Abstract
The game of rendezvous with adversaries is a game on a graph played by two players: Facilitator and Divider. Facilitator has two agents and Divider has a team of \(k \ge 1\) agents. While the initial positions of Facilitator’s agents are fixed, Divider gets to select the initial positions of his agents. Then, they take turns to move their agents to adjacent vertices (or stay put) with Facilitator’s goal to bring both her agents at same vertex and Divider’s goal to prevent it. The computational question of interest is to determine if Facilitator has a winning strategy against Divider with k agents. Fomin, Golovach, and Thilikos [WG, 2021] introduced this game and proved that it is PSPACE-hard and co-W[2]-hard parameterized by the number of agents. This hardness naturally motivates the structural parameterization of the problem. The authors proved that it admits an FPT algorithm when parameterized by the modular width and the number of allowed rounds. However, they left open the complexity of the problem from the perspective of other structural parameters. In particular, they explicitly asked whether the problem admits an FPT or XP-algorithm with respect to the treewidth of the input graph. We answer this question in the negative and show that Rendezvous is co-NP-hard even for graphs of constant treewidth. Further, we show that the problem is co-W[1]-hard when parameterized by the feedback vertex set number and the number of agents, and is unlikely to admit a polynomial kernel when parameterized by the vertex cover number and the number of agents. Complementing these hardness results, we show that the Rendezvous is FPT when parameterized by both the vertex cover number and the solution size. Finally, for graphs of treewidth at most two and girds, we show that the problem can be solved in polynomial time.
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Notes
We use i as well as \(a_1, b_1, c_1\) as running variables in set [n]. We reserve later types of variables for the integer part of elements in sets \(\mathcal {F}\).
It is tempting to imagine that if \((\alpha , a_1)\) is not covered by \(\mathcal {F}^{\prime }\) than Facilitator can move Romeo and Juliet at \(x^{\ell }_{a_1}\) or at \(x^{r}_{a_1}\). However, we argue that Facilitator can move Romeo and Juliet at \(x^{\ell }_{a_2}\) for some \(a_2 \le a_1\) or at \(x^{r}_{a_3}\) for some \(a_3 \ge a_1\).
References
Bringmann, K., Hermelin, D., Mnich, M., Van Leeuwen, E.J.: Parameterized complexity dichotomy for steiner multicut. J. Comput. Syst. Sci. 82(6), 1020–1043 (2016). https://doi.org/10.1016/j.jcss.2016.03.003
Cygan, M., Fomin, F.V., Kowalik, L., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer (2015). https://doi.org/10.1007/978-3-319-21275-3
Diestel, R.: Graph Theory, 4th Edition, vol. 173 of Graduate Texts in Mathematics. Springer (2012)
Fomin, F.V., Golovach, P.A., Thilikos, D.M.: Can Romeo and Juliet meet? Or rendezvous games with adversaries on graphs. Inf. Comput. 293, 105049 (2023). https://doi.org/10.1016/j.ic.2023.105049
Fomin, F.V., Lokshtanov, D., Saurabh, S., Zehavi, M.: Kernelization: Theory of Parameterized Preprocessing. Cambridge University Press (2019). https://doi.org/10.1017/9781107415157
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman (1979)
Acknowledgements
We are grateful for feedback from anonymous reviewers. We especially thank the reviewers of Algorithmica as their feedback helped us to improve the presentation. The first reviewer also suggested a much simpler proof of Lemma 13.
Funding
Neeldhara Misra: The author is grateful for support from DST-SERB and IIT Gandhinagar. This work was partially supported by the ECR grant ECR/2018/002967. Prafullkumar Tale: Part of the work was carried out when the author was a Post-Doctoral Researcher at CISPA Helmholtz Center for Information Security, Germany, supported by the European Research Council (ERC) consolidator Grant No. 725978 SYSTEMATICGRAPH.
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Related Version: A shorter version of this work has been accepted for presentation at the 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS), 2022.
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Misra, N., Mulpuri, M., Tale, P. et al. Romeo and Juliet Meeting in Forest Like Regions. Algorithmica 86, 3465–3495 (2024). https://doi.org/10.1007/s00453-024-01264-x
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DOI: https://doi.org/10.1007/s00453-024-01264-x