Angewandte Mathematik und Informatik Universit?at zu K ? oln Report No. 93.137 On approximately fair cost allocation in Euclidean TSP games by U. Faigle , S. Fekete , W. Hochst?attler , W. Kern 1996 to appear in : Spektrum of OR Ulrich Faigle Walter Kern Department of Applied Mathematics University of Twente P.O.Box 217 NL-7500 AE Enschede The Netherlands S?andor P. Fekete Winfried Hochst?attler ZPR , Zentrum f?ur paralleles Rechnen Universit?at zu K?oln Albertus-Magnus-Platz D-50923 K?oln Germany Telefon -LRB- 0221 -RRB- 470 -LCB- 6024 -LRB- 6023 -RRB- e -LCB- Mail SANDOR@ZPR.UNI-KOELN.DE -LRB- WH@ZPR ... -RRB- On approximately fair cost allocation in Euclidean TSP games Ulrich Faigle University of Twente Twente , Netherlands S?andor P. Fekete ? Mathematisches Institut Universit?at zu K?oln 50923 K?oln , Germany Winfried Hochst?attler y Mathematisches Institut Universit?at zu K?oln 50923 K?oln , Germany Walter Kern University of Twente Twente , Netherlands ? Parts of this work were done while the author was staying at the Department of Applied Mathematics and Statistics , SUNY Stony Brook , NY 11794 -LCB- 3600 , during a visit at RUTCOR , Rutgers Univeristy , and during a stay at Tel Aviv University ; support by NSF Grants ECSE-8857642 and CCR-9204585 . yParts of this work were done while the author was visiting at RUTCOR , Rutgers University , supported by SFB 303 -LRB- Deutsche Forschungsgemeinschaft -RRB- . Euclidean TSP Games 2 Abstract : We consider the problem of allocating the cost of an optimal traveling salesman tour in a fair way among the nodes visited ; in particular , we focus on the case where the distance matrix of the underlying TSP problem satisfies the triangle inequality . We thereby use the model of TSP games in the sense of cooperative game theory . We give examples showing that the core of such games may be empty , even for the case of Euclidean distances . On the positive , we develop an LP-based allocation rule guaranteeing that no coalition pays more than times its own cost , where is the ratio between the optimal TSP-tour and the optimal value of its Held-Karp relaxation , which is also known as the solution over the \ subtour polytope '' . A well known conjecture states that < = 43 . We also exhibit examples showing that this ratio can not be improved below 43 . Zusammenfassung : Wir betrachten die Aufgabe , die Kosten einer optimalen Traveling-Salesman-Tour fair unter den besuchten Knoten zu verteilen ; insbesondere untersuchen wir den Fall , dass die Kostenmatrix des zugrunde liegenden TSP-Problems die Dreiecksungleichung erf?ullt . Dazu wird das Modell von TSP-Spielen im Sinne der kooperativen Spieltheorie benutzt . Wir zeigen anhand eines Beispiels , dass der Core eines solchen Spiels leer sein kann , selbst im Falle euklidischer Distanzen . Andererseits geben wir eine LP-basierte Verteilungsregel an , die garantiert , dass keine Koalition mehr als das - fache ihrer eigenen Kosten bezahlen muss , wobei das Verh?altnis zwischen den Kosten einer optimalen TSP-Tour und dem Optimum der Held-Karp-Relaxation ist , die auch als L?osung ? uber dem \ subtour polytope '' bekannt ist . Es wird allgemein vermutet , dass < = 43 . Abschliessend geben wir eine Klasse von Beispielen an , die beweist , dass keine allgemeine Verteilungsregel f?ur das TSP-game ein generell besseres Verh?altnis zwischen der Belastung einer Koalition und Ihren Kosten garantieren kann , wenn obige Vermutung gilt . Euclidean TSP Games 3 1 Introduction Traveling salesman games -LRB- \ TSP-games '' -RRB- are examples of games in the sense of cooperative game theory , i : e : , mathematical models for cost allocation among individuals in a set N of so-called players that may form coalitions S ae N in order to achieve a common benefit or to provide a service , thereby incurring a cost c -LRB- S -RRB- . In this setting , we define a cooperative game to be a pair -LRB- N ; c -RRB- , where N is a -LRB- finite -RRB- set of players and c : 2N ! < the characteristic function of the game that assigns to every coalition S ? N its cost c -LRB- S -RRB- . N is often called the grand coalition of the game . Example 1.1 -LRB- Minimum Cost Spanning Tree Game -RRB- : Let N = f1 ; 2 ; : : : ; ng and N0 : = N -LSB- f0g and consider the complete undirected graph KN0 with N0 as its the of nodes . Each edge ij , i ; j 2 N0 of KN0 is weighted with a non-negative real number dij . For every S ? N , let c -LRB- S -RRB- = minimum cost of a tree spanning all nodes in S -LSB- f0g : Minimum cost spanning tree games -LRB- \ MCST-games '' -RRB- have received much attention in the literature -LRB- see , e : g : , -LSB- 1 -RSB- , -LSB- 2 -RSB- , or -LSB- 13 -RSB- -RRB- . They arise , for example , in the following modeling context : node represents a supply node for electricity , say . The other nodes -LRB- \ players '' -RRB- i 2 N are