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Winning the War by (Strategically) Losing Battles: Settling the Complexity of Grundy-Values in Undirected Geography
arXiv - CS - Computational Complexity Pub Date : 2021-06-03 , DOI: arxiv-2106.02114
Kyle Burke, Matthew Ferland, Shanghua Teng

We settle two long-standing complexity-theoretical questions-open since 1981 and 1993-in combinatorial game theory (CGT). We prove that the Grundy value (a.k.a. nim-value, or nimber) of Undirected Geography is PSPACE-complete to compute. This exhibits a stark contrast with a result from 1993 that Undirected Geography is polynomial-time solvable. By distilling to a simple reduction, our proof further establishes a dichotomy theorem, providing a "phase transition to intractability" in Grundy-value computation, sharply characterized by a maximum degree of four: The Grundy value of Undirected Geography over any degree-three graph is polynomial-time computable, but over degree-four graphs-even when planar and bipartite-is PSPACE-hard. Additionally, we show, for the first time, how to construct Undirected Geography instances with Grundy value $\ast n$ and size polynomial in n. We strengthen a result from 1981 showing that sums of tractable partisan games are PSPACE-complete in two fundamental ways. First, since Undirected Geography is an impartial ruleset, we extend the hardness of sums to impartial games, a strict subset of partisan. Second, the 1981 construction is not built from a natural ruleset, instead using a long sum of tailored short-depth game positions. We use the sum of two Undirected Geography positions to create our hard instances. Our result also has computational implications to Sprague-Grundy Theory (1930s) which shows that the Grundy value of the disjunctive sum of any two impartial games can be computed-in polynomial time-from their Grundy values. In contrast, we prove that assuming PSPACE $\neq$ P, there is no general polynomial-time method to summarize two polynomial-time solvable impartial games to efficiently solve their disjunctive sum.

中文翻译:

通过(战略上)失败的战斗赢得战争:解决无向地理中 Grundy-Values 的复杂性

我们解决了自 1981 年和 1993 年以来在组合博弈论 (CGT) 中开放的两个长期存在的复杂性理论问题。我们证明了 Undirected Geography 的 Grundy 值(又名 nim-value 或 nimber)是 PSPACE 完全计算的。这与 1993 年的结果形成鲜明对比,即无向地理是多项式时间可解的。通过提炼到一个简单的简化,我们的证明进一步建立了一个二分定理,在 Grundy 值计算中提供了一个“难以处理的相变”,其特征是最大程度为 4:无向地理在任何程度为 3 的图上的 Grundy 值是多项式时间可计算的,但超过四次图 - 即使在平面和二部图 - 是 PSPACE-hard 。此外,我们首次展示,如何使用 Grundy 值 $\ast n$ 和 n 中的大小多项式构造 Undirected Geography 实例。我们加强了 1981 年的结果,表明可处理的党派博弈的总和在两个基本方面是 PSPACE 完备的。首先,由于 Undirected Geography 是一个公正的规则集,我们将求和的难度扩展到公正游戏,这是党派的严格子集。其次,1981 年的构建不是根据自然规则集构建的,而是使用了大量量身定制的短深度游戏位置。我们使用两个 Undirected Geography 位置的总和来创建我们的硬实例。我们的结果也对 Sprague-Grundy 理论(1930 年代)具有计算意义,该理论表明,任何两个公平游戏的析取和的 Grundy 值都可以在多项式时间内从它们的 Grundy 值中计算出来。相比之下,我们证明假设 PSPACE $\neq$ P,
更新日期:2021-06-07
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