Fixed-Parameter Algorithms for Graph Constraint Logic

Authors Tatsuhiko Hatanaka, Felix Hommelsheim, Takehiro Ito , Yusuke Kobayashi , Moritz Mühlenthaler, Akira Suzuki

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Tatsuhiko Hatanaka
  • Graduate School of Information Sciences, Tohoku University, Sendai, Japan
Felix Hommelsheim
  • Fakultät für Mathematik, TU Dortmund University, Germany
Takehiro Ito
  • Graduate School of Information Sciences, Tohoku University, Sendai, Japan
Yusuke Kobayashi
  • Research Institute for Mathematical Sciences, Kyoto University, Japan
Moritz Mühlenthaler
  • Laboratoire G-SCOP, Grenoble INP, Université Grenoble Alpes, France
Akira Suzuki
  • Graduate School of Information Sciences, Tohoku University, Sendai, Japan

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Tatsuhiko Hatanaka, Felix Hommelsheim, Takehiro Ito, Yusuke Kobayashi, Moritz Mühlenthaler, and Akira Suzuki. Fixed-Parameter Algorithms for Graph Constraint Logic. In 15th International Symposium on Parameterized and Exact Computation (IPEC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 180, pp. 15:1-15:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


Non-deterministic constraint logic (NCL) is a simple model of computation based on orientations of a constraint graph with edge weights and vertex demands. NCL captures PSPACE and has been a useful tool for proving algorithmic hardness of many puzzles, games, and reconfiguration problems. In particular, its usefulness stems from the fact that it remains PSPACE-complete even under severe restrictions of the weights (e.g., only edge-weights one and two are needed) and the structure of the constraint graph (e.g., planar AND/OR graphs of bounded bandwidth). While such restrictions on the structure of constraint graphs do not seem to limit the expressiveness of NCL, the building blocks of the constraint graphs cannot be limited without losing expressiveness: We consider as parameters the number of weight-one edges and the number of weight-two edges of a constraint graph, as well as the number of AND or OR vertices of an AND/OR constraint graph. We show that NCL is fixed-parameter tractable (FPT) for any of these parameters. In particular, for NCL parameterized by the number of weight-one edges or the number of AND vertices, we obtain a linear kernel. It follows that, in a sense, NCL as introduced by Hearn and Demaine is defined in the most economical way for the purpose of capturing PSPACE.

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
  • Mathematics of computing → Graph algorithms
  • Combinatorial Reconfiguration
  • Nondeterministic Constraint Logic
  • Fixed Parameter Tractability


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