sites that are to be supplied with electricity . The weight dij reflects the cost of establishing a direct link between nodes i and j . So c -LRB- S -RRB- is the minimal cost the coalition S would incur upon linking itself to without using links to N n S that may already be established . The game theoretic model of central interest here was introduced by Potters et al. -LSB- 25 -RSB- and is based on the same underlying combinatorial structure as the model of a MCST-game . The difference lies in the way the cost of a coalition is assessed . Example 1.2 -LRB- Traveling Salesman Game -RRB- : Let N , N0 , and dij be given as in Example 1.1 and define for every coalition S ? N , c -LRB- S -RRB- = minimum length of a traveling salesman tour visiting all nodes in S -LSB- f0g : A modeling context for Example 1.2 can be the following : node is a server for a new local area network in a larger company . The players in N stand for different departments within the company , acting on independent budgets . c -LRB- S -RRB- is the cost of connecting all departments in S to the server in a ring structure -LRB- token ring or FDDI , say -RRB- . Given a cooperative game -LRB- N ; c -RRB- , one faces the computational problem of finding an allocation vector x 2 < N that is fair in a mathematically well-defined way . A set of mathematical criteria for fairness of an allocation is a solution concept for the game . There have been many solution concepts proposed in cooperative game theory -LRB- see , e : g : , -LSB- 28 -RSB- -RRB- . Euclidean TSP Games 4 The notion of fairness we use here goes back to von Neumann and Morgenstern -LSB- 20 -RSB- and tries to allocate the cost in such a way that no coalition S would be better off playing the game without N nS -LRB- condition -LRB- ii -RRB- below -RRB- . We also want the total cost to be allocated -LRB- condition -LRB- i -RRB- -RRB- . This leads us to the definition of the core of a game -LRB- N ; c -RRB- to be the set core -LRB- c -RRB- of all vectors x 2 < N such that -LRB- i -RRB- P i2N xi > = c -LRB- N -RRB- . -LRB- ii -RRB- P i2S xi < = c -LRB- S -RRB- for all S ? N : Note that core -LRB- c -RRB- is a polyhedron in < N . So a core vector can , in principle , be found by linear programming . Already in the case of MCST-games , however , such an approach is unrealistic because of the exponentially many linear inequalities defining core -LRB- c -RRB- . On the other hand , there is a well-known efficient procedure for computing a core vector x relative to a MCST-game -LRB- see -LSB- 2 -RSB- -RRB- : construct a minimum cost spanning tree T in KN0 and direct the edges of T away from so that the supply node becomes the root of T . Allocate now to each i 2 N as xi the weight of the -LRB- unique -RRB- edge in T that enters node i . It is straightforward to check that this x is indeed a core vector . -LRB- Without going into details , we remark that the core of a MCST-game in its entirety is less straightforward to analyze than this procedure might suggest : it is NP-hard to decide whether a given x 2 < n lies in the core of a given MCST-game -LRB- cf. Faigle et al. -LSB- 8 -RSB- -RRB- . -RRB- In the case of TSP-games , further complications arise . The computation of the value c -LRB- S -RRB- alone is an NP-hard problem -LRB- cf. Garey and Johnson -LSB- 11 -RSB- -RRB- . It has also been observed that the core of a TSP-game may be empty -LRB- see Tamir -LSB- 30 -RSB- -RRB- . -LRB- Tamir 's model actually is slightly more general in that it does not exclude possibly non-hamiltonian graphs -RRB- . In the present paper , we consider the special class of Euclidean TSP-games , where the complete graph KN0 can be represented by points in the Euclidean plane so that the weights dij are just the Euclidean distances between the points i ; j . This class of TSP-games appears quite natural but so far , it has not received much attention in the literature . Kuipers -LSB- 18 -RSB- has shown that TSP-games with 5 or less players have a non-empty core . We show in Section 2 that this result can not be improved even for the class of Euclidean TSP-games . The question , therefore , arises whether allocation vectors x for TSP-games can be computed that are \ approximately '' core vectors . Our notion of an ffl-approximation of the core stipulates that a coalition S should be charged with an allocation that does not exceed the cost c -LRB- S -RRB- by more than a fraction ffl -LRB- see , e : g : , Faigle and Kern -LSB- 7 -RSB- for references for other notions of \ ffl-approximation '' -RRB- . In other words , for ffl > = 0 , we define the ffl-core of the cooperative game -LRB- N ; c -RRB- to be the set ffl-core -LRB- c -RRB- of all vectors x 2 < N satisfying -LRB- i -RRB- P i2N xi > = c -LRB- N -RRB- . -LRB- ii ' -RRB- P i2S xi < = -LRB- 1 + ffl -RRB- c -LRB- S -RRB- for all S ? N : Ideally , we