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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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • M. Bläser
    • 1
  • L. Shankar Ram
    • 1
  1. 1.Institut für Theoretische InformatikETH ZürichZürichSwitzerland

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