Small Approximate Pareto Sets for Bi-objective Shortest Paths and Other Problems

TL;DR

This paper presents a polynomial-time method to compute small approximate Pareto sets for bi-objective problems, achieving near-optimal size within a factor of two, and explores bounds for multi-objective cases.

Contribution

It introduces a polynomial-time algorithm for bi-objective problems that finds an epsilon-Pareto set within a factor of two of the minimum, and proves this factor is tight.

Findings

The epsilon-Pareto set size can be approximated within a factor of 2 in polynomial time.

It is NP-hard to find smaller epsilon-Pareto sets than the established bound.

Bounds are provided for problems with three or more objectives.

Abstract

We investigate the problem of computing a minimum set of solutions that approximates within a specified accuracy $\epsilon$ the Pareto curve of a multiobjective optimization problem. We show that for a broad class of bi-objective problems (containing many important widely studied problems such as shortest paths, spanning tree, and many others), we can compute in polynomial time an $\epsilon$-Pareto set that contains at most twice as many solutions as the minimum such set. Furthermore we show that the factor of 2 is tight for these problems, i.e., it is NP-hard to do better. We present upper and lower bounds for three or more objectives, as well as for the dual problem of computing a specified number $k$ of solutions which provide a good approximation to the Pareto curve.