((v1 , v3 )) = 50, and the weight of any other arc is 10. Then, the cost of the directed 2-centrum CP tour in G for S is 110, for T , it is 60, for S ∪ {i}, it is 150, and for T ∪ {i}, it is 60. It follows that the associated 2-centrum CP game is not submodular. In fact, the associated k-centrum game is not submodular for all k ≥ 2. Note further that if the post office is moved from v0 to either v1 or v3 , the induced k-centrum CP game is not submodular for all k ≥ 2. However, if the location of the post office is moved from node v0 to v2 , then the associated k-centrum CP game is submodular for all k ≥ 2. Thus, using notation introduced in the previous section, graph G is not k-CP globally submodular for any k ≥ 2, but it is k-CP locally submodular with a post office located at v2 . Finally, if we replace edges (v1 , v2 ) and (v2 , v3 ) with a single edge (and thus eliminate node v2 ), then it can be easily shown that the resulting graph is not even k-CP locally submodular for all k ≥ 2. Let us denote the k-centrum CP game induced by a directed graph by (N , c¯ k ). A strongly connected directed graph is said to be weakly cyclic if each arc therein is contained in precisely one directed cycle, and it is said to be k-CP globally submodular if (N , c¯ k ) is submodular for all non-negative arc-weights and all locations of the post office. Proposition 15. A strongly connected directed graph G is k-CP globally submodular for all k ≥ 2 if and only if G is weakly cyclic. Proof. It is easy to show that if G is weakly cyclic then the induced k-centrum CP game (N , c¯ k ) is submodular for k ≥ 2, all non-negative arc-weights and all locations of the post office. On the other hand, suppose (N , c¯ k ) is submodular for k ≥ 2, all non-negative arc-weights, and all locations of the post office. If G is not weakly cyclic, then there exists an arc e ∈ E (G) which is not contained in precisely one directed cycle. Since G is strongly connected, e must be contained in at least one directed cycle, and we conclude that e is contained in two or more directed cycles. Then we can easily construct an example, similar to the one described in Example 14, to show that (N , c¯ k ) is not submodular even for k = 2, and the proof follows. Note that, by Example 14, there exist strongly connected directed graphs which are not weakly cyclic, but, nevertheless, are k-CP locally submodular for k ≥ 2. Thus, as was the case with undirected graphs, the class of k-CP locally submodular graphs properly contains the class of k-CP globally submodular graphs. Finally, consider balancedness of the k-centrum CP game (N , c¯ k ) induced by an arbitrary strongly connected directed graph, and the corresponding classes of k-CP globally and locally balanced graphs. Recall, from Granot et al. [9], that the ordinary CP game induced by an arbitrary connected directed graph is CP-balanced. The proof was shown therein to follow from Owen [16], and the availability of a linear production game formulation of the ordinary CP problem on a directed graph (Orloff [15]). Unfortunately, it appears that such formulation is not available for the k-centrum CP problem defined on a strongly connected directed graph, and we were unable to prove that the k-centrum CP game induced by an arbitrary strongly connected directed graph is always k-CP balanced. We conjecture, however, that this result holds, and therefore, we also conjecture that the class of k-CP globally balanced graphs coincides with the class k-CP locally balanced graphs. Acknowledgments The authors gratefully acknowledge helpful comments by an anonymous referee and by Professor Arie Tamir. Research was supported by Canada Natural Sciences and Engineering Research Council grants (3998 and 4181) and MITAC. References [1] C.G. Bird, On cost allocation for a spanning tree: a game theoretic approach, Networks 6 (1976) 35–50. [2] K. Das, Some algorithms on cactus graphs, Ann. Pure Appl. Math. 2 (2012) 114–128.
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Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam
The k-centrum Chinese Postman delivery problem and a related cost allocation game Daniel Granot a,∗ , Frieda Granot a , Harshavardhan Ravichandran b a
Sauder School of Business, University of British Columbia, Vancouver, B.C., Canada
b
Air New Zealand, Auckland, New Zealand
article
info
Article history: Received 28 October 2013 Received in revised form 18 July 2014 Accepted 20 July 2014 Available online 8 August 2014 Keywords: Chinese Postman problem Chinese Postman game k-centrum Chinese Postman problem Core Submodular game
abstract We analyze the cost allocation cooperative game, denoted by (N , c k ), induced by the kcentrum Chinese Postman (k-centrum CP) delivery problem in connected undirected and strongly connected directed graphs. For the undirected case, we show, for example, that for k = 1, 2, (N , c k ) is submodular for all graphs having non-negative edge-weights, for all locations of the post-office. For k ≥ 3, we prove that (N , c k ) is submodular for all nonnegative edge-weights and for all locations of the post office if and only if the graph is either the cyclic graph on three edges or it is a graph wherein each edge is contained in at most one cycle and these cycles, if any, have only two edges. For the directed graph case we show, for example, that the k-centrum CP game induced by a strongly connected graph G is submodular for all k ≥ 2 if and only if every arc in G is contained in precisely one directed cycle. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Cooperative game theory methodology was extensively applied to analyze cost and revenue allocation problems derived from classical optimization problems. Specifically, under the plausible assumption that the resources in these problems belong to different individuals (players), cooperative game theory was used to suggest fair allocations of the overall cost or revenue that the players can achieve by pooling together their resources. Thus, for example, a Linear Programming (LP) game, e.g., Owen [16], is derived from the classical LP problem by assuming that the right-hand-side vector belongs to different players; minimum cost spanning tree (mcst) games (e.g., Bird [1] and Granot and Huberman [10]) arise from the mcst problem by assuming that different players (e.g., communities) reside in the vertices of the graph, and Chinese Postman (CP) games (e.g., Hamers et al. [12]) arise from the classical CP problem where the cost of a complete tour by the postman has to be allocated among the different players who reside in the edges of the network. A cooperative game formulation of a cost allocation problem is given by the pair (N , c ), where N is the set of players, which, in the CP game, are the edges of the network, and c : 2N → R+ is the (coalitional) cost function, so that for each S ⊆ N, c (S ) is the cost incurred by coalition S if it operates alone. In the CP game, c (S ) is the cost of a minimal or cheapest tour that begins and ends at the post office (v0 ) and visits all the players (edges) in S at least once. The core of a game is the set of feasible allocations for which no coalition (i.e. is better off by seceding subset) of players from the rest of the group. That is, a vector x is in the core of the game (N , c ) if j∈N xj = c (N ) and j∈S xj ≤ c (S ), ∀S ⊂ N. A game with a non-empty core is said to be balanced. It follows from Shapley [18] that a submodular game has a non-empty
∗
Correspondence to: 2053 Main Mall, Vancouver, BC, Canada, V6T 1Z2. Tel.: +1 604 822 8432. E-mail addresses: [email protected] (D. Granot), [email protected] (F. Granot), [email protected] (H. Ravichandran). http://dx.doi.org/10.1016/j.dam.2014.07.021 0166-218X/© 2014 Elsevier B.V. All rights reserved.
