Approximately fair cost allocation in metric traveling salesman games

Abstract

A traveling salesman game is a cooperative game {G}}=(N,cD). 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 ∉ N is called as the home city. Define Core (G)= {xεN:x(N) =cD(N)and∀ ⊆ N,x(S)≤ cD(S)≤} as the core of a traveling salesman game G. Okamoto (Discrete Appl. Math. 138:349-369, [2004]) conjectured that for the traveling salesman game G =(N,cD with D satisfying triangle inequality, the problem of testing whether Core (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 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 εCore(G)={xεin|N|x(N)≥ cD(N) and ∀S⊆ N, x(S)≥ 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 4/3log3(|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 NP -hard to decide whether ε 0-Core G is empty or not. © 2007 Springer Science+Business Media, LLC.