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Journal of Heuristics (2020) 26:743–769

https://doi.org/10.1007/s10732-020-09447-9

A multi-start local search algorithm for the Hamiltonian

completion problem on undirected graphs

Jorik Jooken1·Pieter Leyman1·Patrick De Causmaecker1

Received: 26 April 2019 / Revised: 17 February 2020 / Accepted: 21 June 2020 / Published online: 1 July 2020

© Springer Science+Business Media, LLC, part of Springer Nature 2020

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, con-

centrates 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.

Keywords Metaheuristics ·Matheuristics ·Combinatorial optimisation ·

Hamiltonian completion problem ·Minimum path partition problem

We gratefully acknowledge the support provided by the ORDinL project (FWO-SBO S007318N, Data

Driven Logistics, 1/1/2018–31/12/2021). Pieter Leyman is a Postdoctoral Fellow of the Research

Foundation—Flanders (FWO) with contract number 12P9419N. The computational resources and

services used in this work were provided by the VSC (Flemish Supercomputer Center), funded by the

Research Foundation—Flanders (FWO) and the Flemish Government—Department EWI.

BJorik Jooken

jorik.jooken@kuleuven.be

Pieter Leyman

pieter.leyman@kuleuven.be

Patrick De Causmaecker

patrick.decausmaecker@kuleuven.be

1Department of Computer Science, CODeS, KU Leuven Kulak, Etienne Sabbelaan 53, 8500 Kortrijk,

Belgium

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