One-Exact Approximate Pareto Sets

TL;DR

This paper characterizes when polynomial-time algorithms can compute approximate Pareto sets that are exact in one objective, focusing on multiobjective problems like shortest path and spanning tree, and provides algorithms with optimal bounds for biobjective cases.

Contribution

It establishes a link between singleobjective problem solvability and approximate Pareto set computation, and introduces algorithms with optimal size bounds for biobjective problems.

Findings

Polynomial-time computability of approximate Pareto sets depends on singleobjective problem solvability.

Provides a 2-approximation algorithm for biobjective problems with optimal size bounds.

Shows no constant-factor approximation is possible for three or more objectives.

Abstract

Papadimitriou and Yannakakis show that the polynomial-time solvability of a certain singleobjective problem determines the class of multiobjective optimization problems that admit a polynomial-time computable $(1+\varepsilon, \dots , 1+\varepsilon)$-approximate Pareto set (also called an $\varepsilon$-Pareto set). Similarly, in this article, we characterize the class of problems having a polynomial-time computable approximate $\varepsilon$-Pareto set that is exact in one objective by the efficient solvability of an appropriate singleobjective problem. This class includes important problems such as multiobjective shortest path and spanning tree, and the approximation guarantee we provide is, in general, best possible. Furthermore, for biobjective problems from this class, we provide an algorithm that computes a one-exact $\varepsilon$-Pareto set of cardinality at most twice the cardinality of a smallest such set and show that this factor of 2 is best possible. For three or more objective functions, however, we prove that no constant-factor approximation on the size of the set can be obtained efficiently.