Small Approximate Pareto Sets for Biobjective Shortest Paths and Other Problems
Ilias Diakonikolas, Mihalis Yannakakis · SIAM Journal on Computing · 2009
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 biobjective problems (containing many important widely studied problems such as shortest paths, spanning tree, matching, and many others), we can compute in polynomial time an $\epsilon$-Pareto set that contains at most twice as many solutions as the minimum 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.