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

Burke, Kyle; Ferland, Matthew; Teng, Shang‐Hua

doi:10.1109/focs52979.2021.00119

Cited by 2 publications

(7 citation statements)

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“…If one views the nimber characterization of an impartial game as a reduction from that game to a single pile Nim, then Schaefer et al's complexity results demonstrate that this reduction has intractable constructability. In fact, a recent result [9] proved that the nimber of polynomial-time solvable Undirected Geography -i.e., Generalized Geography on undirected graphs -is also PSPACE-complete to compute. The sharp contrast between the complexity of winnability and nimber computation illustrates a fundamental mathematicalcomputational divergence in Sprague-Grundy Theory [9]: Nimbers can be PSPACE-hard "secrets of deep alternation" even for polynomial-time solvable games.…”

Section: Our Main Contributionsmentioning

confidence: 99%

Nimber-preserving reduction: Game secrets and homomorphic Sprague-Grundy theorem

Burke,

Ferland,

Teng

2024

Theoretical Computer Science

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Section: Our Main Contributionsmentioning

confidence: 99%

Nimber-preserving reduction: Game secrets and homomorphic Sprague-Grundy theorem

Burke,

Ferland,

Teng

2024

Theoretical Computer Science

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“…Obviously, in spite of this algorithmic implication, Sprague-Grundy Theory does not provide a general-purpose polynomial-time solution for all impartial games, as witnessed by many PSPACEhard rulesets, including Node Kayles and Generalized Geography [28,24]. In fact, a recent result [8] proved that the nimber of polynomial-time solvable Undirected Geography-i.e., Generalized Geography on undirected graphs-is PSPACE-complete to compute. The sharp contrast between the complexity of winnability and nimber computation illustrates fundamental mathematical-computational divergence in Sprague-Grundy Theory [8]: Nimbers can be PSPACEhard "secrets of deep alternation" even for polynomial-time solvable games.…”

Section: Our Main Contributionsmentioning

confidence: 99%

“…In fact, a recent result [8] proved that the nimber of polynomial-time solvable Undirected Geography-i.e., Generalized Geography on undirected graphs-is PSPACE-complete to compute. The sharp contrast between the complexity of winnability and nimber computation illustrates fundamental mathematical-computational divergence in Sprague-Grundy Theory [8]: Nimbers can be PSPACEhard "secrets of deep alternation" even for polynomial-time solvable games.…”

Section: Our Main Contributionsmentioning

confidence: 99%

“…It is well known that the Grundy function is intractable to compute in the worst case [28]. As shown in [8], the Grundy function can be PSPACE-hard to compute, even for some tractable games in I P . Figuratively, every impartial game G encodes a secret, nimber(G).…”

Section: Nimber Secrets: a Pspace-complete Homomorphic Encodingmentioning

confidence: 99%

“…For (2), both Undirected Geography [17] and Uncooperative Uno [12] 6 are not Sprague-Grundy complete for I P -unless P = PSPACE-despite their nimber intractability, with Undirected Geography being known to have polynomially high nimber positions [8]. For Undirected Geography, Fraenkel, Scheinerman, and Ullman [17] presented a matching-based characterization to show these games are polynomial-time solvable.…”

Section: Expressiveness Of Intractable Games: Sprague-grundy Complete...mentioning

confidence: 99%

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Nimber-Preserving Reductions and Homomorphic Sprague-Grundy Game Encodings

Burke

Ferland

Teng

2021

Preprint

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The concept of nimbers-a.k.a. Grundy-values or nim-values-is fundamental to combinatorial game theory. Nimbers provide a complete characterization of strategic interactions among impartial games in their disjunctive sums as well as the winnability. In this paper, we initiate a study of nimber-preserving reductions among impartial games. These reductions enhance the winnability-preserving reductions in traditional computational characterizations of combinatorial games. We prove that Generalized Geography is complete for the natural class, I P , of polynomially-short impartial rulesets under nimber-preserving reductions, a property we refer to as Sprague-Grundy-complete. In contrast, we also show that not every PSPACE-complete ruleset in I P is Sprague-Grundy-complete for I P .By considering every impartial game as an encoding of its nimber, our technical result establishes the following striking cryptography-inspired homomorphic theorem: Despite the PSPACE-completeness of nimber computation for I P , there exists a polynomial-time algorithm to construct, for any pair of games G 1 , G 2 in I P , a prime game (i.e. a game that cannot be written as a sum) G of I P , satisfying: nimber(G) = nimber(G 1 ) ⊕ nimber(G 2 ).

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

Cited by 2 publications

References 28 publications

Nimber-preserving reduction: Game secrets and homomorphic Sprague-Grundy theorem

Nimber-preserving reduction: Game secrets and homomorphic Sprague-Grundy theorem

Nimber-Preserving Reductions and Homomorphic Sprague-Grundy Game Encodings

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