would like to find a vector in the ffl-core where ffl is as small as possible . In Section 3 , we prove that TSP-games whose weights satisfy the triangle inequality -LRB- as is the case in Euclidean TSP-games -RRB- always have ffl-approximately fair -LRB- core -RRB- allocations for ffl = 12 . If a well-known conjecture concerning the Held-Karp relaxation of the Euclidean TSP is true , Euclidean TSP Games 5 our allocation rule presented in Section 3 achieves ffl = 13 . Moreover , these allocations can be computed in polynomial time even when the value of the optimal tour is not known . Note that ffl = 13 would be best possible as we show in Section 4 that no allocation rule can guarantee an ffl < 13 . One of these results was already briefly summarized in Faigle and Kern -LSB- 7 -RSB- . Here we give full details and proofs . We end with some open problems and concluding remarks in Section 5 . 2 A minimal example with empty core Kuipers -LSB- 18 -RSB- showed that under quite general assumption -LRB- including the Euclidean case -RRB- , every TSP-game with up to 5 players has a nonempty core . The following example shows that for n = 6 players , there exist Euclidean instances with empty core . Example 2.1 See Figure 1 . Consider an equilateral triangle of side length l = p3 , centered at 0 . Label the vertices by 1 , 2 , 3 . Place three more nodes 4 , 5 , 6 at equal distance d from the center , such that node i lies on the line segment ; i ? 3 . lgdfh0123456P Figure 1 : A minimum Euclidean example with empty core The distance d will be chosen appropriately . Let f denote the distance d56 = d46 = d45 ; clearly , f = p3d . Let h denote the distance d15 = d16 = d24 = d26 = d34 = d35 . Applying Pythagoras ' Theorem to the triangle ? -LRB- 5P 3 -RRB- , we get h = v u u t p3 2 + p3 2 d ! 2 + ? -LRB- 1 ? d -RRB- 2 ? 2 ; we conclude that h = p1 + d + d2 . Finally , let g : = d36 = d25 = d14 , i : e : , g = 1 ? d . As a consequence , the two obvious candidates for an optimal TSP tour of length L have the following lengths L1 and L2 -LRB- see Figure 2 -RRB- : L1 = l + 3g + 2d + h = 3 ? d + p3 + p 1 + d + d2 ; Euclidean TSP Games 6 12345601234560 Figure 2 : The two candidates for optimal TSP tours L2 = 2l + 2g + 2d + f = 2 + 2p3 + p3d : Now suppose x 2 < 6 were a core allocation . In particular , we must have 1 . P6i = 1 xi = L ; 2 . xi + xi +1 + xi +3 + xi +4 < = 2d + 2g + l for i = 1 ; 2 and x3 + x1 + x6 + x4 < = 2d + 2g + l : Adding up the three inequalities , we get L = 6X i = 1 xi < = 3 2 -LRB- 2d + 2g + l -RRB- = 3 + 3 2 p3 : If the optimal tour has length L1 , we get L1 = 3 ? d + p3 + p 1 + d + d2 < = 3 + 3 2 p3 : The last inequality holds if and only if d > = 1 4 -LRB- p3 ? 1 -RRB- > 1 3 : If the optimal tour has length L2 , we get L2 = 2 + 2p3 + p3d < = 3 + 3 2 p3 ; imposing the condition d < = 1p3 ? 1 2 < 1 10 : We conclude that there is no fair cost allocation if neither of these two conditions is satisfied , e : g : , for d = 14 . Euclidean TSP Games 7 3 Approximately fair allocations for Euclidean instances We have seen in the previous section that not all TSP instances allow a fair cost allocation . This makes it desirable to examine approximately fair cost allocations , where the customers can be overcharged by a certain percentage , or the supplier is allowed to run a certain deficit . Interesting allocation rules -LRB- different from core allocations -RRB- have been studied in the context of TSP games , see -LSB- 25 -RSB- . In this section , we will approach the question from a geometric point of view and make use of linear programming duality . There is another good reason for considering approximately fair cost allocations : Since computing the length L of an optimal TSP tour is NP-hard , we can not expect to find an efficient way of computing cost allocations , whether they are fair or not . These computational difficulties make it desirable to consider performance bounds on cost allocations which can be computed in polynomial time . In section 3.2 below , we will introduce a modified game with cost function c -LRB- HK -RRB- < = c , for which we can efficiently compute core allocation vectors z . It is known that c -LRB- HK -RRB- > = 23 cChr , where cChr denotes the cost function corresponding to the well-known Christofides heuristic . Hence , by scaling the vector z by a factor = cChr -LRB- N -RRB- cHK -LRB- N -RRB- < = 32 , we get an '' - approximately fair allocation for the original TSP game with '' < = 12 -LCB- see Theorem 3.6 below for details . In case an optimal global tour of length L is known , we may of course scale the vector z by a factor of = L cHK -LRB- N -RRB- < = to obtain an '' < = '' < = 12 . 