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core, in addition to having other nice properties. A game is said to be submodular if and only if its characteristic function c is submodular. A function c is said to be submodular if and only if c (S ∪ {i}) − c (S ) ≤ c (T ∪ {i}) − c (T ), ∀i ∈ N , with T ⊆ S ⊆ N \ {i}, or, equivalently, if and only if c (S ∪ T ) + c (S ∩ T ) ≤ c (S ) + c (T ), ∀S , T ⊆ N. Finally, a game (N , c ) is said to be totally balanced if the core of every subgame (S , cS ) of (N , c ) is not empty, where cS is restricted to subsets of S. It is well known that submodular games are totally balanced. In general, a CP game can have an empty core. However, Hamers et al. [12] have shown that a CP game induced by a weakly Eulerian graph has a non-empty core, where a graph G = (V (G), E (G)) is called weakly Eulerian if its bi-connected components1 are Eulerian subgraphs. Further, Hamers [11] has shown that a CP game is submodular if it is induced by a weakly cyclic connected graph, where a graph is weakly cyclic if every edge therein is contained in at most one cycle. A graph G is called CP balanced (resp., totally balanced, submodular) if the CP game (N , c ) induced by G is balanced (resp., totally balanced, submodular) for all non-negative edge-weights and all locations of the post office node in G. Granot et al. [9] have characterized CP balanced, CP totally balanced, and CP submodular graphs. For example, it was shown therein that a connected undirected graph G is CP balanced (resp., CP submodular), if and only if G is weakly Eulerian (resp., weakly cyclic), and that the CP game induced by an arbitrary strongly connected directed graph is balanced for all non-negative arc-weights and all locations of the post office node in G. For related studies see, e.g., Granot and Hamers [7], Granot et al. [8] and Granot and Granot [6]. In this paper we study the cost allocation problem associated with the k-centrum Chinese Postman (k-centrum CP) problem, wherein we minimize the sum of the weights of the k largest edges in the tour. k-centrum optimization problems are optimization problems wherein only the largest k-terms in the objective function are optimized; see, e.g., Ogryczak and Tamir [14]. For k = 1, k-centrum optimization problems are referred to as bottleneck problems; see, e.g., Garfinkel and Rao [5] for the bottleneck transportation problem, or Gabow and Tarjan [3] for the bottleneck spanning tree problem in a directed graph and the bottleneck maximum cardinality matching problem. For the k-centrum shortest path problem for an arbitrary k, see, e.g., Garfinkel et al. [4], and for k-centrum optimization problems in location, see, e.g., Tamir [19]. In general, a k-centrum (i.e., k-sum) optimization problem can be solved in polynomial time whenever an associated minsum problem can be solved in polynomial time (Punnen and Aneja [17]). Let (N , c k ) denote the k-centrum CP game induced by a connected undirected graph G. We will refer to G as k-CP balanced (resp., totally balanced, submodular) if the k-centrum CP game induced by G is balanced (resp., totally balanced, submodular) for all non-negative edge-weights and all locations of the post office node. We prove in this paper that for k = 1, 2, (N , c k ) is submodular for all undirected connected graphs with non-negative edge-weights and for all locations of the post office, and that G is k-CP submodular for k ≥ 3 if and only if G is either the cyclic graph on three edges or G is a weakly cyclic graph wherein the cycles, if any, have two edges. Characterizations of k-CP balanced and k-CP totally balanced graphs are also provided. In the directed case we show, for example, that a strongly connected graph G is k-CP submodular for all k ≥ 2 if and only if each arc in G is contained in precisely one directed cycle. 2. Notation and definitions Let G = (V (G), E (G)) be an undirected (directed) graph where V (G) and E (G) denote the set of nodes and the set of edges (arcs) of G. A (directed) walk in G is a finite sequence of nodes and edges (arcs) of the form v1 , e1 , v2 , . . . , em , vm+1 with m ≥ 0, v1 , . . . , vm+1 ∈ V (G), e1 , . . . , em ∈ E (G), such that ej = {vj , vj+1 }(ej = (vj , vj+1 )) for all j ∈ {1, . . . , m}. Such a walk is said to be closed if v1 = vm+1 . A simple (directed) path in G is a (directed) walk in which all vertices (except, possibly v1 and vm+1 ) are distinct. A simple, closed (directed) path containing at least one edge (arc) is called a simple (directed) cycle. An undirected (directed) graph G is (strongly) connected if there is a (directed) path (from) between any node to any other node in G. A set of edges E¯ in (a connected graph) G is a (minimal) edge-cutset of G, if its removal, but not the removal of any proper subset thereof, disconnects G. A path, P, in G will be referred to as a bridge, if the degrees of all internal nodes of P in G are 2 and every edge in P is a (minimal) edge-cutset of G. We denote by v0 ∈ V (G) the post office in G. We define an S-tour w.r.t. v0 associated with S ⊆ E (G) as a closed (directed) walk that starts at v0 , traverses all the edges (arcs) in S at least once and ends in v0 . We can associate with an S-tour, an Eulerian multigraph HS which contains t copies of the edges (arcs) that are traversed t times in the S-tour, where t ∈ {0, 1, 2, . . . , |E (G)|}. The set of Eulerian multigraphs associated with the set S ⊆ E (G) is denoted by D(S ). Let w : E (G) → R+ be the travel cost function associated with the edges (arcs) of the graph. Then the k-centrum travel cost w k (HS ) of an S-tour (v0 , e1 , v1 , . . . , vr −1 , er , v0 ) is given by the sum of the nk most expensive edges (arcs) in the tour, where nk = min{r , k}. Formally, let τk = (E (G), (G, v0 ), w) denote the delivery problem, where E (G) is the set of players, (G, v0 ) is the undirected graph with v0 as the post office, and w is the edge travel cost function. Then the delivery game, denoted by (N , c k ), where N = E (G), corresponding to the delivery problem τk is defined for all S ⊆ E (G) by c k (S ) = min w k (HS ). HS ∈D(S )
We refer to the above delivery game as the k-centrum Chinese Postman (CP) game. The definition of the k-centrum CP game in the directed case is derived similarly and denoted as (N , c¯ k ). 1 A bi-connected component of a graph G is a maximal subgraph, Q , of G such that every pair of distinct edges in Q is contained in a simple cycle in Q .
