# Approximate Fair Cost Allocation in Metric Traveling Salesman Games

• M. Bläser
• L. Shankar Ram
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3879)

## Abstract

A traveling salesman game is a cooperative game $$\mathcal{G} = (N, c_{D})$$. Here N, the set of players is the set of cities (or the vertices of the complete graph) and cD is the characteristic function where D is the underlying cost matrix. For all S ⊆ N, define cD(S) to be the cost of a minimum cost Hamiltonian tour through the vertices of S∪ {0} where 0 ∉ N is called as the home city. Define Core $$(\mathcal{G})= \{x \in \Re^{N} :x(N) = c_{D}(N)\ {\rm and}\ \forall S\subseteq N, x(S) \leq c_{D} (S)n\}$$ as the core of a traveling salesman game ($$\mathcal{G}$$). Okamoto [15] conjectured that for the traveling salesman game $$\mathcal{G} = (N, c_{D})$$ with D satisfying triangle inequality, the problem of testing whether Core ($$\mathcal{G}$$) is empty or not is NPhard. We prove that this conjecture is true. This result directly implies the NPhardness for the general case when D is asymmetric. We also study approximate fair cost allocations for these games. For this, we introduce the cycle cover games and show that the core of a cycle cover game is non–empty by finding a fair cost allocation vector in polynomial time. For a traveling salesman game, let $$\epsilon-{\rm Core} (\mathcal{G})=\{x \in \Re^{N} :x(N) = c_{D}(N) {\rm and}\ \forall S\subseteq N, x(S) \leq {\epsilon}\cdot c_{D}(S)\}$$ be an ε–approximate core, for a given ε > 1. By viewing an approximate fair cost allocation vector for this game as a sum of exact fair cost allocation vectors of several related cycle cover games, we provide a polynomial time algorithm demonstrating the non–emptiness of the log2(|N|–1)– approximate core by exhibiting a vector in this approximate core for the asymmetric traveling salesman game. We also show that there exists an ε0 > 1 such that it is NPhard to decide whether ε0–Core($$\mathcal{G})$$ is empty or not.

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### References

1. 1.
Berman, P., Karpinski, M., Scott, A.D.: Approximation hardness and satisfiability of bounded occurrence instances of SAT. Technical Report TR03-022, ECCC (2003)Google Scholar
2. 2.
Bird, C.G.: On cost allocation for a spanning tree: a game theoretic approach. Networks 6, 335–350 (1976)
3. 3.
Deng, X., Ibaraki, T., Nagamochi, H.: Algorithmic aspects of the core of combinatorial optimization games. Math. Oper. Res. 24, 751–766 (1999)
4. 4.
Edmonds, J., Johnson, L.: Matching: A well-solved class of integer linear programs. In: Proceedings of Calgary International conference on combinatorial structures and their applications, Gordon and Breach, pp. 89–92 (1970)Google Scholar
5. 5.
Engebretsen, L.: An explicit lower bound for TSP with distances one and two. Algorithmica 35(4), 301–319 (2003)
6. 6.
Faigle, U., Fekete, S.P., Hochstättler, W., Kern, W.: On approximately fair cost allocation in Euclidean TSP games. OR Spektrum 20, 29–37 (1998)
7. 7.
Faigle, U., Kern, W.: On the core of ordered submodular cost games. Math. Program. 87, 483–499 (2000)
8. 8.
Frieze, A., Galbiati, G., Maffioli, F.: On the worst–case performance of some algorithms for the asymmetric travelling salesman problem. Networks 12, 23–39 (1982)
9. 9.
Goemans, M.X., Skutella, M.: Cooperative facility location games. In: Proc. 11th SODA, pp. 76–85 (2000)Google Scholar
10. 10.
Granot, D., Hamers, H., Tijs, S.: On some balanced, totally balanced and submodular delivery games. Math. Program. 86, 355–366 (1999)
11. 11.
Granot, D., Huberman, G.: Minimum cost spanning tree games. Math. Program. 21, 1–18 (1981)
12. 12.
Kaplan, H., Lewenstein, M., Shafrir, N., Sviridenko, M.: A 2/3 approximation for maximum asymmetric TSP by decomposing directed regular multi graphs. In: Proc. of the 44th Annual IEEE Symposium on Foundations of Computer Science (2003)Google Scholar
13. 13.
Kolen, A.: Solving covering problems and the uncapacitated plant location algorithms. Eur. J. Oper. Res. 12, 266–278 (1983)
14. 14.
Kuhn, H.W.: The Hungarian method for the assignment problem. Nav. Res. Logist. Q. 2, 83–97 (1955)
15. 15.
Okamoto, Y.: Traveling salesman games with the Monge property. Disc. Appl. Math. 138, 349–369 (2004)
16. 16.
Papadimitriou, C., Yannakakis, M.: The traveling salesman problem with distances one and two. Math. Oper. Res. 18, 1–11 (1993)
17. 17.
Potters, J.A.M., Curiel, I.J., Tijs, S.H.: Traveling salesman games. Math. Program. 53, 199–211 (1992)
18. 18.
Shapley, L., Shubik, M.: The assignment game I: the core. Int. J. Game Theory 1, 111–130 (1972)
19. 19.
Tamir, A.: On the core of a traveling salesman cost allocation game. Oper. Res. Lett. 8, 31–34 (1988)
20. 20.
Tovey, C.A.: A simplified NP-complete satisfiability problem. Disc. Appl. Math. 8, 85–89 (1984)