# A multi-start local search algorithm for the Hamiltonian completion problem on undirected graphs

## TL;DR

This paper introduces a multi-start local search algorithm for the Hamiltonian Completion Problem on undirected graphs, integrating exact solutions for trees and demonstrating high-quality results on a new benchmark.

## Contribution

It presents a novel local search algorithm for general undirected graphs that combines exact tree solutions and applies to related problems like minimum path partition.

## Key findings

- Achieves high-quality solutions compared to state-of-the-art solvers.
- Effectively solves HCP on general undirected graphs.
- Demonstrates versatility by also addressing the minimum path partition problem.

## Abstract

This paper proposes a local search algorithm for a specific combinatorial optimisation problem in graph theory: the Hamiltonian Completion Problem (HCP) on undirected graphs. In this problem, the objective is to add as few edges as possible to a given undirected graph in order to obtain a Hamiltonian graph. This problem has mainly been studied in the context of various specific kinds of undirected graphs (e.g. trees, unicyclic graphs and series-parallel graphs). The proposed algorithm, however, concentrates on solving HCP for general undirected graphs. It can be considered to belong to the category of matheuristics, because it integrates an exact linear time solution for trees into a local search algorithm for general graphs. This integration makes use of the close relation between HCP and the minimum path partition problem, which makes the algorithm equally useful for solving the latter problem. Furthermore, a benchmark set of problem instances is constructed for demonstrating the quality of the proposed algorithm. A comparison with state-of-the-art solvers indicates that the proposed algorithm is able to achieve high-quality results.