Deciding the Winner of an Arbitrary Finite Poset Game Is PSPACE-Complete

Grier, Daniel

doi:10.1007/978-3-642-39206-1_42

Cited by 12 publications

(15 citation statements)

References 12 publications

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“…From a technical standpoint, both Theorems 1.5 and 1.8 are proved roughly as follows. We start with a theorem in prior work stating that it is PSPACE-hard to decide the winner in a related class of games: Hex from an arbitrary starting position [Rei81], or a certain class of poset games [Gri13]. We then apply transformations that introduce the necessary strategy-stealing properties to these games while arguing that the winner in the original game is implicitly encoded by the first player's winning move(s).…”

Section: Our Resultsmentioning

confidence: 99%

Strategy-Stealing is Non-Constructive

Bodwin¹,

Grossman²

2019

Preprint

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In many combinatorial games, one can prove that the first player wins under best play using a simple but non-constructive argument called strategy-stealing. This work is about the complexity behind these proofs: how hard is it to actually find a winning move in a game, when you know by strategy-stealing that one exists? We prove that this problem is PSPACE-Complete already for Minimum Poset Games and Symmetric Maker-Maker Games, which are simple classes of games that capture two of the main types of strategy-stealing arguments in the current literature.

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Section: Our Resultsmentioning

confidence: 99%

Strategy-Stealing is Non-Constructive

Bodwin¹,

Grossman²

2019

Preprint

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“…In view of this, the above method for ordered joins seems unsatisfying at first. However, even the simple case of computing the outcome class of S i 1 is known to be a PSPACE-complete problem [9], and thus the question at hand is a PSPACE-hard problem. Consequently, one should expect only minor improvements in this general setting.…”

Section: Computational Aspectsmentioning

confidence: 99%

The Ordered Join of Impartial Games

Gavrilović,

Thumm

2021

Preprint

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“…On the intractable spectrum, for any PSPACE-complete game with polynomial game-tree height-e.g., Node-Kayles [34], Generalized Geography [34,23], Col [10,2] and many games based on logic, topology, network sciences, etc, [34,9,36,19,8]-the answer to the second question is always a YES. However, this complexity-theoretical polynomial-time reduction is not extendable from PSPACE-complete games to games with potentially lower complexity.…”

Section: Open Question 3 (Tractable Structures) For Any Ruleset Does ...mentioning

confidence: 99%

“…Poset games with the greatest element may have reachable game positions without the greatest element: playing these special games requires later moves on "normal" Poset games, outside the greatest-element family. Indeed, Grier [19] proves that deciding winnability of normal Poset games is PSPACE-complete.…”

Section: Open Question 3 (Tractable Structures) For Any Ruleset Does ...mentioning

confidence: 99%

Winning the War by (Strategically) Losing Battles: Settling the Complexity of Grundy-Values in Undirected Geography

Burke¹,

Ferland²,

Teng³

2021

Preprint

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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 & bipartite-is PSPACE-hard. Additionally, we show, for the first time, how to construct Undirected Geography instances with Grundy value * n and size polynomial in n.We strengthen a result from 1981 showing that sums of tractable partisan games are PSPACEcomplete 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 = P, there is no general polynomial-time method to summarize two polynomial-time solvable impartial games to efficiently solve their disjunctive sum.

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Deciding the Winner of an Arbitrary Finite Poset Game Is PSPACE-Complete

Cited by 12 publications

References 12 publications

Strategy-Stealing is Non-Constructive

Strategy-Stealing is Non-Constructive

The Ordered Join of Impartial Games

Winning the War by (Strategically) Losing Battles: Settling the Complexity of Grundy-Values in Undirected Geography

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