3.1 Geometric cost allocation We encounter methods of allocating the cost of transportation in many instances in everyday life . Two of the easiest allocation rules are also the most common ones : A taxi charges by the distance that is traveled by an individual customer . On the other hand , public mass transportation typically charges a flat fee for anyone who uses it , regardless of the distance . The practicality of these two allocation rules relies on the fact that the number of players in a game is either extremely small , i : e : , 1 , or arbitrarily large . -LRB- Insufficiencies of the latter assumption are reflected by the deficits of most public transportation . -RRB- In many cities , there are attempts to refine the fares by using a \ zone structure '' : The region is subdivided into traffic zones , and a customer is charged a certain amount for crossing from one zone into the other . -LRB- See Figure 3 for a practical example . -RRB- For our purposes , a subdivision of the plane into fixed zones -LRB- e : g : , concentric circles around the depot -RRB- is much too crude . Instead , we have to take into account the relative position of the customers . In the following , we will describe a geometric cost allocation method that follows this idea . Definition 3.1 For a given set of vertices in the plane , a moat is a simply closed strip of constant width that separates two nonempty complementary subsets of the vertices . The inside of the moat is the region containing the vertex , the other region we call the outside . A moat packing is a collection of moats with pairwise disjoint interior . The cost of a moat packing is twice the sum over all widths . Note that any tour has to cross every moat twice , hence the cost of a TSP tour is greater than or equal to the cost of a moat packing . Figure 4 shows a moat packing for an instance of cities in the American midwest , distances are taken from Nemhauser and Wolsey , -LSB- 19 -RSB- , p. 530 . Euclidean TSP Games 8 Figure 3 : The fare zones of Hamburg , reproduced with kind permission of the Hamburger Verkehrsverbund HVV 3.2 Moats and the subtour polytope The cost of any moat packing can be allocated as follows : Distribute twice the width of any moat among the vertices on the outside -LRB- in an arbitrary way -RRB- . It is intuitively clear that if the total cost of the moat packing is distributed this way , the resulting distribution is such that no coalition pays more than its TSP cost . -LRB- We will prove that formally later on . -RRB- Note , however that by distributing the cost of a moat packing we will in general not get an allocation vector of the TSP game , since the cost of a moat packing is in general strictly less than the cost of an optimal TSP tour . The situation where the two costs -LRB- maximum moat packing and minimum TSP tour -RRB- coincide occurs precisely if there is a moat packing and a tour such that : 1 . no part of the given tour is left uncovered , and 2 . each moat is crossed by the tour exactly twice . In the general situation , i : e : , when the maximum moat packing has a cost strictly less than the optimum TSP tour , we can still get an allocation by simply scaling the cost distribution above appropriately . This yields an approximately fair allocation for the TSP game , which will be discussed in detail below . The allocated cost for any moat packing is twice the sum of the moat widths . Moat widths are required to be nonnegative and the sum of the widths of moats separating two vertices must not exceed their distance . This motivates the following linear program : max B@2 X fS ; Sg2M wfSSg 1 CA Euclidean TSP Games 9 Denver Wichita Omaha Minneapolis St. Louis Oklahoma City Dallas Chicago Kansas City Springfield Figure 4 : A moat packing for an instance of 10 cities in the midwestern United States Euclidean TSP Games 10 -LRB- M -RRB- subject to the constraints wfSSg > = for all fS ; Sg 2 M ; X ij2ffi -LRB- S -RRB- wfSSg < = dij for all i ; j 2 N0 : Here M denotes the set of all nontrivial partitions fS ; Sg of N0 -LRB- assuming that the vertex is contained in S -RRB- and ffi -LRB- S -RRB- is the set of all edges that join a vertex from S to a vertex from S. Note that there are exponentially many variables . Furthermore , there may be solutions to the linear program which do not correspond to a moat packing , since there may be positive wfS1S1g , wfS2S2g with S1 \ S2 < > ; and S1 6 ? S2 and S2 6 ? S1 , in which case the two corresponding moats are forced to intersect . However , for any instance of -LRB- M -RRB- , there is an optimal solution for which the sets with wfSSg > have the special structure of a nested family : Definition 3.2 A family of partitions fS1 ; S1g ; : : : ; fSk ; Skg is called nested , if for any two partitions fSi ; Sig and fSj ; Sjg , we have Si \ Sj = ; or Si ? Sj or Sj ? Si . For details on nested families , see Pulleyblank -LSB- 26 -RSB- ; a proof for the above claim can be found in Cornu?ejols , Fonlupt and Naddef -LSB- 5 -RSB- . It is not hard to see that a solution