A Canceling Heuristic for the Directed Traveling Salesman Problem

TL;DR

This paper introduces a new heuristic for the Traveling Salesman Problem that leverages flow-canceling techniques to improve suboptimal tours efficiently.

Contribution

It develops a novel heuristic connecting Minimum Cost Flow Problems with the TSP to enhance tour optimization.

Findings

Flow-canceling reduces the number of subtours effectively.

A lightweight patching step achieves high success in gap closure.

The heuristic improves solutions with low computational overhead.

Abstract

The Traveling Salesman Problem (TSP) is one of the classic and hard problems in combinatorial optimization. We develop a new heuristic that uses a connection between Minimum Cost Flow Problems and the TSP to improve on a given suboptimal tour, such as a local optimum found using a classic heuristic. Minimum Cost Flow Problems can be solved efficiently through linear programming or combinatorial algorithms based on cycle canceling. We investigate the potential of flow-canceling in the context of the TSP. Through a restriction of the search space to cycles and circulations that alternate between arcs in- and outside of the tour, practical results exhibit that only a low number of subtours is created, and a lightweight patching step suffices for a high success rate and gap closure towards an optimum.