Tolerance based algorithms for the ATSP

Abstract

In this paper we use arc tolerances, instead of arc costs, to improve Branch-and-Bound type algorithms for the Asymmetric Traveling Salesman Problem (ATSP). We derive new tighter lower bounds based on exact and approximate bottleneck upper tolerance values of the Assignment Problem (AP). It is shown that branching by tolerances provides a more rational branching process than branching by costs. Among others, we show that branching on an arc with the bottleneck upper tolerance value is the best choice, while such an arc appears quite often in a shortest cycle of the current AP relaxation. This fact shows why branching on shortest cycles was always found as a best choice. Computational experiments confirm our theoretical results. © Springer-Verlag 2004.