with the structure of a nested family allows a moat packing . The details of using a moat packing for allocating the optimal value of -LRB- M -RRB- will be discussed in the following section . We will assume that triangle inequality and -LRB- for the sake of simplicity -RRB- symmetry hold for the distance function . We assume the distance function to be defined for all ij ; i ; j 2 N0 . How can we solve -LRB- M -RRB- in polynomial time ? Consider its dual , i : e : , the following linear problem : min X ij2 -LRB- N02 -RRB- dijxij -LRB- D -RRB- subject to the constraints xij > = for all i ; j ; X ij2ffi -LRB- S -RRB- xij > = 2 for all fS ; Sg 2 M ; Proposition 3.3 Let d : N20 ! < + be an instance of the symmetric TSP satisfying the triangle inequality dij + djk > = dik for all i ; j ; k . Then the corresponding program -LRB- D -RRB- has an optimal solution x 2 < -LRB- N02 -RRB- satisfying 1 . xij < = 1 for all ij 2 ? N02 ? , 2 . NX j = 0 j < > i xij = 2 for all i 2 N0 . Proof . Consider an optimal solution to -LRB- D -RRB- and suppose there was a vertex i with P j < > i xij > 2 : If there was only one vertex k < > i with xki > 0 , we would have xik > 2 and we could lower xik and stay feasible . Since dik > = 0 , this would not increase the objective value . So assume there are two vertices k1 and k2 with xik1 > and xik2 > 0 . In that case we can lower both xik1 and xik2 and raise xk1k2 by the same value and maintain feasibility . Since by triangle inequality Euclidean TSP Games 11 dk1i + dik2 > = dk1k2 , this does not increase the objective value . So we may assume that for any vertex i , we have P j < > i xij = 2 . Furthermore , there can be no edge k1k2 with xk1k2 > 1 , since otherwise the total weight of the edges leaving the set of vertices k1 ; k2 would be deficient , i : e : , we would get the violated constraint X ij2ffi -LRB- fk1 ; k2g -RRB- xij = X i6 = k1 xik1 + X i6 = k2 xik2 ? 2xk1k2 = 4 ? 2xk1k2 < 2 : This means we may consider the following linear program -LRB- HK -RRB- instead of -LRB- D -RRB- : min X ij2 -LRB- N02 -RRB- dijxij -LRB- HK -RRB- subject to the constraints xij > = for all i ; j ; xij < = 1 for all i ; j ; X ij2ffi -LRB- S -RRB- xij > = 2 for all fS ; Sg 2 M ; X j < > i xij = 2 for all i 2 N0 : The feasible region for this second linear program is known as the subtour polytope Sn , the program itself as Held-Karp relaxation : Any of the constraints corresponding to a moat variable is a so-called subtour elimination constraint . These subtour elimination constraints were first introduced by Dantzig , Fulkerson and Johnson -LSB- 6 -RSB- . Gr?otschel and Padberg showed that they are facet-inducing for n > = 4 . Gr?otschel , Lov?asz and Schrijver -LSB- 14 -RSB- , and Karp and Papadimitriou -LSB- 17 -RSB- , showed that a polynomial method for solving the separation problem for a polytope yields a polynomial method for optimization by means of the ellipsoid method . Padberg and Hong -LSB- 22 -RSB- -LRB- see also Padberg and Wolsey -LSB- 24 -RSB- , and Padberg and Rao -LSB- 23 -RSB- -RRB- demonstrated how to solve separation for the subtour polytope in polynomial time by using the method of Gomory and Hu -LSB- 13 -RSB- for finding the minimum cost cut in a graph . Thus we know that optimization over the subtour polytope is possible in polynomial time by means of the ellipsoid algorithm . For a comprehensive study of optimizing over the subtour polytope , see Boyd -LSB- 3 -RSB- and also Boyd and Pulleyblank -LSB- 4 -RSB- . Summarizing , we state : Theorem 3.4 We can determine an optimal moat packing in polynomial time . Since the feasible region Qn of the Traveling Salesman Polytope is contained in Sn , any optimal solution to minimizing over Sn is a lower bound for the optimal value of TSP . It was proved by Wolsey -LSB- 32 -RSB- and by Shmoys and Williamson -LSB- 27 -RSB- that for any distance function d satisfying the triangle inequality , this bound can be at worst 2/3 of the optimum : Theorem 3.5 -LRB- -LSB- 32 -RSB- , -LSB- 27 -RSB- -RRB- If the distances satisfy the triangle inequality then the optimum value of -LRB- HK -RRB- is at least 23 of the length of a shortest tour . Euclidean TSP Games 12 It is a well-known open conjecture that the factor of 2/3 can be replaced by 3/4 . This is known as the Held-Karp Conjecture . It should be noted that Shmoys and Williamson have shown an even stronger version of Theorem 3.5 : Theorem 3.6 -LRB- -LSB- 27 -RSB- -RRB- If the distances satisfy the triangle inequality then the optimum value of -LRB- HK -RRB- is at least 23 of the length of a tour obtained by the method of Christofides . This means we can guarantee allocation of 2/3 of the cost of an approximate tour where both the allocation and the tour have been computed in polynomial time . 