Enumeration of Pareto optimal multi-criteria spanning trees - A proof of the incorrectness of Zhou and Gen's proposed algorithm

The minimum spanning tree (MST) problem is a well-known optimization problem of major significance in operational research. In the multi-criteria MST (mc-MST) problem, the scalar edge weights of the MST problem are replaced by vectors, and the aim is to find the complete set of Pareto optimal minimum-weight spanning trees. This problem is NP-hard and so approximate methods must be used if one is to tackle it efficiently. In an article previously published in this journal, a genetic algorithm (GA) was put forward for the mc-MST. To evaluate the GA, the solution sets generated by it were compared with solution sets from a proposed (exponential time) algorithm for enumerating all Pareto optimal spanning trees. However, the proposed enumeration algorithm that was used is not correct for two reasons: (1) It does not guarantee that all Pareto optimal minimum-weight spanning trees are returned. (2) It does not guarantee that those trees that are returned are Pareto optimal. In this short paper we prove these two theorems. © 2002 Elsevier Science B.V. All rights reserved.SCOPUS: ar.jinfo:eu-repo/semantics/publishe

Multi-Objective Combinatorial Optimization: Problematic and Context

International audienceThe present chapter aims to serve as a brief introduction for the rest of the chapters in this volume. The main goal is to provide a general overview of multi-objective combinatorial optimization, including its main basic definitions and some notions regarding the incorporation of user's preferences. Additionally, we also present short descriptions of some of the most popular multi-objective evolutionary algorithms in current use. Since performance assessment is a critical task in multi-objective optimization, we also present some performance indicators, as well as some discussion on statistical validation in a multi-objective optimization context. The aim of this chapter is not to be comprehensive, but simply to touch on the main fundamental topics that are required to understand the material that is presented in the rest of the book