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 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 NP–hard to decide whether ε ₀–Core($\mathcal{G})$ is empty or not.