一正合近似Pareto集
One-Exact Approximate Pareto Sets
摘要：
Papadimitriou和Yannakakis证明了某个单目标问题的多项式时间可解性决定了允许多项式时间可计算的$(1+\varepsilon, \dots , 1+\varepsilon)$ -近似Pareto集（也称为$\varepsilon$ -Pareto集）的多目标优化问题的类别。同样，在这篇文章中，我们刻画了一类具有多项式时间可计算近似$\varepsilon$ -Pareto集的问题，该集在一个目标中是精确的，由一个适当的singleobjective问题的有效可解性。这个类包括重要的问题，如多目标最短路径和生成树，和近似保证，我们提供的是，在一般情况下，最好的可能。此外，从这个类的双目标问题，我们提供了一个算法，计算一个精确的$\varepsilon$ -帕累托集的基数最多两倍的基数最小的这样的集合，并表明，这个因素2是最好的可能。对于三个或更多的目标函数，但是，我们证明，没有常数因子近似的大小的集合可以有效地获得。
英文摘要：
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.
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