Mathematical Programming: Series A and B
The traveling salesman problem with distances one and two
Mathematics of Operations Research
Cooperative facility location games
SODA '00 Proceedings of the eleventh annual ACM-SIAM symposium on Discrete algorithms
Algorithmic Aspects of the Core of Combinatorial Optimization Games
Mathematics of Operations Research
Approximation Algorithms for Asymmetric TSP by Decomposing Directed Regular Multigraphs
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
Traveling salesman games with the Monge property
Discrete Applied Mathematics
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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 cD is the characteristic function where D is the underlying cost matrix. For all S⊆N, define cD(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 ε0 1 such that it is NP–hard to decide whether ε0–Core($\mathcal{G})$ is empty or not.