Abstract
In the Token Jumping problem we are given a graph \(G = (V,E)\) and two independent sets S and T of G, each of size \(k \ge 1\). The goal is to determine whether there exists a sequence of k-sized independent sets in G, \(\langle S_0, S_1, \ldots , S_\ell \rangle\), such that for every i, \(|S_i| = k\), \(S_i\) is an independent set, \(S = S_0\), \(S_\ell = T\), and \(|S_i \varDelta S_{i+1}| = 2\). In other words, if we view each independent set as a collection of tokens placed on a subset of the vertices of G, then the problem asks for a sequence of independent sets which transforms S to T by individual token jumps which maintain the independence of the sets. This problem is known to be PSPACE-complete on very restricted graph classes, e.g., planar bounded degree graphs and graphs of bounded bandwidth. A closely related problem is the Token Sliding problem, where instead of allowing a token to jump to any vertex of the graph we instead require that a token slides along an edge of the graph. Token Sliding is also known to be PSPACE-complete on the aforementioned graph classes. We investigate the parameterized complexity of both problems on several graph classes, focusing on the effect of excluding certain cycles from the input graph. In particular, we show that both Token Sliding and Token Jumping are fixed-parameter tractable on \(C_4\)-free bipartite graphs when parameterized by k. For Token Jumping, we in fact show that the problem admits a polynomial kernel on \(\{C_3,C_4\}\)-free graphs. In the case of Token Sliding, we also show that the problem admits a polynomial kernel on bipartite graphs of bounded degree. We believe both of these results to be of independent interest. We complement these positive results by showing that, for any constant \(p \ge 4\), both problems are W[1]-hard on \(\{C_4, \dots , C_p\}\)-free graphs and Token Sliding remains W[1]-hard even on bipartite graphs.
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Notes
Informally, it means that they are very unlikely to admit an FPT algorithm.
The girth of a graph is the length of its shortest cycle.
References
Bartier, V., Bousquet, N., Dallard, C., Lomer, K., Mouawad, A.E.: On girth and the parameterized complexity of token sliding and token jumping. In: 31st International Symposium on Algorithms and Computation (ISAAC 2020), vol. 181, pp. 44:1–44:17. https://doi.org/10.4230/LIPIcs.ISAAC.2020.44
Belmonte, R., Kim, E.J., Lampis, M., Mitsou, V., Otachi, Y., Sikora, F.: Token sliding on split graphs. In: 36th International Symposium on Theoretical Aspects of Computer Science, STACS 2019, March 13–16, 2019, Berlin, Germany, pp. 13:1–13:17 (2019). https://doi.org/10.4230/LIPIcs.STACS.2019.13
Bonamy, M., Bousquet, N.: Token sliding on chordal graphs. CoRR (2016). arXiv:1605.00442
Bonnet, É., Bousquet, N., Charbit, P., Thomassé, S., Watrigant, R.: Parameterized complexity of independent set in H-free graphs. In: Paul, C., Pilipczuk, M. (eds.), 13th International Symposium on Parameterized and Exact Computation, IPEC 2018, August 20–24, 2018, Helsinki, Finland, LIPIcs, vol. 115, pp. 17:1–17:13. Schloss Dagstuhl—Leibniz-Zentrum für Informatik (2018). https://doi.org/10.4230/LIPIcs.IPEC.2018.17
Bonsma, P.S., Kaminski, M., Wrochna, M.: Reconfiguring independent sets in claw-free graphs. In: Algorithm Theory—SWAT 2014—14th Scandinavian Symposium and Workshops, Copenhagen, Denmark, July 2–4, 2014. Proceedings, pp. 86–97 (2014)
Bousquet, N., Mary, A., Parreau, A.: Token jumping in minor-closed classes. In: Fundamentals of Computation Theory—21st International Symposium, FCT 2017, Bordeaux, France, September 11–13, 2017, Proceedings, pp. 136–149 (2017). https://doi.org/10.1007/978-3-662-55751-8_12
Brewster, R.C., McGuinness, S., Moore, B., Noel, J.A.: A dichotomy theorem for circular colouring reconfiguration. Theor. Comput. Sci. 639, 1–13 (2016)
Cereceda, L., van den Heuvel, J., Johnson, M.: Connectedness of the graph of vertex-colourings. Discrete Math. 308(56), 913–919 (2008)
Cygan, M., Fomin, F.V., Kowalik, L., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, Berlin (2015). https://doi.org/10.1007/978-3-319-21275-3
Demaine, E.D., Demaine, M.L., Fox-Epstein, E., Hoang, D.A., Ito, T., Ono, H., Otachi, Y., Uehara, R., Yamada, T.: Polynomial-time algorithm for sliding tokens on trees. In: Algorithms and Computation. Lecture Notes in Computer Science, vol. 8889, pp. 389–400. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-13075-0_31
Fox-Epstein, E., Hoang, D.A., Otachi, Y., Uehara, R.: Sliding token on bipartite permutation graphs. In: Algorithms and Computation—26th International Symposium, ISAAC 2015, Nagoya, Japan, December 9–11, 2015, Proceedings, pp. 237–247 (2015)
