# Approximately Fair Cost Allocation in Metric Traveling Salesman Games

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## 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.

## Keywords

Cooperative games Fair cost allocations Traveling salesman game Approximate fair cost allocation Combinatorial optimization## Preview

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