A pareto-beneficial sub-tree mutation for the multi-criteria minimum spanning tree problem

Abstract

We contribute to the efficient approximation of the Pareto-set for the classical N Phard multi-objective minimum spanning tree problem (moMST) adopting evolutionary computation. More precisely, by building upon preliminary work, we analyse the neighborhood structure of Pareto-optimal spanning trees and design several highly biased sub-graph-based mutation operators founded on the gained insights. In a nutshell, these operators replace (un)connected sub-trees of candidate solutions with locally optimal sub-trees. The latter (biased) step is realized by applying Kruskal's single-objective MST algorithm to a weighted sum scalarization of a sub-graph.We prove runtime complexity results for the introduced operators and investigate the desirable Pareto-beneficial property. This property states that mutants cannot be dominated by their parent. Moreover, we perform an extensive experimental benchmark study to showcase the operator's practical suitability. Our results confirm that the subgraph based operators beat baseline algorithms from the literature even with severely restricted computational budget in terms of function evaluations on four different classes of complete graphs with different shapes of the Pareto-front.

mcMST: A Toolbox for the Multi-Criteria Minimum Spanning Tree Problem

“…Beside a multi-objective version of Prims algorithm (see, e.g., Knowles and Corne (2001)) the package offers evolutionary multi-objective algorithms (EMOAs), e.g., an Prüfer-encoding (Prüfer 1918) based EMOA as proposed by Zhou and Gen (1999) and an EMOA based on direct encoding and a Pareto-beneficial subtree mutation operator (Bossek and Grimme 2017). Further, a simple and generic enumeration algorithm is included, which is useful to compute the exact front of graph problems (not limited to mcMST) of small instance size by exhaustive enumeration.…”

The First Proven Performance Guarantees for the Non-Dominated Sorting Genetic Algorithm II (NSGA-II) on a Combinatorial Optimization Problem

“…We refer to Ehrgott [Ehrgott, 2005] for an extensive discussion of the problem and the different algorithmic approaches to it. Being NP-complete, many heuristic approaches have been developed [Arroyo et al, 2008], including many based on evolutionary algorithms [Knowles and Corne, 2000;Knowles and Corne, 2001;Bossek and Grimme, 2017;Parraga-Alava et al, 2017;Majumder et al, 2020].…”