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Romeo and Juliet Is EXPTIME-Complete

Authors Harmender Gahlawat , Jan Matyáš Křišťan, Tomáš Valla

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
Keywords
  • Rendezvous Games on graphs
  • EXPTIME-completeness
  • Dynamic Separators

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Abstract

Romeo and Juliet is a two player Rendezvous game played on graphs where one player controls two agents, Romeo (ℛ) and Juliet (𝒥) who aim to meet at a vertex against k adversaries, called dividers, controlled by the other player. The optimization in this game lies at deciding the minimum number of dividers sufficient to restrict ℛ and 𝒥 from meeting in a graph, called the dynamic separation number. We establish that Romeo and Juliet is EXPTIME-complete, settling a conjecture of Fomin, Golovach, and Thilikos [Inf. and Comp., 2023] positively. We also consider the game for directed graphs and establish that although the game is EXPTIME-complete for general directed graphs, it is PSPACE-complete and co-W[2]-hard for directed acyclic graphs.

Cite As Get BibTex

Harmender Gahlawat, Jan Matyáš Křišťan, and Tomáš Valla. Romeo and Juliet Is EXPTIME-Complete. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 54:1-54:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.MFCS.2024.54

Author Details

Harmender Gahlawat
  • G-SCOP, Grenoble-INP, France
Jan Matyáš Křišťan
  • Faculty of Information Technology, Czech Technical University in Prague, Czech Republic
Tomáš Valla
  • Faculty of Information Technology, Czech Technical University in Prague, Czech Republic

Funding

This work was supported by the Czech Science Foundation Grant no. 24-12046S. This work was supported by the Grant Agency of the Czech Technical University in Prague, grant No. SGS23/205/OHK3/3T/18.

References

  1. M. Aigner and M. Fromme. A game of cops and robbers. Discrete Applied Mathematics, 8(1):1-12, 1984. Google Scholar
  2. S. Alpern. The rendezvous search problem. SIAM Journal on Control and Optimization, 33(3):673-683, 1995. Google Scholar
  3. S. Alpern and S. Gal. The theory of search games and rendezvous, volume 55. Springer Science & Business Media, 2006. Google Scholar
  4. J. Barát. Directed path-width and monotonicity in digraph searching. Graphs and Combinatorics, 22(2):161-172, 2006. Google Scholar
  5. D. Berwanger, A. Dawar, P. Hunter, S. Kreutzer, and J. Obdržálek. The dag-width of directed graphs. Journal of Combinatorial Theory Series B, 104(4):900-923, 2012. Google Scholar
  6. A. Bonato. An invitation to pursuit-evasion games and graph theory, volume 97. American Mathematical Society, 2022. Google Scholar
  7. A. Bonato and P. Pralat. Graph searching games and probabilistic methods. Chapman and Hall/CRC, 2017. Google Scholar
  8. P. Bradshaw, S. A. Hosseini, and J. Turcotte. Cops and robbers on directed and undirected abelian Cayley graphs. European Journal of Combinatorics, 97, 2021. URL: https://doi.org/10.1016/j.ejc.2021.103383.
  9. S. Das, H. Gahlawat, U. k. Sahoo, and S. Sen. Cops and robber on some families of oriented graphs. Theoretical Computer Science, 888:31-40, 2021. Google Scholar
  10. A. Dessmark, P. Fraigniaud, D. R. Kowalski, and A. Pelc. Deterministic rendezvous in graphs. Algorithmica, 46:69-96, 2006. Google Scholar
  11. F. Fomin, P. Golovach, A. Hall, M. Mihalák, E. Vicari, and P. Widmayer. How to guard a graph? Algorithmica, 61(4):839-856, 2011. Google Scholar
  12. F. V. Fomin, P. A. Golovach, and D. M. Thilikos. Can romeo and juliet meet? or rendezvous games with adversaries on graphs. Information and Computation, 293:105049, 2023. Google Scholar
  13. F. V. Fomin and D. M. Thilikos. An annotated bibliography on guaranteed graph searching. Theoretical Computer Science, 399(3):236-245, 2008. Google Scholar
  14. F.V. Fomin, P.A. Golovach, and D. Lokshtanov. Guard games on graphs: Keep the intruder out! Theoretical Computer Science, 412(46):6484-6497, 2011. Google Scholar
  15. T. Johnson, N. Robertson, P. D. Seymour, and R. Thomas. Directed tree-width. Journal of Combinatorial Theory Series B, 82(1):138-154, 2001. Google Scholar
  16. W. B. Kinnersley. Cops and robbers is exptime-complete. Journal of Combinatorial Theory Series B, 111:201-220, 2015. Google Scholar
  17. P. Loh and S. Oh. Cops and robbers on planar directed graphs. Journal of Graph Theory, 86(3):329-340, 2017. Google Scholar
  18. N. Misra, M. Mulpuri, P. Tale, and G. Viramgami. Romeo and juliet meeting in forest like regions. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS), 2022. Google Scholar
  19. H. Nagamochi. Cop-robber guarding game with cycle robber-region. Theoretical Computer Science, 412:383-390, 2011. Google Scholar
  20. R. Nowakowski and P. Winkler. Vertex-to-vertex pursuit in a graph. Discrete Mathematics, 43(2-3):235-239, 1983. Google Scholar
  21. A. Quilliot. Thése d'Etat. PhD thesis, Université de Paris VI, 1983. Google Scholar
  22. R. Šámal, R. Stolař, and T. Valla. Complexity of the cop and robber guarding game. In Combinatorial Algorithms: 22nd International Workshop, IWOCA 2011, Victoria, BC, Canada, July 20-22, 2011, Revised Selected Papers 22, pages 361-373. Springer, 2011. Google Scholar
  23. R. Šámal and T. Valla. The guarding game is E-complete. Theoretical Computer Science, 521:92-106, 2014. Google Scholar
  24. A. Ta-Shma and U. Zwick. Deterministic rendezvous, treasure hunts, and strongly universal exploration sequences. ACM Transactions on Algorithms (TALG), 10(3):1-15, 2014. Google Scholar
  25. T. V. Thirumala Reddy, D. Sai Krishna, and C. Pandu Rangan. The guarding problem - complexity and approximation. In Lecture Notes in Computer Science, volume 5874, pages 460-470. Springer, 2009. Google Scholar
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