Abstract
Papadimitriou and Yannakakis (Proceedings of the 41st annual IEEE symposium on the Foundations of Computer Science (FOCS), pp 86–92, 2000) show that the polynomial-time solvability of a certain auxiliary 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 multiobjective optimization problems having a polynomial-time computable approximate varepsilon -Pareto set that is exact in one objective by the efficient solvability of an appropriate auxiliary 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 optimization 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 cardinality of the set can be obtained efficiently.
Highlights
In many cases, real-world optimization problems involve several conflicting objectives, e.g., the minimization of cost and time in transportation systems or the maximization of profit and security in investments
We consider general multiobjective optimization problems with an arbitrary, fixed number of objectives and show that, for any such problem, there exist polynomially-sized ε-Pareto sets that are exact in one objective function
This article addresses the task of computing approximate Pareto sets for multiobjective optimization problems
Summary
Real-world optimization problems involve several conflicting objectives, e.g., the minimization of cost and time in transportation systems or the maximization of profit and security in investments. Several results in the literature, show that multiobjective optimization problems are hard to solve exactly [4,5,8] and, in addition, the cardinalities of the set of nondominated points (the nondominated set) and the set of efficient solutions (the efficient set) may be exponentially large for discrete multiobjective optimization problems (and are typically infinite for continuous problems). This impairs the applicability of exact solution methods to real-life optimization problems and provides a strong motivation for studying approximations of multiobjective optimization problems
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