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The multigraph that optimizes the above function for a certain value of k is denoted by HS∗k . It is important to note that HS contains 0, 1 or 2 copies of every edge in E (G). Clearly, c k is subadditive, i.e., c k (S ) + c k (T ) ≥ c k (S ∪ T ), ∀S , T ⊆ E (G) such that S ∩ T = ∅. Moreover, since w is non-negative, c k is also monotonic, i.e., c k (S ) ≤ c k (T ), ∀S ⊆ T ⊆ E (G). ∗k
3. Submodularity of the k-centrum CP game induced by undirected graphs We first consider in this section the cases when k = 1, 2, and prove that for these cases the k-centrum CP game, (N , c k ), k = 1, 2, is submodular for all connected undirected graphs with non-negative edge-weights and all locations of the post office. Note that when k = 1, the k-centrum CP problem is called the bottleneck CP problem. Proposition 1. Any connected graph is 1-CP submodular. 1 ′ Proof. Let i ∈ E (G) and S , T be such that T ⊆ S ⊆ E (G) \ {i}. Consider the walk H ′ defined as H ′ = HS∗1 + HT∗∪{ i} . Clearly, H
is a multigraph associated with a feasible tour for S ∪ {i}, i.e., H ′ ∈ D(S ∪ {i}). Therefore, c 1 (S ∪ {i}) ≤ w 1 (H ′ ), where w 1 (H ′ ) is the 1-centrum CP cost of H ′ . Now, as k = 1, w 1 (H ′ ) = max{c 1 (S ), c 1 (T ∪ {i})}. If w 1 (H ′ ) = c 1 (S ) then c 1 (S ∪{i})− c 1 (S ) ≤ w 1 (H ′ )− c 1 (S ) = 0. However, by monotonicity, c 1 (S ∪{i})− c 1 (S ) ≥ 0, which implies that c 1 (S ∪ {i}) − c 1 (S ) = 0. Since c 1 (T ∪ {i}) − c 1 (T ) ≥ 0, we derive that c 1 (S ∪ {i}) − c 1 (S ) ≤ c 1 (T ∪ {i}) − c 1 (T ). On the other hand, if w 1 (H ′ ) = c 1 (T ∪ {i}), then c 1 (S ∪ {i}) ≤ w 1 (H ′ ) = c 1 (T ∪ {i}), and since c 1 (S ) ≥ c 1 (T ) we conclude that c 1 (S ∪ {i}) − c 1 (S ) ≤ c 1 (T ∪ {i}) − c 1 (T ), and the proof is complete. Proposition 2. Any connected graph is 2-CP submodular. Proof. Let S , T ⊆ E (G), and let w : E (G) → R+ . If S ∩ T = ∅, then the subadditivity of c 2 implies its submodularity for S and T . Hence, we will only consider the case when S ∩ T ̸= ∅. Let HS∗2 and HT∗2 denote optimal 2-centrum S-tour and T -tour, respectively. For simplicity of presentation, we assume that all tours must contain at least 2 edges. Hence, we can express c 2 (S ) as c 2 (S ) = w(s1 ) + w(s2 ) where w(s1 ) ≥ w(s2 ) ≥ 0, i.e., s1 , s2 ∈ HS∗2 are the largest and second largest edges in HS∗2 , respectively. Similarly c 2 (T ) = w(t1 ) + w(t2 ), where w(t1 ) ≥ w(t2 ) ≥ 0. We assume, without loss of generality, that w(s1 ) ≥ w(t1 ). Now, consider H ′ = HS∗2 + HT∗2 . Since we have assumed that w(s1 ) ≥ w(t1 ), the 2-centrum CP cost, w 2 (H ′ ), of the tour associated with H ′ , satisfies w 2 (H ′ ) = w(s1 ) + max{w(t1 ), w(s2 )}. Case 1: w(s2 ) ≥ w(t1 ). Then, c 2 (S ∪ T ) ≤ w 2 (H ′ ) = w(s1 )+w(s2 ) = c 2 (S ), and since, by monotonicity, c 2 (S ∩ T ) ≤ c 2 (T ), we derive that c 2 (S ∪ T ) + c 2 (S ∩ T ) ≤ c 2 (S ) + c 2 (T ). Case 2: w(s1 ) ≥ w(t1 ) > w(s2 ). Since w(s1 ) > w(s2 ), the multiplicity of w1 in HS∗2 is exactly 1. Thus, s1 is not an edgecutset. Then, we claim that s1 ∈ S. Indeed, if s1 ̸∈ S and s1 is not an edge-cutset, then it is possible to construct an S-tour that does not pass through s1 and whose cost is at most 2w(s2 ), contradicting the optimality of HS∗2 . Subcase 2a: s1 ∈ S ∩ T . Then, s1 = t1 . If w(s2 ) ≥ w(t2 ), then w 2 (H ′ ) = w(s1 ) + w(s2 ), implying that c 2 (S ∪ T ) ≤ w(s1 ) + w(s2 ) = c 2 (S ). By monotonicity, c 2 (S ∩ T ) ≤ c 2 (T ), which implies that c 2 is submodular. On the other hand, if w(t2 ) > w(s2 ), w 2 (H ′ ) = w(t1 ) + w(t2 ), and thus, c 2 (S ∪ T ) ≤ w(t1 ) + w(t2 ) = c 2 (T ). Since c (S ∩ T ) ≤ c 2 (S ), we conclude that c 2 is submodular. Subcase 2b: s1 ̸∈ S ∩ T . Then, since w(t1 ) > w(s2 ), t1 ̸∈ S ∩ T and t1 ̸∈ HS∗2 . If w(t1 ) = w(t2 ), then t2 ̸∈ S ∩ T , t2 ̸∈ HS∗2 , and we have c 2 (S ∩ T ) ≤ c 2 (S \ {s1 }) ≤ 2w(s2 ) ≤ w(s2 ) + w(t2 ), c 2 (S ∪ T ) ≤ w(s1 ) + w(t1 ), which implies that c 2 (S ∪ T ) + c 2 (S ∩ T ) ≤ c 2 (S ) + c 2 (T ). If, on the other hand, w(t1 ) > w(t2 ), then we conclude that t1 is also not an edge-cutset, and therefore t1 ∈ T . We have, c 2 (S ∩ T ) ≤ min{c 2 (S \ {s1 }), c 2 (T \ {t1 })} ≤ min{2w(s2 ), 2w(t2 )} = 2 min{w(s2 ), w(t2 )} ≤ w(s2 ) + w(t2 ), c 2 (S ∪ T ) ≤ w(s1 ) + w(t1 ), which implies that c 2 (S ∪ T ) + c 2 (S ∩ T ) ≤ c 2 (S ) + c 2 (T ). We conclude that (N , c 2 ) is a submodular game for all connected undirected graphs, for all non-negative edge-weights, and for all locations of the post-office, and the proof is complete. We now demonstrate, in Example 3 below, that for k ≥ 3 the k-centrum CP game (N , c k ) may not be submodular. Example 3. Consider the graph drawn in Fig. 1, and suppose that w(e1 ) = 5, w(e2 ) = 6 and w(e3 ) = 7, and that player i resides in ei , i = 1, 2, 3. For the 3-centrum CP game (N , c 3 ) associated with Fig. 1 we have: c 3 ({1, 2, 3}) = 18, c 3 ({1, 3}) = 12, c 3 ({1, 2}) = 11 and c 3 ({1}) = 10, which can be easily shown to imply that c 3 is not submodular. In fact, since c k ({1, 2, 3}) = 23 for k > 3, it follows that (N , c k ) is not submodular for all k ≥ 3. Moreover, it can be easily shown that (N , c k ) is not even balanced for all k ≥ 4.
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Fig. 1. (N , c k ) is not submodular.
Fig. 2. A C3 with an edge attached.
4. k-CP submodular graphs for k ≥ 3 Recall that by Granot et al. [9], the ordinary CP game (N , c ) is CP-submodular if and only if (N , c ) is induced by a connected weakly cyclic graph, where a graph is weakly cyclic if each edge therein is contained in at most one cycle. This result should be contrasted with Propositions 1 and 2, in the previous section, and Proposition 4 below. Proposition 4. The k-centrum CP game induced by a connected graph G is submodular for all k ≥ 3 if and only if G is either the cyclic graph on three edges, denoted by C3 , or G is weakly cyclic and every cycle therein consists of two edges. Proof. It can be easily verified by inspection that C3 is k-CP submodular for all k ≥ 3. Next, let G be a weakly cyclic graph whose cyclic subgraphs, if any, are C2 , i.e., consist of two edges, and consider i, S , T such that T ⊆ S ⊆ N \ {i}. We first show that HT∗k ⊆ HS∗k , i.e., all the edges in HT∗k , the multigraph corresponding to c k (T ), can be assumed to be also present in HS∗k . Clearly, all edges in HT∗k which are not contained in any cycle in G are also contained in HS∗k . Consider any subgraph, C2 , of G, with edges a and b and assume, without loss of generality, that HS∗k contains a with a multiplicity of 2 (as the case when a and b both are present in HS∗k with a multiplicity of 1 is trivial). If a does not contribute towards w k (HS∗k ), and w(a) ≤ w(b), then HT∗k can be assumed to contain a with multiplicity 2, and if w(a) > w(b) then there exists an alternate optimum where HS∗k traverses both a and b once, and then HT∗k ⊆ HS∗k . If, on the other hand, a does contribute to the cost w k (HS∗k ), then clearly w(a) ≤ w(b). This implies that HT∗k must also contain a with multiplicity 2, and thus, we conclude that HT∗k ⊆ HS∗k . Therefore, c k (S ∪ {i}) − c k (S ) ≤ c k (T ∪ {i}) − c k (T ), and thus, G is k-CP submodular for all k. To prove the other direction, consider first the graph shown in Fig. 2. Let S = {PA, AB, BC }, T = {BC } and i = PB. Clearly, T ⊆ S ⊆ N \ {i}. For k = 3, HS∗3 takes the route P–A–B–C–B–A–P with c 3 (S ) = 12 and HT∗3 takes the route P–B–C–B–P with c 3 (T ) = 11. We see that edge PB lies in HT∗k but not in HS∗k . Therefore, the marginal cost of adding PB to S is greater than that of adding it to T . Hence, C3 with even one edge attached to one of its nodes is not submodular for k = 3. One could similarly show that the 3-centrum CP game induced by a 1-sum of two C3 cycles is not necessarily submodular, where 1-sum of two graphs G1 and G2 is obtained by coalescing one vertex in G1 with another in G2 . Finally, consider a graph G which is a cycle of length m. Let the cycle be defined by v1 , e1 , v2 , e2 , . . . , vm , em , v1 and let v1 be the post office. Set w(e1 ) = w(e2 ) = · · · = w(em−2 ) = 10, w(em−1 ) = 1 and w(em ) = 11. Also, let S = {e1 , e2 , . . . , em−1 }, T = {em−1 } and i = em . Clearly, T ⊆ S ⊆ N \ {i}. Consider the case when k = m − 1. Clearly, HS∗k takes the path e1 , e2 , . . . , em−2 , em−1 , em−1 , em−2 , . . . , e2 , e1 with a cost of c k (S ) = 10k. However, HT∗k takes the path em , em−1 , em−1 , em with a cost of c k (T ) = 23 for k = 3, and c k (T ) = 24 for all k ≥ 4. Again we see that edge i lies in HT∗k but not in HS∗k , and therefore, from the marginal cost definition of submodularity, it follows that the k-centrum CP game induced by G is not submodular. We can conclude that the k-centrum CP game induced by a weakly cyclic graph G that is not a tree, or C3 , or a weakly cyclic graph with only cycles of size 2, is not submodular for some k, and the proof is complete. 