Bi-objective search problems are a useful generalization of shortest path search. This paper reviews some recent contributions for the solution of this problem with emphasis on the efficiency of the dominance checks required for pruning, and introduces a new algorithm that improves time efficiency over previous proposals. Experimental results are presented to show the performance improvement using a set of standard problems over bi-objective road maps.

Abstract This paper addresses the problem of approximating the set of all solutions for Multi-objective Markov Decision Processes. We show that in the vast majority of interesting cases, the number of solutions is exponential or even infinite. In order to overcome this difficulty we propose to approximate the set of all solutions by means of a limited precision approach based on White’s multi-objective value-iteration dynamic programming algorithm. We prove that the number of calculated solutions is tractable and show experimentally that the solutions obtained are a good approximation of the true Pareto front.

We present a parallel algorithm for finding all Pareto optimal paths from a specified source in a graph. The algorithm is label-setting, i.e., it only performs work on distance labels that are optimal. The main result is that the added complexity when going from one to multiple objectives is completely parallelizable. The algorithm is based on a multiobjective generalization of a priority queue. Such a Pareto queue can be efficiently implemented for two dimensions. Surprisingly, the parallel biobjective approach yields an algorithm performing asymptotically less work than the previous sequential algorithms. We also discuss generalizations for d ≥ 3 objective functions and for single target search.

This article considers the performance of the MOA* multiobjective search algorithm with heuristic information. It is shown that in certain cases blind search can be more efficient than perfectly informed search, in terms of both node and label expansions. A class of simple graph search problems is defined for which the number of nodes grows linearly with problem size and the number of nondominated labels grows quadratically. It is proved that for these problems the number of node expansions performed by blind MOA* grows linearly with problem size, while the number of such expansions performed with a perfectly informed heuristic grows quadratically. It is also proved that the number of label expansions grows quadratically in the blind case and cubically in the informed case.

This paper studies the complexity of two different algorithms proposed as extensions of A* for multiobjective search: MOA* and NAMOA*. It is known that, for any given problem, NAMOA* requires the consideration of no more alternatives than MOA* when provided with the same heuristic information.

The article describes and analyzes NAMOA * , an algorithm for multiobjective heuristic graph search problems. The algorithm is presented as an extension of A * , an admissible scalar shortest path algorithm. Under consistent heuristics A * is known to improve its efficiency with more informed heuristics, and to be optimal over the class of admissible algorithms in terms of the set of expanded nodes and the number of node expansions. Equivalent beneficial properties are shown to prevail in the new algorithm.

Frontier search is a new search technique that achieves important memory savings over previous best-first search algorithms. This paper describes an extension of frontier search to bicriterion graph search problems that achieves also important memory savings. The new algorithm is evaluated using a set of random problems.