3.3 Tours and Moats We now proceed to describe how to obtain allocation vectors from moat packings . The idea is to distribute twice the width of every moat in an arbitrary way -LRB- e : g : , uniformly -RRB- among the vertices on its outside . Let w ? be an optimal solution of the linear program -LRB- M -RRB- and define xi : = 2 ? X fS ; ? Sg2M i2S ; 02 ? S w?S ; ? S jSj : By linear programming duality this vector is in the core of the game associated with the linear program -LRB- HK -RRB- in a natural way : Define the cost of a coalition S ? N by c -LRB- HK -RRB- -LRB- S -RRB- : = min X i ; j dijzij subject to the constraints zij > = for all i ; j ; P ij2ffiS -LRB- T -RRB- zij > = 2 for all 2 T ae S -LSB- f0g : -LRB- Here ffiS -LRB- T -RRB- denotes the set of edges joining T to -LRB- S -LSB- f0g n T :-RRB- -RRB- The general idea to apply linear programming duality to combinatorial games goes back to -LSB- 21 -RSB- . The vector x 2 < N is easily seen to be a core vector of the game defined by the cost function c -LRB- HK -RRB- : Indeed , by linear programming duality we have X i2N xi = 2 ? X i2N X fS ; ? Sg2M i2S ; 02 ? S w?S ; ? S jSj = 2 ? X fS ; Sg2M X i2S 1 jSjw?S ; ? S = 2 ? X fS ; Sg2M w?S ; ? S = c -LRB- HK -RRB- -LRB- N -RRB- : To see that this is in fact a fair allocation vector , recall that for a coalition S ? N , its cost c -LRB- HK -RRB- -LRB- S -RRB- is defined as the optimum of an LP . Its dual is given by c -LRB- HK -RRB- -LRB- S -RRB- = max 2 ? X 62T aeS wT ; -LRB- S -LSB- f0g -RRB- nT subject to the constraints X 62T aeS ffiS -LRB- T -RRB- 3ij wT ; -LRB- S -LSB- f0g -RRB- nT < = dij 8i ; j wT ; -LRB- S -LSB- f0g -RRB- nT > = : Euclidean TSP Games 13 Now an optimum solution w ? of -LRB- M -RRB- induces a feasible solution of the LP above by ~ wT : = X S \ U = T w?U . Hence we get c -LRB- HK -RRB- -LRB- S -RRB- > = 2 ? X 62T aeS ~ wT = 2 ? X -LRB- i ; T -RRB- i2T?S 1 jT j ~ wT = X i2S 2 ? X i2U w?U jU \ Sj > = X i2S 2 ? X i2U w?U jU j = X i2S xi : Figure 5 : A moat packing Note that we can choose any distribution of the cost of a moat among the outside vertices , i : e : , instead of taking xi : = 2 ? X fS ; ? Sg2M i2S ; 02 ? S wS ; ? S jSj ; we could choose any xi : = 2 ? X i2S 2 ? S > = SiwS ; ? S with P i2S > = Si = 1 ; < = > = Si < = 1 without changing the validity of any of the above statements . If the optimal tour is not an optimal solution of -LRB- HK -RRB- then for any moat packing either at least one moat is traversed more than two times -LRB- see Figure 5 -RRB- or the optimal tour runs through territory which is not covered by any moat . Euclidean TSP Games 14 Proposition 3.7 We would like to note that even if c -LRB- N -RRB- > c -LRB- HK -RRB- -LRB- N -RRB- , the core may be nonempty . The easiest example for this situation arises from Example 2.1 , see Figures 1 and 5 : It can be shown that an optimal moat packing has cost 3 -LRB- f2 + g + l2 -RRB- +2 -LRB- d ? f2 -RRB- = -LRB- 3 ? d -RRB- + -LRB- 3 + d -RRB- p32 = : L3 . Now consider the case where d satisfies one of the bounds given in Example 2.1 with equal - ity ; in that case , we have a tour of length L = 3 + 3 p32 > L3 , meaning that we can not distribute the cost of the tour by the Held-Karp allocation rule . However , one can check that allocating p3 to each of the players on the outside vertices and 1 ? p32 to each of the players on the inside vertices is a fair allocation . 3.4 Approximately fair allocations Summarizing the results of the previous two sections , we state : Theorem 3.8 For TSP games with triangle inequality , there is a vector x 2 < N , which can be computed in polynomial time and satisfies the following conditions : -LRB- i -RRB- x is an '' - approximative core allocation for '' = 12 . -LRB- ii -RRB- If the \ 34 - conjecture '' on the Held-Karp bound is true , x is an '' - approximative core allocation for '' = 13 . -LRB- iii -RRB- x '' - approximately allocates the cost of an approximative TSP tour obtained by the Christofides heuristic for '' = 12 . Proof . Let x as above denote the fair allocation of the associated LP-game . Let c ? -LRB- HK -RRB- : = c -LRB- HK -RRB- -LRB- N -RRB- denote the optimum value of -LRB- HK -RRB- and let L be the length of a shortest tour . Obviously , ^ x : = L c ? x is a vector in < N satisfying 1 . ^ x -LRB- N -RRB- = L and 2 . ^ x -LRB- S -RRB- = L c ? x -LRB- S -RRB- < = L c ? L -LRB- S -RRB- < = 3 2L -LRB- S -RRB- : We will see in the following Section 4 that it is impossible to achieve a better general bound than '' = 13 , even in the case of Euclidean distances . Remark 3.9 Potters , Curiel and Tijs -LSB- 25 -RSB- have shown that in the case of a distance function induced by the Euclidean metric for a planar arrangement of points , any convex arrangement of players , i : e : , a set of points which lie on the boundary of their convex hull , guarantees a fair cost allocation . As it was shown by Fekete and Pulleyblank -LSB- 9 -RSB- , any