DeepAI

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

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 ∗ 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 ≠ P, there is no general polynomial-time method to summarize two polynomial-time solvable impartial games to efficiently solve their disjunctive sum.

10/20/2018

### Simple Games versus Weighted Voting Games: Bounding the Critical Threshold Value

A simple game (N,v) is given by a set N of n players and a partition of ...
09/12/2021

### Nimber-Preserving Reductions and Homomorphic Sprague-Grundy Game Encodings

The concept of nimbers–a.k.a. Grundy-values or nim-values–is fundamental...
11/07/2020

### Quantum Combinatorial Games: Structures and Computational Complexity

Recently, a standardized framework was proposed for introducing quantum-...
06/12/2015

### New Results for Domineering from Combinatorial Game Theory Endgame Databases

We have constructed endgame databases for all single-component positions...
04/08/2019

### A Manifold of Polynomial Time Solvable Bimatrix Games

This paper identifies a manifold in the space of bimatrix games which co...
03/29/2023

### Planar 3-way Edge Perfect Matching Leads to A Holant Dichotomy

We prove a complexity dichotomy theorem for a class of Holant problems o...
06/17/2022

### Universal Complexity Bounds Based on Value Iteration and Application to Entropy Games

We develop value iteration-based algorithms to solve in a unified manner...