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Computational Complexity of Swish Is Solved

Authors Takashi Horiyama , Takehiro Ito , Jun Kawahara , Shin-ichi Minato , Akira Suzuki , Ryuhei Uehara , Yutaro Yamaguchi

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LIPIcs.FUN.2026.25.pdf
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Document Identifiers

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Graph algorithms analysis
Keywords
  • Swish
  • Computational complexity
  • Matching
  • Parity-constrained cycles

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Abstract

Swish is a card game in which players are given cards having symbols (hoops and balls), and find a valid superposition of cards, called a "swish." Dailly, Lafourcade, and Marcadet (FUN 2024) studied a generalized version of Swish and showed that the problem is solvable in polynomial time with one symbol per card, while it is NP-complete with three or more symbols per card. In this paper, we resolve the previously open case of two symbols per card, which corresponds to the original game. We show that Swish is NP-complete for this case. Specifically, we prove the NP-hardness when the allowed transformations of cards are restricted to a single (horizontal or vertical) flip or 180-degree rotation, and extend the results to the original setting allowing all three transformations. In contrast, when neither transformation is allowed, we present a polynomial-time algorithm. Combining known and our results, we establish a complete characterization of the computational complexity of Swish with respect to both the number of symbols per card and the allowed transformations.

Cite As Get BibTex

Takashi Horiyama, Takehiro Ito, Jun Kawahara, Shin-ichi Minato, Akira Suzuki, Ryuhei Uehara, and Yutaro Yamaguchi. Computational Complexity of Swish Is Solved. In 13th International Conference on Fun with Algorithms (FUN 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 366, pp. 25:1-25:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.FUN.2026.25

Author Details

Takashi Horiyama
  • Faculty of Information Science and Technology, Hokkaido University, Sapporo, Japan
Takehiro Ito
  • Graduate School of Information Sciences, Tohoku University, Sendai, Japan
Jun Kawahara
  • Graduate School of Informatics, Kyoto University, Kyoto, Japan
Shin-ichi Minato
  • Graduate School of Informatics, Kyoto University, Kyoto, Japan
Akira Suzuki
  • Center for Data-driven Science and Artificial Intelligence, Tohoku University, Sendai, Japan
Ryuhei Uehara
  • School of Information Science, Japan Advanced Institute of Science and Technology, Nomi, Japan
Yutaro Yamaguchi
  • Graduate School of Information Science and Technology, Osaka University, Suita, Japan

Funding

General: JSPS/MEXT KAKENHI Grant Numbers JP20H00605, JP20H05964, JP24H00690, JP25H01114
  • Horiyama, Takashi: JSPS KAKENHI Grant Number JP23K24806.
  • Ito, Takehiro: JSPS KAKENHI Grant Number JP24H00686.
  • Suzuki, Akira: JSPS KAKENHI Grant Number JP25K14980.
  • Yamaguchi, Yutaro: JST CRONOS Japan Grant Number JPMJCS24K2.

Acknowledgements

We would like to thank the reviewers for enjoying the paper and giving helpful comments.

References

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