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)


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 c D is the characteristic function where D is the underlying cost matrix. For all S ⊆ N, define c D (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 ε 0 > 1 such that it is NPhard to decide whether ε 0–Core(\(\mathcal{G})\) is empty or not.


Cooperative Game Assignment Game Home Node Hamiltonian Tour Cycle Cover 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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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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