Efficiently Computing Succinct Trade-Off Curves

Abstract

Trade-off (aka Pareto) curves are typically used to represent the trade-off among different objectives in multiobjective optimization problems. Although trade-off curves are exponentially large for typical combinatorial optimization problems (and infinite for continuous problems), it was observed in [PY1] that there exist polynomial size ε approximations for any ε >0, and that under certain general conditions, such approximate ε-Pareto curves can be constructed in polynomial time. In this paper we seek general-purpose algorithms for the efficient approximation of trade-off curves using as few points as possible. In the case of two objectives, we present a general algorithm that efficiently computes an ε-Pareto curve that uses at most 3 times the number of points of the smallest such curve; we show that no algorithm can be better than 3-competitive in this setting. If we relax ε to any ε′ > ε, then we can efficiently construct an ε′-curve that uses no more points than the smallest ε-curve. With three objectives we show that no algorithm can be c-competitive for any constant c unless it is allowed to use a larger ε value. We present an algorithm that is 4-competitive for any ε′ > (1 + ε)² − 1. We explore the problem in high dimensions and give hardness proofs showing that (unless P=NP) no constant approximation factor can be achieved efficiently even if we relax ε by an arbitrary constant.