arXiv 2109.100762021-09math.OCcs.DS

一类一般多参数最优化问题的近似算法

An Approximation Algorithm for a General Class of Multi-Parametric Optimization Problems

Stephan Helfrich, Arne Herzel, Stefan Ruzika, Clemens Thielen

摘要：

在广泛研究的一类多参数最优化问题中，每个解的目标值是实值参数的仿射函数。然后，目标是提供一个最优解集，即，包含通过固定参数向量获得的每个非参数问题的最优解的集合。然而，对于许多多参数最优化问题，最小基数的最优解集可以包含超多项式多个解。因此，对于这些问题，即使$\textsf{P}=\textsf{NP}$也不存在多项式时间精确算法。我们提出了一种近似方法，适用于一般类的多参数最优化问题，并输出一组解决方案的基数多项式的实例大小和近似保证的倒数。该方法将用于非参数最优化问题的近似算法提升到它们的参数版本，并且提供任意接近于用于非参数问题的近似算法的近似保证的近似保证。如果非参数问题可以在多项式时间内精确求解，或者如果FPTAS可用，则我们的算法是FPTAS。此外，我们表明，对于任何给定的近似保证，近似集的最小基数是，在一般情况下，不$\ell$ -逼近的任何自然数$\ell$小于或等于参数的数量，我们讨论了我们的结果的应用程序经典的多参数组合优化问题。特别地，我们得到了多参数最小$s$ - $t$ -割问题的FPTAS，多参数背包问题的FPTAS，以及多参数独立系统最大化问题的近似算法。

英文摘要：

In a widely-studied class of multi-parametric optimization problems, the objective value of each solution is an affine function of real-valued parameters. Then, the goal is to provide an optimal solution set, i.e., a set containing an optimal solution for each non-parametric problem obtained by fixing a parameter vector. For many multi-parametric optimization problems, however, an optimal solution set of minimum cardinality can contain super-polynomially many solutions. Consequently, no polynomial-time exact algorithms can exist for these problems even if $\textsf{P}=\textsf{NP}$. We propose an approximation method that is applicable to a general class of multi-parametric optimization problems and outputs a set of solutions with cardinality polynomial in the instance size and the inverse of the approximation guarantee. This method lifts approximation algorithms for non-parametric optimization problems to their parametric version and provides an approximation guarantee that is arbitrarily close to the approximation guarantee of the approximation algorithm for the non-parametric problem. If the non-parametric problem can be solved exactly in polynomial time or if an FPTAS is available, our algorithm is an FPTAS. Further, we show that, for any given approximation guarantee, the minimum cardinality of an approximation set is, in general, not $\ell$-approximable for any natural number $\ell$ less or equal to the number of parameters, and we discuss applications of our results to classical multi-parametric combinatorial optimizations problems. In particular, we obtain an FPTAS for the multi-parametric minimum $s$-$t$-cut problem, an FPTAS for the multi-parametric knapsack problem, as well as an approximation algorithm for the multi-parametric maximization of independence systems problem.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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