5. k-CP total balancedness and k-CP balancedness For k ∈ {1, 2}, any graph is k-CP submodular, and thus it is trivially k-CP totally balanced and k-CP balanced. We prove in this section that graph G is k-CP totally balanced for all k ≥ 3 if and only if G is weakly cyclic. Recall that a graph G is weakly cyclic if and only if every edge in E (G) belongs to at most one cycle. Alternatively, G is weakly cyclic iff every bi-connected component is a cycle. Hence, it can be seen that every weakly cyclic graph can be constructed by joining simple cycles together. These cycles can either have a node in common or they can have a bridge (line graph) in between them. A weakly cyclic graph is also called a cactus graph (see, e.g., Das [2] for a recent reference). Lemma 5. If G is not weakly cyclic, then there exists a k for which the k-CP game induced by it is not k-CP totally balanced.
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Proof. Consider an undirected connected graph G which is not weakly cyclic. As trees are, by definition, weakly cyclic, G cannot be a tree, and hence, G is not acyclic. Therefore, there exist bi-connected components in G, and since G is not weakly cyclic, at least one of those components is not a cycle. In that component, there must exist a pair of nodes, say u and v , with at least three edge-disjoint paths between them. Then, as in Example 3, we could choose, say, u as the post office and introduce edge-weights which would result with a k-CP game defined on a subgraph of this component whose core is empty, implying that G is not k-CP totally balanced. Lemma 6. If G is weakly cyclic, then the k-CP game induced by G is totally balanced for every k. Proof. We first show that the cyclic graph, C , is k-CP totally balanced, which will follow if we show that C is k-CP balanced for an arbitrary number of public edges and arbitrary non-negative edge-weights. Let C = (V (C ), E (C )) be a cyclic graph of arbitrary size with non-negative edge-weights in which some of the edges are public. Denote by T a public path consisting of contiguous public edges. If exists, let e1 (T ) (resp., e2 (T )) denote the non-public edge which, counter-clockwise from the post office, is just before (resp., after) the public path T . Further, for each non-public edge e, let S (e) (resp., R(e)) denote the coalition of all players including (the one residing at) e, which are counter-clockwise (resp., clockwise) between the post office and e. Let w k (S (e)) (resp., w k (R(e))) denote the k-centrum cost of these coalitions when the tours they employ consist of a counter-clockwise (resp., clockwise) walk from the post office to e and returning clockwise (resp., counter-clockwise) to the post office. Let Q be an arbitrary subset of N = E (C ). We will show that the subgame, (Q , cQk ), of the k-centrum CP game (N , c k ) induced by C is balanced. For that purpose, we regard the edges in N \ Q as public. Case 1. The tour corresponding to cQk (Q ) is the cycle tour which traverses all edges in C . Let T1 , . . . , Tγ denote the public paths in C which contribute to cQk (Q ). Assume that these paths are arranged in a counterclockwise manner from the post office. Thus, T1 is, counter-clockwise, the closest such public path to the post office. Procedure 1 below constructs a core vector for the subgame (Q , cQk ) in Case 1. Procedure 1 Step 0: For every edge e ∈ Q , if the cost w(e) enters the calculation of cQk (Q ), set x(e) = w(e). For all other e ∈ Q set x(e) = 0. Let x = (x(e)). Clearly x(Q ) = Σ (x(e) : e ∈ Q ) ≤ cQk (Q ).
Step 1: For Stages j = 1, . . . , γ , for i = 1, 2, if ei (Tj ) exists, let ∆(ei (Tj )) = min{w k (S (ei (Tj ))) − x(S (ei (Tj ))), cQk (Q ) − x(Q )}. Increment x(ei (Tj )) by
∆(ei (Tj )). If ∆(ei (Tj )) = cQk (Q ) − x(Q ), then stop. End.
