Approximately 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 $0\not \in N$ is called as the home city. Define Core$({\mathcal{G}})=\{x\in \Re^{|N|}:x(N)=c_{D}(N) \mbox{and} \forall S\subseteq N,x(S)\le c_{D}(S)\}$ as the core of a traveling salesman game ${\mathcal{G}}$. Okamoto (Discrete Appl. Math. 138:349–369, [2004]) 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 $\mathsf{NP}$-hard. We prove that this conjecture is true. This result directly implies the $\mathsf{NP}$-hardness for the general case when D is asymmetric. We also study approximately 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 $\epsilon\mbox{-Core}({\mathcal{G}})=\{x\in \Re^{|N|}:x(N)\ge c_{D}(N)$ and ∀S⊆N, x(S)≤e⋅cD(S)} be an e-approximate core, for a given e>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 improve it further by finding a $(\frac{4}{3}\log_{3}(|N|)+c)$-approximate core in polynomial time for some constant c. We also show that there exists an e0>1 such that it is $\mathsf{NP}$-hard to decide whether e0-Core $({\mathcal{G}})$ is empty or not.