A Support-Based Algorithm for the Bi-Objective Pareto Constraint

Bi-Objective Combinatorial Optimization problems are ubiquitous in real-world applications and designing approaches to solve them efficiently is an important research area of Artificial Intelligence. In Constraint Programming, the recently introduced bi-objective Pareto constraint allows one to solve bi-objective combinatorial optimization problems exactly. Using this constraint, every non-dominated solution is collected in a single tree-search while pruning sub-trees that cannot lead to a non-dominated solution. This paper introduces a simpler and more efficient filtering algorithm for the bi-objective Pareto constraint. The efficiency of this algorithm is experimentally confirmed on classical bi-objective benchmarks. Bi-Objective Combinatorial Optimization (BOCO) aims at optimizing two objective functions simultaneously. Since these objective functions are often conflicting, there is usually no perfect solution that is optimal for both objectives at the same time. In this context, decision makers are looking for all the “best compromises” between the objectives to choose a posteriori the solution that best fits their needs. Hence, the notion of optimal solution is replaced by the notion of efficiency and we are searching for the set of all the efficient solutions (usually called efficient set or Pareto frontier) instead of one single solution (Ehrgott 2005). During the past years, many approaches were developed to tackle BOCO problems exactly. However, many of them were developed in the context of Mathematical Programming (Mavrotas 2007; Ralphs, Saltzman, and Wiecek 2006) and only a few can be applied efficiently in Constraint Programming. Among these approaches, the -constraint is probably the most widely used (Haimes, Lasdon, and Wismer 1971; Le Pape et al. 1994; Van Wassenhove and Gelders 1980). The idea is to decompose the original problem into a sequence of subproblems to optimize with regard to the first objective function. At each iteration, a new subproblem is generated by constraining the second objective to take a better value than its value in the optimal solution of the previously solved subproblem. Notice that the number of single∗This work was (partially) supported by the ARC grant 13/18054 from Communauté française de Belgique. Copyright c © 2014, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. objective problems to solve is linear in the efficient set’s cardinality (Haimes, Lasdon, and Wismer 1971). The bi-objective Pareto constraint is an alternative (and more efficient (Gavanelli 2002)) approach to compute the efficient set exactly. The idea behind this constraint is to build an approximation of the efficient set incrementally during the search and to use this approximation to detect and to prune sub-trees that can only lead to solutions that are less efficient than the ones already contained in the approximation. Eventually, the approximation becomes the efficient set and its optimality is proven when the search is completed. The algorithm of the Pareto constraint relies on two operations. The first operation is used to update the approximation (by inserting new solutions in it) while the second consists to use this approximation to reduce the search space of the problem. In this work, we show how to use specific BOCO properties to improve the efficiency of the biobjective Pareto constraint. Precisely, we show that both operations (update and filtering) can benefit from each other in an iterative way to build a simpler and more efficient algorithm for the constraint. This document is structured as follows. First, we briefly introduce constraint programming and its main concepts. Then, we formalize multi-objective combinatorial optimization in the context of constraint programming and present some important definitions. The third section is dedicated to the Pareto constraint in its general multi-objective form. Our support-based algorithm is presented in the fourth section. Section five directly follows with our experiments and results on two classical benchmarks i.e. the bi-objective knapsack problem and the bi-objective travelling salesman problem. Finally, the last section offers some conclusions and perspectives. Constraint Programming Background Constraint Programming is a powerful paradigm to solve constraint satisfaction problems and combinatorial optimization problems. A constraint programming problem is usually defined by a set of variables with their respective domain (i.e. the set of values that can be assigned to a variable), and a set of constraints on these variables. The objective is to find an assignment of the variables that respects all the constraints of the problem. The constraint programming process interleaves a tree-search exploration (common in arProceedings of the Twenty-Eighth AAAI Conference on Artificial Intelligence