关于鲁棒生成树问题的可逼近性
On the approximability of robust spanning tree problems
摘要：
本文讨论了边费用不确定的最小生成树问题。为了模拟的不确定性，一个离散的场景集指定和一个强大的框架，采用选择一个解决方案。讨论了最小-最大、最小-最大后悔和两阶段最小-最大问题。所有这些问题的复杂性和可逼近性进行了探讨。证明了对于任何$ε>0$，具有非负边代价的min-max和min-max后悔方案都很难在$O(\log^{1-ε} n)$内逼近，除非NP问题具有拟多项式时间算法.类似地，2阶段最小-最大问题不能在$O(\log n)$内近似，除非NP中的问题具有准多项式时间算法。本文还提出了求解min-max和两阶段min-max问题的性能比为$O(\log^2 n)$的随机LP基近似算法。
英文摘要：
In this paper the minimum spanning tree problem with uncertain edge costs is discussed. In order to model the uncertainty a discrete scenario set is specified and a robust framework is adopted to choose a solution. The min-max, min-max regret and 2-stage min-max versions of the problem are discussed. The complexity and approximability of all these problems are explored. It is proved that the min-max and min-max regret versions with nonnegative edge costs are hard to approximate within $O(\log^{1-ε} n)$ for any $ε>0$ unless the problems in NP have quasi-polynomial time algorithms. Similarly, the 2-stage min-max problem cannot be approximated within $O(\log n)$ unless the problems in NP have quasi-polynomial time algorithms. In this paper randomized LP-based approximation algorithms with performance ratio of $O(\log^2 n)$ for min-max and 2-stage min-max problems are also proposed.
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