Solving a "Hard" Problem to Approximate an "Easy" One: Heuristics for Maximum Matchings and Maximum Traveling Salesman Problems

TL;DR

This paper introduces a heuristic method leveraging geometric duality to efficiently approximate maximum matchings and traveling salesman problems on large instances, achieving near-optimal solutions in near-linear time.

Contribution

It presents a novel heuristic approach based on geometric duality that effectively solves large-scale MWMP and MTSP instances with high accuracy and efficiency.

Findings

Solutions within less than 1% of the optimum

Near-linear time computational performance

High-accuracy solutions for large instances

Abstract

We consider geometric instances of the Maximum Weighted Matching Problem (MWMP) and the Maximum Traveling Salesman Problem (MTSP) with up to 3,000,000 vertices. Making use of a geometric duality relationship between MWMP, MTSP, and the Fermat-Weber-Problem (FWP), we develop a heuristic approach that yields in near-linear time solutions as well as upper bounds. Using various computational tools, we get solutions within considerably less than 1% of the optimum. An interesting feature of our approach is that, even though an FWP is hard to compute in theory and Edmonds' algorithm for maximum weighted matching yields a polynomial solution for the MWMP, the practical behavior is just the opposite, and we can solve the FWP with high accuracy in order to find a good heuristic solution for the MWMP.