Approximating Multi-criteria Max-TSP

The traveling salesman problem (TSP) is one of the most fundamental problems in combinatorial optimization. Given a graph, the goal is to find a Hamiltonian cycle of minimum or maximum weight. We consider finding Hamiltonian cycles of maximum weight (Max-TSP). An instance of Max-TSP is a complete graph G = (V,E) with edge weights w : E → N. The goal is to find a Hamiltonian cycle of maximum weight. The weight of a Hamiltonian cycle (or, more general, of a subset of E) is the sum of the weights of its edges. If G is undirected, we speak of Max-STSP (symmetric TSP). If G is directed, we have Max-ATSP (asymmetric TSP). Both Max-STSP and Max-ATSP are NP-hard and APX-hard. Thus, we are in need of approximation algorithms. The currently best approximation algorithms for Max-STSP and Max-ATSP achieve approximation ratios of 61/81 and 2/3, respectively [2, 5]. Cycle covers are an important tool for designing approximation algorithms for the TSP. A cycle cover of a graph is a set of vertex-disjoint cycles such that every vertex is part of exactly one cycle. Hamiltonian cycles are special cases of cycle covers that consist of just one cycle. Thus, the weight of a maximum-weight cycle cover is an upper bound for the weight of a maximum-weight Hamiltonian cycle. In contrast to Hamiltonian cycles, cycle covers of minimum or maximum weight can be computed efficiently using matching algorithms [1].