Approximating the Bipartite TSP and Its Biased Generalization

Abstract

We examine a generalization of the symmetric bipartite traveling salesman problem (TSP) with quadrangle inequality, by extending the cost function of a Hamiltonian tour to include a bias factor β ≥ 1. The bias factor is known and given as a part of the input. We propose a novel heuristic procedure for building Hamiltonian cycles in bipartite graphs, and show that it is an approximation algorithm for the generalized problem with an approximation ratio of $1+\frac{1+\lambda}{\beta+\lambda}$, where λ is a real parameter dependent on the problem instance. This expression is bounded above by a constant 2, for any positive real λ and β ≥ 1, which improves a previously reported approximation ratio of 16/7. As a part of a composite heuristic, the proposed procedure can contribute to an approximation ratio of $1+\frac{2}{\zeta+\beta(2-\zeta)}$, where ζ is an approximation ratio for the metric TSP.