We claim that the vector x produced by Procedure 1 is a core vector for the k-CP subgame (Q , cQk ) induced by C . Indeed, as far as violations of proper core constraints, it is sufficient to show that the coalitions of the form S (e) and R(e), for all nonpublic edges e, are not violated by x. Now, by the incrementation carried out in Step 1, we ensure that coalitions of the form S (e) are not violated by x. Next, suppose, on the contrary, that for j = l, the incrementation at Step 1, wherein, say, the allocation to e1 (Tl ) has been increased, resulted in a violation for coalition R(e1 (Tl )). Thus, if x denotes the vector after x(e1 (Tl )) was incremented, then x(R(e1 (Tl ))) > cQk (R(e1 (Tl ))). However, consider stage l − 1 in Step 1 of Procedure 1, and assume,
for simplicity of exposition, that e2 (Tl−1 ) exists. Then, by Step 1, x(S (e2 (Tl−1 ))) = w k (S (e2 (Tl−1 ))). Denote by A the set of all players residing in non-public edges, located strictly between e2 (Tl−1 ) and e1 (Tl ), and whose cost has entered cQk (Q ). Then,
Σ (x(e) : e ∈ A) = Σ (w(e) : e ∈ A), and x(Q ) > cQk (R(e1 (Tl ))) + Σ (w(e) : e ∈ A) + w k (S (e2 (Tl−1 ))) > cQk (Q ), implying that cQk (Q ) − x(Q ) < 0, which, by Step 1, cannot happen. A similar conclusion is reached if we consider a violation which could possibly occur when x(e2 (Tl )) was incremented. Thus, none of the coalitions of the form R(e) is violated by x. Finally, we need to show that x(Q ) = cQk (Q ). Suppose Step 1 terminated at Stage γ . If at this stage, for some i, ∆(ei (Tγ )) = cQk (Q ) − x(Q ), we are done. Otherwise, suppose, on the contrary, that ∆(ei (Tγ )) < cQk (Q ) − x(Q ), i = 1, 2.
• If e2 (Tγ ) exists, let R∗ ≡ R(e2 (Tγ ) \ {e2 (Tγ )}). If R∗ = ∅, regard x(R∗ ) as 0. Then, at this stage we have x(Q ) = x(R∗ ) + x(S (e2 (Tγ ))) = Σ (w(e) : e ∈ R∗ ) + w k (S (e2 (Tγ ))) > cQk (Q ), contradicting the assumption that ∆(ei (Tγ )) < cQk (Q ) − x(Q ), i = 1, 2. • If e2 (Tγ ) does not exist, then wk (S (e1 (Tγ ))) is the cost of a feasible tour for all players Q , which, since we are in Case 1, is larger than cQk (Q ), contradicting the assumption that ∆(e1 (Tγ )) = w k (S (e1 (Tγ ))) − x(Q ) < cQk (Q ) − x(Q ). Case 2. The tour corresponding to cQk (Q ) is not the cycle tour which traverses all edges in C .
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In this case, the cheapest k-centrum tour for Q takes the form S (ea ) + R(eb ) for some non-public edges ea and eb , and there are no non-public edges (i.e., edges in Q ) strictly between ea and eb . In Case 2 we follow Procedure 2 outlined below. Procedure 2 Step 0: For each edge e ∈ Q , let w ∗ (e) denote the total contribution of e to the cost cQk (Q ). Thus, w ∗ (e) is either 0, or w(e), or 2w(e). Let x(e) = w ∗ (e) for each such edge e and x(e) = 0 otherwise. Step 1: For each public path T contained in the tour of S (ea ) (resp., R(eb )), which contributes a total cost c (T ) to cQk (Q ), let e(T ) = e2 (T ) (resp., e(T ) = e1 (T )) denote the non-public edge immediately following T counterclockwise in S (ea ) (resp., clockwise in R(eb )). Increment x(e(T )) by c (T ). End. It is easy to show that the vector x derived by Procedure 2 in Case 2 is a core allocation vector for the subgame (Q , cQk ). We conclude that the k-centrum CP game induced by the cyclic graph C , with non-negative edges and wherein an arbitrary number of the edges is public, is CP balanced, which, in turn, implies that C is k-CP totally balanced. The result that a weakly cyclic graph G is k-CP totally balanced follows immediately from the result that every cyclic component of G is k-CP totally balanced, and that for every edge, e, which is not contained in a cycle in G (i.e., e is contained in a bridge), the cost that e contributes to c k (N ), if any, can be allocated to e itself, without violating core constraints, and the proof of Lemma 6 is complete. Combining Lemmas 5 and 6 we derive the following corollary: Corollary 7. Graph G is k-CP totally balanced for all k ≥ 3 if and only if G is weakly cyclic. Reflecting on Proposition 4 and Corollary 7, and recalling the corresponding results derived in Granot et al. [9], we recognize the effect of the introduction of the k-centrum optimization criterion on the CP-game model. Namely, while in the ordinary case, graphs G are CP submodular if and only if they are CP totally balanced if and only if they are weakly cyclic, the introduction of the k-centrum criterion eliminates this equivalence. Namely, under this criterion, for k ≥ 3, graph G is k-CP totally balanced if and only if it is weakly cyclic, while for k ≥ 3, the class of k-CP submodular graphs is a proper subset of the class of weakly cyclic graphs. Let us turn now to the characterization of graphs which are k-CP balanced for k ≥ 3. Recall that graph G is said to be k-CP balanced if the k-centrum CP game induced by G has a non-empty core for all non-negative edge-weights and for all locations of the post office. We show here that graph G is k-CP balanced if and only if none of its blocks contains an edgecutset whose cardinality is odd and less than k. A block is either a bi-connected component, or a graph consisting of a single bi-connected component. It is easy to show that a graph is k-CP balanced if and only if all its blocks are k-CP balanced, and that a bridge is k-CP balanced. Proposition 8. A block, G = (V (G), E (G)), that does not contain an edge-cutset whose cardinality is odd and less than k is k-CP balanced. Proof. If there are no odd edge-cutsets, the graph is Eulerian, and as such, it contains a closed walk that starts at the post office, traverses every edge in E (G) exactly once, and ends at the post office. Then clearly, c k (N ) = w(e1 )+w(e2 )+· · ·+w(enk ) where e1 , . . . , enk are the nk largest edges in E (G) such that w(e1 ) ≥ w(e2 ) ≥ · · · ≥ w(enk ) ≥ 0. Consider the allocation x(ei ) = w(ei ), x(ei ) = 0,
∀ i ≤ nk ∀ i > nk .
Such an allocation satisfies the relation ei ∈E (G) x(ei ) = c k (N ) and the constraints ei ∈S x(ei ) ≤ c k (S ), ∀S ⊂ E (G), and hence lies in the core. If the graph has odd edge-cutsets, then one of the edges in every one of those cuts must be traversed twice in order to complete the tour. But, we know that these odd edge-cutsets have a cardinality of at least k. Therefore, we can always traverse only the cheapest edge among the edges in the cut more than once without increasing the cost of the k-centrum tour. Hence, the allocation described above is in the core for this case as well.
