Approximately Fair Cost Allocation in Metric Traveling Salesman Games

Markus Bläser, L. Shankar Ram

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)≤ε⋅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 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 ε 0>1 such that it is \(\mathsf{NP}\) -hard to decide whether ε 0-Core \(({\mathcal{G}})\) is empty or not.