Choquet-based optimisation in multiobjective shortest path and spanning tree problems

Abstract

International audienceThis paper is devoted to the search of Choquet-optimal solutions in finite graph problems with multiple objectives. The Choquet integral is one of the most sophisticated preference models used in decision theory for aggregating preferences on multiple objectives. We first present a condition on preferences (name hereafter preference for interior points) that characterizes preferences favouring compromise solutions, a natural attitude in various contexts such as multicriteria optimisation, robust optimisation and optimisation with multiple agents. Within Choquet expected utility theory, this condition amounts to using a submodular capacity and a convex utility function. Under these assumptions, we focus on the fast determination of Choquet-optimal paths and spanning trees. After investigating the complexity of these problems, we introduce a lower bound for the Choquet integral, computable in polynomial time. Then, we propose different algorithms using this bound, either based on a controlled enumeration of solutions (ranking approach) or an implicit enumeration scheme (branch and bound). Finally, we provide numerical experiments that show the actual efficiency of the algorithms on multiple instances of different sizes

Maximization of the Choquet integral over a convex set and its application to resource allocation problems

“…The main body of research in the area has been dedicated to solving the multicriteria versions of combinatorial optimization problems. Thus, Galand et al (2010aGaland et al ( , 2010b look at the minimal spanning tree and shortest path problems in graphs, where every edge has several weights. The resulting tree (path) is therefore characterized by some vector of lengths.…”

“…As stated by Galand et al (2010b) the search for extremal values of the Choquet integral can be viewed as an approach to multiobjective optimization problems. Instead of explicit generation of the whole Pareto set we use the decision maker's preferences to discriminate between various options directly.…”

A Case of Pathology in Multiobjective Heuristic Search

“…In the first place, many graph search problems can benefit directly from multiobjective analysis (De Luca Cardillo & Fortuna, 2000;Gabrel & Vanderpooten, 2002;Refanidis & Vlahavas, 2003;Müller-Hannemann & Weihe, 2006;Dell'Olmo, Gentili, & Scozzari, 2005;Ziebart, Dey, & Bagnell, 2008;Wu, Campbell, & Merz, 2009;Delling & Wagner, 2009;Fave, Canu, Iocchi, Nardi, & Ziparo, 2009;Mouratidis, Lin, & Yiu, 2010;Caramia, Giordani, & Iovanella, 2010;Boxnick, Klöpfer, Romaus, & Klöpper, 2010;Klöpper, Ishikawa, & Honiden, 2010;Wu, Campbell, & Merz, 2011;Machuca & Mandow, 2011). On the other hand, other multicriteria preference models used in graph search typically look for a subset of Pareto-optimal solutions (Mandow & Pérez de la Cruz, 2003;Perny & Spanjaard, 2005;Galand & Perny, 2006;Galand & Spanjaard, 2007;Galand, Perny, & Spanjaard, 2010). Therefore, improvements in performance of multiobjective algorithms can guide the development of efficient algorithms for other multicriteria decision rules.…”

Ordered Weighted Average optimization in Multiobjective Spanning Tree Problem

“…Thus alternative formulations that originally do not exhibit such good properties, may now outperform them. In Galand et al (2010) OWASTP was addressed using Choquet optimization and Galand and Spanjaard (2012) presented a first ordered median Mixed Integer Linear Programming (MILP) formulation. Our goal in this paper is to exploit properties of alternative formulations for OWASTP.…”