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

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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.
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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
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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... HCP can for example be used to solve a special case of the traveling salesman problem (Applegate et al., 2006) or to assign frequencies to transmitters (Franzblau and Raychaudhuri, 2002). Recently, a matheuristic was developed by Jooken et al. (2020), which attempts to solve this problem heuristically by using a multi-start local search algorithm. HCP can also be solved by first converting it to a traveling salesman problem instance (TSP) and then using a TSP solver such as Concorde (Applegate, 2003) or Lin-Kernighan-Helsgaun (Helsgaun, 2000) to get an exact or heuristic solution respectively. ...
... HCP can also be solved by first converting it to a traveling salesman problem instance (TSP) and then using a TSP solver such as Concorde (Applegate, 2003) or Lin-Kernighan-Helsgaun (Helsgaun, 2000) to get an exact or heuristic solution respectively. In this paper we will compare the multi-start local search algorithm (MSLS) (Jooken et al., 2020) with the Concorde TSP solver (Concorde). ...
... We have evolved more challenging instances than standard graph generators, which allows us to find instances that MSLS solves faster than Concorde and vice versa. We also show that MSLS does not outperform Concorde in many cases, which is in contrast to the conclusions drawn in Jooken et al. (2020). One can only draw reasonable conclusions about the performance of an algorithm when testing on a broad and diverse set of instances, which is demonstrated in this paper. ...
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... HCP can also be solved by first converting it to a travelling salesman problem (TSP) and then using a TSP solver such as Concorde or Lin-Kernighan-Helsgaun to get an exact or heuristic solution respectively. In this paper we will compare the multi-start local search algorithm (MSLS) [8] with the Concorde TSP solver (Concorde) [1]. ...
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