such arrangement has a moat packing with cost equal to the length of the optimal tour , implying that this special case is covered by our above approach . Remark 3.10 We would like to point out that the above worst case estimate for the ratio L c ? -LRB- HK -RRB- is far from the average case ratio . It can be shown -LRB- cf. -LSB- 29 -RSB- , -LSB- 12 -RSB- -RRB- -RRB- that if p1 ; : : : ; pn ; : : : are independently uniformly distributed in the unit square , and L -LRB- p1 ; : : : ; pn -RRB- denotes the length Euclidean TSP Games 15 of a shortest tour through p1 ; : : : ; pn and c ? -LRB- HK -RRB- -LRB- p1 ; : : : ; pn -RRB- denotes the optimum value of the Held-Karp relaxation , then there exist constants TSP and -LRB- HK -RRB- , such that L -LRB- p1 ; : : : ; pn -RRB- pn n ! 1 ?! TSP a.s. and c ? -LRB- HK -RRB- -LRB- p1 ; : : : ; pn -RRB- pn n ! 1 ?! -LRB- HK -RRB- a.s. Numerical experiments are reported -LSB- 12 -RSB- saying that TSP ss :709 and -LRB- HK -RRB- ss :7 . This means that in the average one can expect '' - approximately fair allocations for '' ss :013 . 4 A lower bound on '' We have seen in the previous section that -LRB- subject to the conjecture on the Held-Karp bound -RRB- there is a way to guarantee '' < = 13 . In the following , we present an example with '' arbitrarily close to this bound of 13 . Theorem 4.1 The bound '' < = 13 resulting from the conjecture on the Held-Karp bound is best possible . Proof . Consider a set of n points , equally distributed on 3 columns , as shown in Figure 6 . Let h denote the height of the columns . Since the distance d between two columns can be made arbitrarily small -LRB- provided we place a sufficient number of points on each column -RRB- , a shortest tour has length approximately equal to 4h . On the other hand , any coalition consisting of two of the three columns is willing to pay at most 2h +4 d. Hence any potential core vector must satisfy x -LRB- N -RRB- < = 32 -LRB- 2h + 4d -RRB- = 3h + 6d . This shows that there are no '' - approximately fair allocations for '' < = 4h 3h + 6d ? 1 . If we let d tend to 0 , we get the desired result . dh0 Figure 6 : An example with '' ?! 13 Combined with our results in Section 3 , this bound implies a previous result due to Goemans -LSB- 12 -RSB- , stating that there are TSP instances for which the ratio between the length of a shortest tour and the optimal value of the Held-Karp relaxation is arbitrarily close to 43 . Euclidean TSP Games 16 5 Concluding remarks A proof of the \ 34-conjecture '' on the Held-Karp relaxation would completely settle the question about the worst case analysis of the '' - approximability problem . On the other hand , it may be possible to find completely different methods of constructing approximately fair allocations with guaranteed worst case approximation error '' = 13 . For example , Faigle and Kern -LSB- 7 -RSB- propose some allocation rules based on the minimum spanning tree allocation and the distance from the supply node . An empirical study of randomly generated small instances -LRB- n < = 10 -RRB- seems to indicate that the Held-Karp allocation rule is by far preferable to other heuristic rules based on minimum spanning trees and relative distance functions , cf. Hunting -LSB- 15 -RSB- . Another interesting question concerns the average case behavior . We conjecture that if the points are independently and uniformly generated , say , in the unit square , then the probability of the core being empty tends to zero as n ?! 1 , where n is the number of players . One way of approaching this problem may be using the so-called zero-or-one law from probability theory , see Feller -LSB- 10 -RSB- Volume II , Theorem 3 , Chapter IV .6 . This is easily seen to imply the following : Let X0 , X1 , X2 ; : : : be independent random variables that are uniformly distributed in the unit square . Let A ? ? -LSB- 0 ; 1 -RSB- 2 ? 1 be defined as A : = f -LRB- a0 ; a1 ; a2 ; : : :-RRB- j the TSP game has nonempty core for almost all begin sequencesg : Then either P -LRB- A -RRB- = or P -LRB- A -RRB- = 1 , where P is the infinite product measure on ? -LSB- 0 ; 1 -RSB- 2 ? 