Proposition 9. A block that contains an edge-cutset whose cardinality is odd and less than k is not k-CP balanced. Proof. Let the edges in this odd edge-cutset be e1 , e2 , . . . , e2l+1 , where 2l + 1 < k and w(e1 ) > w(e2 ) > · · · > w(e2l+1 ) > 0. Let this edge-cutset divide the graph G into two components, A (with the post office v0 in it), and B. Also set w(e) = 0 for all edges e not in the edge-cut.
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Suppose that x is a core element of (N , c k ). It is easy to see that all the players in A do not pay anything (i.e. x(A) = 0), because they all have zero edge-weights and can visit every edge in A at zero cost without crossing the edge-cutset. Now consider the subset of players S = {e2l+1 }∪ E (B). HS∗k traverses e2l+1 , visits all the edges in B and traverses back using e2l+1 as it is the cheapest edge in the edge-cutset. Therefore, x(e2l+1 ) + x(B) ≤ 2w(e2l+1 ). But, x(N ) = x(A) + x(e1 ) + · · · + x(e2l )+ x(e2l+1 )+ x(B) = w(e1 )+· · ·+w(e2l )+ 2w(e2l+1 ) and x(A) = 0. Therefore, x(e1 )+· · ·+ x(e2l ) ≥ w(e1 )+· · ·+w(e2l ). Consider another subset of players T = {e1 , e2 , . . . , e2l }∪ E (B). We can see that x(e1 )+· · ·+ x(e2l )+ x(B) ≤ w(e1 )+· · ·+ w(e2l ), which is only possible when x(B) = 0. Since x(A) = x(B) = 0, x(N ) = x(e1 ) + · · · + x(e2l+1 ). But we know that, x(e1 ) + x(e2 ) ≤ w(e1 ) + w(e2 ) x(e2 ) + x(e3 ) ≤ w(e2 ) + w(e3 )
... x(e2l+1 ) + x(e1 ) ≤ w(e2l+1 ) + w(e1 ). Summing up, we get, x(e1 ) + · · · + x(e2l+1 ) = x(N ) ≤ w(e1 ) + · · · + w(e2l+1 ) < c (N ), which implies that there exists no core allocation. This completes the proof. The results can be summarized as follows: Corollary 10. Graph G is k-CP balanced for all k ≥ 3 if and only if G is weakly Eulerian. Recalling that for ordinary CP games, G is CP balanced if and only if it is weakly Eulerian (Granot et al. [9]), we conclude from Corollary 10 that for k ≥ 3, the introduction of the k-centrum optimization criterion has not altered the class of graphs which induce CP games with a non-empty core for all non-negative edge-weights and all locations of the post office. 6. Globally and locally k-CP balanced, totally balanced and submodular graphs Following the lead of Herer and Penn [13] and Granot et al. [9], our k-CP balanced, totally balanced, and submodular graphs are required to induce, respectively, CP balanced, totally balanced and submodular games for all locations of the post office and all non-negative edge-weights. However, to the extent that the location of the post office in G can be advantageously chosen, requiring balancedness, total balancedness and submodularity to hold for all nodes of G may be a bit too restrictive. Accordingly, following Granot and Hamers [7], we will briefly consider in this section graphs G for which the k-centrum CP problem induces CP balanced, totally balanced, and submodular games for all non-negative edge weights and only for some location of the post office, and to which we will refer to, respectively, as k-CP locally balanced, locally totally balanced, and locally submodular graphs. By comparison, we will refer to the classes of graphs discussed in previous sections, where balancedness, total balancedness, and submodularity were required to hold for all node locations of the post office (and all non-negative edge-weights), as k-CP globally balanced, globally totally balanced and globally submodular graphs, respectively. Proposition 11. G is a k-CP globally balanced graph for all k ≥ 3 if and only if G is a k-CP locally balanced graph for all k ≥ 3. Proof. Clearly, if G is a k-CP globally balanced graph, it is also k-CP locally balanced. So, let G be a k-CP locally balanced graph, and assume, on the contrary, that it is not k-CP globally balanced. Then, by Corollary 10, G is not weakly Eulerian, which implies that at least one of the bi-connected components of G, say, M, is not Eulerian. Now, since M is not Eulerian, it must contain an odd edge-cutset, and using the same proof technique as in Proposition 9, we can demonstrate that G is not k-CP locally balanced for any location of the post office, which contradicts our assumption that G is k-CP locally balanced. Proposition 12. G is k-CP globally totally balanced for all k ≥ 3 if and only if G is k-CP locally totally balanced for all k ≥ 3. Proof. If G is k-CP globally totally balanced for all k ≥ 3 then, by definition, it must also be k-CP locally totally balanced for all k ≥ 3. Thus, by Corollary 7, the proof will follow if we show that graph G which is not weakly cyclic is not k-CP locally totally balanced for some k ≥ 3. So, suppose G is k-CP locally totally balanced, and assume, on the contrary, that G is not weakly cyclic. Then, there must exist three node-disjoint paths between two nodes, say, w1 and w2 , in G. We can then use exactly the same proof technique as in Granot and Hamers [7, Theorem 2.2], to demonstrate that for some edge-weights, the k-centrum CP game induced by G is not totally balanced for some k ≥ 3 and for all locations of the post office, contradicting our assumption that G is k-CP locally totally balanced. We will demonstrate below that, by contrast with the previous two cases, the class of k-CP locally submodular graphs properly contains the class of k-CP globally submodular graphs. Recall, from Proposition 4, that a k-CP globally submodular graph is either graph C3 , or a weakly cyclic graph wherein every cycle consists of two edges. Since graph G which is not weakly cyclic is not even k-CP locally totally balanced, as shown in the proof of Proposition 12, a k-CP locally submodular graph must be weakly cyclic. Further, it essentially follows from the proof of Proposition 4 that any weakly cyclic graph with a cycle of at least 4 edges is not k-CP locally submodular for some k ≥ 3. However, consider Fig. 2, wherein it is shown that
