Asymptotic Optimality of the Greedy Patching Heuristic for Max TSP in Doubling Metrics

TL;DR

This paper proves that in doubling metric spaces, a simple greedy patching heuristic asymptotically finds optimal solutions for the Max TSP by iteratively merging cycles with minimal weight loss.

Contribution

It establishes the asymptotic optimality of the greedy patching heuristic for Max TSP in doubling metrics, a previously unproven property.

Findings

Greedy patching heuristic is asymptotically optimal in doubling metrics.

Starting from maximum-weight cycle cover, the heuristic effectively merges cycles.

The approach minimizes weight loss at each merging step.

Abstract

The maximum traveling salesman problem (Max~TSP) consists of finding a Hamiltonian cycle with the maximum total weight of the edges in a given complete weighted graph. We prove that, in the case when the edge weights are induced by a metric space of bounded doubling dimension, asymptotically optimal solutions of the problem can be found by the simple greedy patching heuristic. Taking as a start point a maximum-weight cycle cover, this heuristic iteratively patches pairs of its cycles into one minimizing the weight loss at each step.