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

@article{Burke2022WinningTW, title={Winning the War by (Strategically) Losing Battles: Settling the Complexity of Grundy-Values in Undirected Geography}, author={Kyle G. Burke and Matthew Ferland and Shang-Hua Teng}, journal={2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)}, year={2022}, pages={1217-1228} }

We settle two long-standing complexity-theoretical questions—open since 1981 and 1993—in combinatorial game theory (CGT). We prove that the Grundy value 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 sharp “phase transition to intractability”: The Grundy value of the game over any…

## 3 Citations

### Nimber-Preserving Reduction: Game Secrets And Homomorphic Sprague-Grundy Theorem

- MathematicsFUN
- 2022

The concept of nimbers – a.k.a. Grundy-values or nim-values – is fundamental to combinatorial game theory. Beyond the winnability, nimbers provide a complete characterization of strategic…

### Quantum-Inspired Combinatorial Games: Algorithms and Complexity

- Computer ScienceFUN
- 2022

It is proved that both Nim and Undirected Geography make a complexity leap over NP, when starting with superpositions: the former becomes Σ p 2 -hard and the latter becomes PSPACE-complete.

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

- MathematicsArXiv
- 2021

It is proved that Generalized Geography is complete for the natural class, I, of polynomially-short impartial rulesets under nimber-preserving reductions, a property the authors refer to as Sprague-Grundy-complete.

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