# Approximate Fair Cost Allocation in Metric Traveling Salesman Games

## 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 *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 NP–*hard*. We prove that this conjecture is true. This result directly implies the NP–*hardness* 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 log_{2}(|*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 NP–*hard* to decide whether *ε* _{0}–Core(\(\mathcal{G})\) is empty or not.

## Keywords

Cooperative Game Assignment Game Home Node Hamiltonian Tour Cycle Cover## Preview

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