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Complexity of Retrograde and Helpmate Chess Problems: Even Cooperative Chess Is Hard

Authors Josh Brunner, Erik D. Demaine , Dylan Hendrickson , Julian Wellman

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LIPIcs.ISAAC.2020.17.pdf
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ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
Keywords
  • hardness
  • board games
  • PSPACE

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Abstract

We prove PSPACE-completeness of two classic types of Chess problems when generalized to n × n boards. A "retrograde" problem asks whether it is possible for a position to be reached from a natural starting position, i.e., whether the position is "valid" or "legal" or "reachable". Most real-world retrograde Chess problems ask for the last few moves of such a sequence; we analyze the decision question which gets at the existence of an exponentially long move sequence. A "helpmate" problem asks whether it is possible for a player to become checkmated by any sequence of moves from a given position. A helpmate problem is essentially a cooperative form of Chess, where both players work together to cause a particular player to win; it also arises in regular Chess games, where a player who runs out of time (flags) loses only if they could ever possibly be checkmated from the current position (i.e., the helpmate problem has a solution). Our PSPACE-hardness reductions are from a variant of a puzzle game called Subway Shuffle.

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Josh Brunner, Erik D. Demaine, Dylan Hendrickson, and Julian Wellman. Complexity of Retrograde and Helpmate Chess Problems: Even Cooperative Chess Is Hard. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 17:1-17:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.ISAAC.2020.17

Author Details

Josh Brunner
  • Computer Science and Artificial Intelligence Laboratory, MIT, Cambridge, MA, USA
Erik D. Demaine
  • Computer Science and Artificial Intelligence Laboratory, MIT, Cambridge, MA, USA
Dylan Hendrickson
  • Computer Science and Artificial Intelligence Laboratory, MIT, Cambridge, MA, USA
Julian Wellman
  • Computer Science and Artificial Intelligence Laboratory, MIT, Cambridge, MA, USA

Acknowledgements

This work was initiated during open problem solving in the MIT class on Algorithmic Lower Bounds: Fun with Hardness Proofs (6.892) in Spring 2019. We thank the other participants of that class - in particular, John Urschel - for related discussions and providing an inspiring atmosphere.

References

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