Solving the multi-objective Hamiltonian cycle problem using a Branch-and-Fix based algorithm

Maialen Murua

doi:10.1016/j.jocs.2022.101578

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Solving the multi-objective Hamiltonian cycle problem using a Branch-and-Fix based algorithm

2022-04-01

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Maialen Murua *

Diego Galar

Roberto Santana

DOI：10.1016/j.jocs.2022.101578

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Abstract

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The Hamiltonian cycle problem consists of finding a cycle in a given graph that passes through every single vertex exactly once, or determining that this cannot be achieved. In this investigation, a graph is considered with an associated set of matrices. The entries of each of the matrix correspond to a different weight of an arc. A multi-objective Hamiltonian cycle problem is addressed here by computing a Pareto set of solutions that minimize the sum of the weights of the arcs for each objective. Our heuristic approach extends the Branch-and-Fix algorithm, an exact method that embeds the problem in a stochastic process. To measure the efficiency of the proposed algorithm, we compare it with a multi-objective genetic algorithm in graphs of a different number of vertices and density. The results show that the density of the graphs is critical when solving the problem. The multi-objective genetic algorithm performs better (quality of the Pareto sets) than the proposed approach in random graphs with high density; however, in these graphs it is easier to find Hamiltonian cycles, and they are closer to the multi-objective traveling salesman problem. The results reveal that, in a challenging benchmark of Hamiltonian graphs with low density, the proposed approach significantly outperforms the multi-objective genetic algorithm.

Keywords:

Graph theory

Multi-objective optimization

Discrete optimization problems

Hamiltonian cycle problem

Branching algorithm