Gharibian, S., Sikora, J.: Ground state connectivity of local Hamiltonians. In: Automata, Languages, and Programming—42nd International Colloquium, ICALP 2015, Kyoto, Japan, July 6–10, 2015, Proceedings, Part I, pp. 617–628 (2015). https://doi.org/10.1007/978-3-662-47672-7_50
Gopalan, P., Kolaitis, P.G., Maneva, E.N., Papadimitriou, C.H.: The connectivity of Boolean satisfiability: computational and structural dichotomies. SIAM J. Comput. 38(6), 2330–2355 (2009)
Hearn, R.A., Demaine, E.D.: PSPACE-completeness of sliding-block puzzles and other problems through the nondeterministic constraint logic model of computation. Theor. Comput. Sci. 343(1–2), 72–96 (2005). https://doi.org/10.1016/j.tcs.2005.05.008
Ito, T., Demaine, E.D., Harvey, N.J.A., Papadimitriou, C.H., Sideri, M., Uehara, R., Uno, Y.: On the complexity of reconfiguration problems. Theor. Comput. Sci. 412(12–14), 1054–1065 (2011). https://doi.org/10.1016/j.tcs.2010.12.005
Ito, T., Kamiński, M., Demaine, E.D.: Reconfiguration of list edge-colorings in a graph. Discrete Appl. Math. 160(15), 2199–2207 (2012)
Ito, T., Kaminski, M., Ono, H., Suzuki, A., Uehara, R., Yamanaka, K.: On the parameterized complexity for token jumping on graphs. In: Theory and Applications of Models of Computation—11th Annual Conference, TAMC 2014, Chennai, India, April 11–13, 2014. Proceedings, pp. 341–351 (2014)
Ito, T., Kami\(\acute{n}\)ski, M., Ono, H.: Fixed-parameter tractability of token jumping on planar graphs. In: Algorithms and Computation, Lecture Notes in Computer Science, pp. 208–219. Springer (2014)
Ito, T., Nooka, H., Zhou, X.: Reconfiguration of vertex covers in a graph. IEICE Trans. 99–D(3), 598–606 (2016)
Johnson, W.W., Story, W.E.: Notes on the “15” puzzle. Am. J. Math. 2(4), 397–404 (1879)
Kaminski, M., Medvedev, P., Milanic, M.: Complexity of independent set reconfigurability problems. Theor. Comput. Sci. 439, 9–15 (2012). https://doi.org/10.1016/j.tcs.2012.03.004
Kamiński, M., Medvedev, P., Milanič, M.: Complexity of independent set reconfigurability problems. Theor. Comput. Sci. 439, 9–15 (2012)
Kendall, G., Parkes, A.J., Spoerer, K.: A survey of NP-complete puzzles. ICGA J. 13–34 (2008)
Kim, J.H.: The Ramsey number \(R(3, t)\) has order of magnitude \(t^2/\log t\). Random Struct. Algorithms 7(3), 173–207 (1995). https://doi.org/10.1002/rsa.3240070302
Lokshtanov, D., Mouawad, A.E.: The complexity of independent set reconfiguration on bipartite graphs. ACM Trans. Algorithms 15(1), 7:1–7:19 (2019). https://doi.org/10.1145/3280825
Lokshtanov, D., Mouawad, A.E., Panolan, F., Ramanujan, M.S., Saurabh, S.: Reconfiguration on sparse graphs. In: Algorithms and Data Structures—14th International Symposium, WADS 2015, Victoria, BC, Canada, August 5–7, 2015. Proceedings, pp. 506–517 (2015)
Lubiw, A., Pathak, V.: Flip distance between two triangulations of a point set is NP-complete. Comput. Geom. 49, 17–23 (2015). https://doi.org/10.1016/j.comgeo.2014.11.001
Mouawad, A.E., Nishimura, N., Pathak, V., Raman, V.: Shortest reconfiguration paths in the solution space of Boolean formulas. In: Automata, Languages, and Programming—42nd International Colloquium, ICALP 2015, Kyoto, Japan, July 6–10, 2015, Proceedings, Part I, pp. 985–996 (2015)
Mouawad, A.E., Nishimura, N., Raman, V.: Vertex cover reconfiguration and beyond. In: Algorithms and Computation—25th International Symposium, ISAAC 2014, Jeonju, Korea, December 15–17, 2014, Proceedings, pp. 452–463 (2014)
Nishimura, N.: Introduction to reconfiguration. Algorithms 11(4), 52 (2018). https://doi.org/10.3390/a11040052
van den Heuvel, J.: The complexity of change. Surv. Combin. 2013(409), 127–160 (2013)
Wrochna, M.: Reconfiguration in bounded bandwidth and tree depth. CoRR (2014). arXiv:1405.0847
Wrochna, M.: Homomorphism reconfiguration via homotopy. In: 32nd International Symposium on Theoretical Aspects of Computer Science, STACS 2015, March 4–7, 2015, Garching, Germany, pp. 730–742 (2015)
Zuckerman, D.: Linear degree extractors and the inapproximability of max clique and chromatic number. Theory Comput. 3(1), 103–128 (2007). https://doi.org/10.4086/toc.2007.v003a006
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A preliminary version of this paper appeared in the Proceedings of the 31st International Symposium on Algorithms and Computation (ISAAC 2020).
The first two authors are supported by ANR project GrR (ANR-18-CE40-0032). The third author is supported in part by the Slovenian Research Agency (Research Project N1-0102).
The fifth author is supported by URB project “A theory of change through the lens of reconfiguration.”
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Bartier, V., Bousquet, N., Dallard, C. et al. On Girth and the Parameterized Complexity of Token Sliding and Token Jumping. Algorithmica 83, 2914–2951 (2021). https://doi.org/10.1007/s00453-021-00848-1
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DOI: https://doi.org/10.1007/s00453-021-00848-1