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the k-centrum CP game induced by graph G consisting of C3 with one appended edge is not submodular when the post office is located at node P. Then, it can be easily shown that if the post office is moved from node P to node B then the k-centrum CP game induced by G is submodular for all k. Indeed, it can be similarly easily shown that if G consists of a 1-sum of two C3 graphs, with a common vertex v , then G, with a post office at v , is k-CP locally submodular, and, in general, we derive the following characterization: Proposition 13. Graph G is k-CP locally submodular for all k ≥ 3 if and only if G is a 1-sum of its subgraphs G1 = (V1 , E1 ), G2 = (V2 , E2 ), . . . , Gn = (Vn , En ), such that each subgraph Gi is either a C3 graph or a weakly cyclic graph in which all cycles, if any, consist of two edges, ∩i=1,...,n V (Gi ) = {v0 }, and thus v0 is a node-cutset of G, and the post office is located at v0 . 7. k-CP submodularity and k-CP balancedness of directed graphs We study in this section k-centrum CP games induced by strongly connected directed graphs. Now, it can be easily verified that the proof of Proposition 1, which establishes that any connected undirected graph is 1-CP submodular, is also valid for strongly connected directed graphs. That is, the 1-centrum CP game induced by an arbitrary strongly connected directed graph is submodular for all non-negative arc-weights and all locations of the post office. However, Proposition 2 is not valid for directed graphs. Indeed: Example 14. Consider graph G = (V (G), E (G)) consisting of the cyclic graph on four vertices, directed, say, counterclockwise, and an additional edge. That is, V (G) = {v0 , v1 , v2 , v3 }, E (G) = {(v0 , v1 ), (v1 , v2 ), (v2 , v3 ), (v3 , v0 ), (v1 , v3 )}, the edge (v1 , v3 ) is directed from v1 to v3 , and the post office is located at v0 . Let S = {(v0 , v1 ), (v1 , v2 ), (v2 , v3 ), (v3 , v0 )}, T = {(v0 , v1 )}, and i = (v1 , v3 ). Suppose the weights are as follows: w((v1 , v2 )) = 100, w((v1 , v3 )) = 50, and the weight of any other arc is 10. Then, the cost of the directed 2-centrum CP tour in G for S is 110, for T , it is 60, for S ∪ {i}, it is 150, and for T ∪ {i}, it is 60. It follows that the associated 2-centrum CP game is not submodular. In fact, the associated k-centrum game is not submodular for all k ≥ 2. Note further that if the post office is moved from v0 to either v1 or v3 , the induced k-centrum CP game is not submodular for all k ≥ 2. However, if the location of the post office is moved from node v0 to v2 , then the associated k-centrum CP game is submodular for all k ≥ 2. Thus, using notation introduced in the previous section, graph G is not k-CP globally submodular for any k ≥ 2, but it is k-CP locally submodular with a post office located at v2 . Finally, if we replace edges (v1 , v2 ) and (v2 , v3 ) with a single edge (and thus eliminate node v2 ), then it can be easily shown that the resulting graph is not even k-CP locally submodular for all k ≥ 2. Let us denote the k-centrum CP game induced by a directed graph by (N , c¯ k ). A strongly connected directed graph is said to be weakly cyclic if each arc therein is contained in precisely one directed cycle, and it is said to be k-CP globally submodular if (N , c¯ k ) is submodular for all non-negative arc-weights and all locations of the post office. Proposition 15. A strongly connected directed graph G is k-CP globally submodular for all k ≥ 2 if and only if G is weakly cyclic. Proof. It is easy to show that if G is weakly cyclic then the induced k-centrum CP game (N , c¯ k ) is submodular for k ≥ 2, all non-negative arc-weights and all locations of the post office. On the other hand, suppose (N , c¯ k ) is submodular for k ≥ 2, all non-negative arc-weights, and all locations of the post office. If G is not weakly cyclic, then there exists an arc e ∈ E (G) which is not contained in precisely one directed cycle. Since G is strongly connected, e must be contained in at least one directed cycle, and we conclude that e is contained in two or more directed cycles. Then we can easily construct an example, similar to the one described in Example 14, to show that (N , c¯ k ) is not submodular even for k = 2, and the proof follows. Note that, by Example 14, there exist strongly connected directed graphs which are not weakly cyclic, but, nevertheless, are k-CP locally submodular for k ≥ 2. Thus, as was the case with undirected graphs, the class of k-CP locally submodular graphs properly contains the class of k-CP globally submodular graphs. Finally, consider balancedness of the k-centrum CP game (N , c¯ k ) induced by an arbitrary strongly connected directed graph, and the corresponding classes of k-CP globally and locally balanced graphs. Recall, from Granot et al. [9], that the ordinary CP game induced by an arbitrary connected directed graph is CP-balanced. The proof was shown therein to follow from Owen [16], and the availability of a linear production game formulation of the ordinary CP problem on a directed graph (Orloff [15]). Unfortunately, it appears that such formulation is not available for the k-centrum CP problem defined on a strongly connected directed graph, and we were unable to prove that the k-centrum CP game induced by an arbitrary strongly connected directed graph is always k-CP balanced. We conjecture, however, that this result holds, and therefore, we also conjecture that the class of k-CP globally balanced graphs coincides with the class k-CP locally balanced graphs. Acknowledgments The authors gratefully acknowledge helpful comments by an anonymous referee and by Professor Arie Tamir. Research was supported by Canada Natural Sciences and Engineering Research Council grants (3998 and 4181) and MITAC. References [1] C.G. Bird, On cost allocation for a spanning tree: a game theoretic approach, Networks 6 (1976) 35–50. [2] K. Das, Some algorithms on cactus graphs, Ann. Pure Appl. Math. 2 (2012) 114–128.
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