An exponential lower bound for Cunningham’s rule

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• 作者：David Avis ; Oliver Friedmann
• 关键词：Simplex method ; Cunningham’s rule ; Parity games ; Acyclic unique sink orientations ; Markov decision processes
• 刊名：Mathematical Programming
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：161
• 期：1-2
• 页码：271-305
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Calculus of Variations and Optimal Control; Optimization; Mathematics of Computing; Numerical Analysis; Combinatorics; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Phys
• 出版者：Springer Berlin Heidelberg
• ISSN：1436-4646
• 卷排序：161

文摘

In this paper we give an exponential lower bound for Cunningham’s least recently considered (round-robin) rule as applied to parity games, Markov decision processes and linear programs. This improves a recent subexponential bound of Friedmann for this rule on these problems. The round-robin rule fixes a cyclical order of the variables and chooses the next pivot variable starting from the previously chosen variable and proceeding in the given circular order. It is perhaps the simplest example from the class of history-based pivot rules. Our results are based on a new lower bound construction for parity games. Due to the nature of the construction we are also able to obtain an exponential lower bound for the round-robin rule applied to acyclic unique sink orientations of hypercubes (AUSOs). Furthermore these AUSOs are realizable as polytopes. We believe these are the first such results for history based rules for AUSOs, realizable or not. The paper is self-contained and requires no previous knowledge of parity games.