Winning the War by (Strategically) Losing Battles: Settling the Complexity of GrundyValues in Undirected Geography
Abstract
We settle two longstanding complexitytheoretical questionsopen since 1981 and 1993in combinatorial game theory (CGT). We prove that the Grundy value (a.k.a. nimvalue, or nimber) of Undirected Geography is PSPACEcomplete to compute. This exhibits a stark contrast with a result from 1993 that Undirected Geography is polynomialtime solvable. By distilling to a simple reduction, our proof further establishes a dichotomy theorem, providing a "phase transition to intractability" in Grundyvalue computation, sharply characterized by a maximum degree of four: The Grundy value of Undirected Geography over any degreethree graph is polynomialtime computable, but over degreefour graphseven when planar and bipartiteis PSPACEhard. 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 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 shortdepth game positions. We use the sum of two Undirected Geography positions to create our hard instances. Our result also has computational implications to SpragueGrundy Theory (1930s) which shows that the Grundy value of the disjunctive sum of any two impartial games can be computedin polynomial timefrom their Grundy values. In contrast, we prove that assuming PSPACE $\neq$ P, there is no general polynomialtime method to summarize two polynomialtime solvable impartial games to efficiently solve their disjunctive sum.
 Publication:

arXiv eprints
 Pub Date:
 June 2021
 arXiv:
 arXiv:2106.02114
 Bibcode:
 2021arXiv210602114B
 Keywords:

 Computer Science  Computational Complexity;
 Computer Science  Artificial Intelligence;
 Computer Science  Discrete Mathematics;
 Computer Science  Data Structures and Algorithms;
 Mathematics  Combinatorics;
 91A46;
 F.1.3;
 F.2.2;
 G.2.1;
 G.2.2