1 . Acknowledgements : The second author would like to thank Bill Pulleyblank for many helpful discussions about moat packings . We appreciate the friendly permission of the Hamburger Verkehrsverbund HVV to use Figure 3 . References -LSB- 1 -RSB- H.Aarts and T.S.H.Driessen , The irreducible core of a minimum cost spanning tree game , Zeitschrift f?ur Operations Research , 38 -LRB- 1993 -RRB- , 163 -LCB- 174 . -LSB- 2 -RSB- C.G.Bird On cost allocation for a spanning tree : a game theoretic approach , Networks , 6 -LRB- 1976 -RRB- , 335 -LCB- 350 . -LSB- 3 -RSB- S.C.Boyd , The subtour polytope of the traveling salesman polytope , Ph.D. . Thesis , University of Waterloo , Waterloo , Ontario , 1986 . -LSB- 4 -RSB- S.C.Boyd , W.R.Pulleyblank , Optimization over the subtour polytope of the traveling salesman polytope , Mathematical Programming , 49 -LRB- 1990 -RRB- , 163 -LCB- 187 . -LSB- 5 -RSB- G.Cornu?ejols , J.Fonlupt , D.Naddef , The traveling salesman problem on a graph and some related integer polyhedra , Mathematical Programming , 33 -LRB- 1985 -RRB- , 1 -LCB- 27 . -LSB- 6 -RSB- G.B.Dantzig , D.R.Fulkerson , S.M.Johnson , Solutions of a large-scale traveling salesman problem , Operations Research , 2 -LRB- 1954 -RRB- , 393 -LCB- 410 . -LSB- 7 -RSB- U.Faigle and W.Kern , On some approximately balanced combinatorial cooperative games , Zeitschrift f?ur Operations Research , 38 -LRB- 1993 -RRB- , 141 -LCB- 152 . -LSB- 8 -RSB- U.Faigle , S.P.Fekete , W.Hochst?attler and W.Kern , On the complexity of testing membership in the core of min-cost spanning tree games , International Journal of Game Theory , -LRB- to appear 1996 -RRB- . -LSB- 9 -RSB- S.P.Fekete , Geometry and the Traveling Salesman Problem , Ph.D. . Thesis , University of Waterloo , Waterloo , Ontario , 1992 . Euclidean TSP Games 17 -LSB- 10 -RSB- W.Feller , An introduction to Probability Theory and its Applications , Wiley , 1971 . -LSB- 11 -RSB- M.R.Garey and D.S.Johnson , Computers and Intractability . A Guide to the Theory of NP - completeness , W.H. Freeman , New York , 1979 . -LSB- 12 -RSB- M.X.Goemans , Analysis of LP-relaxations for a class of connectivity problems , Ph.D. . Thesis , MIT , Boston , Massachusetts , 1990 . -LSB- 13 -RSB- D.Granot and G.Hubermann , Minimum cost spanning tree games , Math . Progr. , 21 -LRB- 1981 -RRB- , 1 -LCB- 18 . -LSB- 14 -RSB- M.Gr?otschel , L.Lov?asz , A.Schrijver , The ellipsoid method an its consequences , Combinatorica , 1 -LRB- 1981 -RRB- , 169 -LCB- 197 . -LSB- 15 -RSB- M.Hunting , De least-core epsilon voor euclidische TSP-spelen met weinig spelers , Diploma thesis , University of Twente , December 1993 . -LSB- 16 -RSB- M.J?unger , W.R.Pulleyblank , Geometric Duality and Combinatorial Optimization , in : Jahrbuch ? Uberblicke Mathematik , S.D.Chatterji . B.Fuchssteiner , U.Kulisch , R.Liedl eds. , Vieweg , 1993 . -LSB- 17 -RSB- R.Karp , C.Papadimitriou , On linear characterizations of combinatorial optimization problems , Proceedings of the Twenty-first Annual Symposium on the Foundations of Computer Science , IEEE Press , New York , 1980 , 1 -LCB- 9 . -LSB- 18 -RSB- J.Kuipers , A note on the 5-person Traveling Salesman game , Technical Report , Univ. of Limburg , The Netherlands , 1991 . -LSB- 19 -RSB- G.L.Nemhauser , L.A.Wolsey , Integer and Combinatorial Optimization , Wiley , New York , 1988 . -LSB- 20 -RSB- J.v.Neumann and O.Morgenstern , Theory of Games and Economic Behavior , Princeton University Press , Princeton , 1944 . -LSB- 21 -RSB- G.Owen , On the core of linear production games , Math . Progr. , 9 -LRB- 1975 -RRB- , 358 -LCB- 370 . -LSB- 22 -RSB- M.W.Padberg , S.Hong , On the symmetric traveling salesman problem : a combinatorial study , Mathematical Programming Study , 12 -LRB- 1980 -RRB- , 78 -LCB- 107 . -LSB- 23 -RSB- M.W.Padberg , M.R.Rao , Odd Minimum Cut-Sets and b-Matchings , Mathematics of Operations Research , 7 -LRB- 1982 -RRB- , 67 -LCB- 80 . -LSB- 24 -RSB- M.W.Padberg , L.A.Wolsey , Trees and Cuts , Annals of Discrete Mathematics , 17 -LRB- 1983 -RRB- , 511 -LCB- 517 . -LSB- 25 -RSB- J.Potters , I.Curiel , S.Tijs , Traveling Salesman games , Math . Progr. , 53 -LRB- 1992 -RRB- , 199 -LCB- 211 . -LSB- 26 -RSB- W.R.Pulleyblank , Faces of Matching Polyhedra , Ph.D. . Thesis , University of Waterloo , Waterloo , Ontario , 1973 . -LSB- 27 -RSB- D.Shmoys , D.Williamson , Analyzing the Held-Karp TSP bound : a monotonicity property with applications , Information Processing Letters , 35 -LRB- 1990 -RRB- , 281 -LCB- 285 . -LSB- 28 -RSB- M.Shubik , Game theory models and methods in political economy . In : Handbook of Mathematical Economics , vol . 1 -LRB- K.J. Arrows et al. , eds . -RRB- , North-Holland , New York , -LRB- 1981 -RRB- , 285 -LCB- 330 . -LSB- 29 -RSB- J.M.Steele , Probabilistic and worst case analysis of classical problems of combinatorial optimization in Euclidean space , Mathematics of OR , 15 -LRB- 4 -RRB- 1990 , 749 -LCB- 770 . -LSB- 30 -RSB- A.Tamir , On the core of a traveling salesman cost allocation game , OR Letters , 9 -LRB- 1989 -RRB- , 31 -LCB- 34 . -LSB- 31 -RSB- A.Tamir , On the core of network synthesis games , Math . Progr. , 50 -LRB- 1991 -RRB- , 123 -LCB- 135 . -LSB- 32 -RSB- L.A.Wolsey , Heuristic Analysis , linear programming and branch and bound , Mathematical Programming Study , 13 -LRB- 1980 -RRB- 121 -LCB- 134 .

O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O B I O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O B I B I O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O

cooperative game
core
cost allocation
traveling salesman