A 3-Approximation Algorithm for a Particular Case of the Hamiltonian p-Median Problem

TL;DR

This paper presents a polynomial-time 3-approximation algorithm for a specific case of the Hamiltonian p-median problem, demonstrating practical efficiency and better performance than exact algorithms for large p.

Contribution

Introduces a novel $O(n^6)$ 3-approximation algorithm for a particular case of the Hamiltonian p-median problem, with empirical validation showing improved performance over exact methods.

Findings

Algorithm achieves a 3-approximation ratio.

Better practical ratios observed in experiments.

Outperforms exact algorithms for large p.

Abstract

Given a weighted graph $G$ with $n$ vertices and $m$ edges, and a positive integer $p$, the Hamiltonian $p$-median problem consists in finding $p$ cycles of minimum total weight such that each vertex of $G$ is in exactly one cycle. We introduce an $O(n^6)$ 3-approximation algorithm for the particular case in which $p \leq \lceil \frac{n-2\lceil \frac{n}{5} \rceil}{3} \rceil$. An approximation ratio of 2 might be obtained depending on the number of components in the optimal 2-factor of $G$. We present computational experiments comparing the approximation algorithm to an exact algorithm from the literature. In practice much better ratios are obtained. For large values of $p$, the exact algorithm is outperformed by our approximation algorithm.