Abstract
Suppose that we are given two independent sets I 0 and I r of a graph such that ∣ I 0 ∣ = ∣ I r ∣, and imagine that a token is placed on each vertex in I 0. Then, the token jumping problem is to determine whether there exists a sequence of independent sets which transforms I 0 into I r so that each independent set in the sequence results from the previous one by moving exactly one token to another vertex. Therefore, all independent sets in the sequence must be of the same cardinality. This problem is PSPACE-complete even for planar graphs with maximum degree three. In this paper, we first show that the problem is W[1]-hard when parameterized only by the number of tokens. We then give an FPT algorithm for general graphs when parameterized by both the number of tokens and the maximum degree. Our FPT algorithm can be modified so that it finds an actual sequence of independent sets between I 0 and I r with the minimum number of token movements.
This work is partially supported by JSPS KAKENHI Grant Numbers 25106504 (Ito), 25104521 (Ono), 24106004 (Ono and Uehara), 24.3660 (Suzuki) and 25106502 (Yamanaka).
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References
Bonamy, M., Johnson, M., Lignos, I., Patel, V., Paulusma, D.: On the diameter of reconfiguration graphs for vertex colourings. Electronic Notes in Discrete Mathematics 38, 161–166 (2011)
Bonsma, P., Cereceda, L.: Finding paths between graph colourings: PSPACE-completeness and superpolynomial distances. Theoretical Computer Science 410, 5215–5226 (2009)
Cereceda, L., van den Heuvel, J., Johnson, M.: Finding paths between 3-colourings. J. Graph Theory 67, 69–82 (2011)
Gopalan, P., Kolaitis, P.G., Maneva, E.N., Papadimitriou, C.H.: The connectivity of Boolean satisfiability: computational and structural dichotomies. SIAM J. Computing 38, 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. Theoretical Computer Science 343, 72–96 (2005)
Hearn, R.A., Demaine, E.D.: Games, Puzzles, and Computation. A K Peters (2009)
Ito, T., Demaine, E.D.: Approximability of the subset sum reconfiguration problem. To appear in J. Combinatorial Optimization, doi:10.1007/s10878-012-9562-z
Ito, T., Demaine, E.D., Harvey, N.J.A., Papadimitriou, C.H., Sideri, M., Uehara, R., Uno, Y.: On the complexity of reconfiguration problems. Theoretical Computer Science 412, 1054–1065 (2011)
Ito, T., Kamiński, M., Demaine, E.D.: Reconfiguration of list edge-colorings in a graph. Discrete Applied Mathematics 160, 2199–2207 (2012)
Ito, T., Kawamura, K., Ono, H., Zhou, X.: Reconfiguration of list L(2,1)-labelings in a graph. In: Chao, K.-M., Hsu, T.-S., Lee, D.-T. (eds.) ISAAC 2012. LNCS, vol. 7676, pp. 34–43. Springer, Heidelberg (2012)
Ito, T., Kawamura, K., Zhou, X.: An improved sufficient condition for reconfiguration of list edge-colorings in a tree. IEICE Trans. on Information and Systems E95-D, 737–745 (2012)
Kamiński, M., Medvedev, P., Milanič, M.: Shortest paths between shortest paths. Theoretical Computer Science 412, 5205–5210 (2011)
Kamiński, M., Medvedev, P., Milanič, M.: Complexity of independent set reconfigurability problems. Theoretical Computer Science 439, 9–15 (2012)
Makino, K., Tamaki, S., Yamamoto, M.: An exact algorithm for the Boolean connectivity problem for k-CNF. Theoretical Computer Science 412, 4613–4618 (2011)
Mouawad, A.E., Nishimura, N., Raman, V., Simjour, N., Suzuki, A.: On the parameterized complexity of reconfiguration problems. In: Gutin, G., Szeider, S. (eds.) IPEC 2013. LNCS, vol. 8246, pp. 281–294. Springer, Heidelberg (2013)
Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press (2006)
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Ito, T., Kamiński, M., Ono, H., Suzuki, A., Uehara, R., Yamanaka, K. (2014). On the Parameterized Complexity for Token Jumping on Graphs. In: Gopal, T.V., Agrawal, M., Li, A., Cooper, S.B. (eds) Theory and Applications of Models of Computation. TAMC 2014. Lecture Notes in Computer Science, vol 8402. Springer, Cham. https://doi.org/10.1007/978-3-319-06089-7_24
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DOI: https://doi.org/10.1007/978-3-319-06089-7_24
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