Abstract
We give a novel general approach for solving NP-hard optimization problems that combines dynamic programming and fast matrix multiplication. The technique is based on reducing much of the computation involved to matrix multiplication. We show that our approach works faster than the usual dynamic programming solution for any vertex subset problem on graphs of bounded branchwidth. In particular, we obtain the fastest algorithms for Planar Independent Set of runtime \(O(2^{2.52 \sqrt{n}})\), for Planar Dominating Set of runtime exact \(O(2^{3.99 \sqrt{n}})\) and parameterized \(O(2^{11.98 \sqrt{k}}) \cdot n^{O(1)}\), and for Planar Hamiltonian Cycle of runtime \(O(2^{5.58 \sqrt{n}})\). The exponent of the running time is depending heavily on the running time of the fastest matrix multiplication algorithm that is currently o(n 2.376).
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References
Alber, J., Bodlaender, H.L., Fernau, H., Kloks, T., Niedermeier, R.: Fixed parameter algorithms for dominating set and related problems on planar graphs. Algorithmica 33, 461–493 (2002)
Alber, J., Niedermeier, R.: Improved tree decomposition based algorithms for domination-like problems. In: Rajsbaum, S. (ed.) LATIN 2002. LNCS, vol. 2286, pp. 613–627. Springer, Heidelberg (2002)
Alon, N., Galil, Z., Margalit, O.: On the exponent of the all pairs shortest path problem. Journal of Computer and System Sciences 54, 255–262 (1997)
Cook, W., Seymour, P.: Tour merging via branch-decomposition. INFORMS Journal on Computing 15, 233–248 (2003)
Coppersmith, D.: Rectangular matrix multiplication revisited. Journal of Complexity 13, 42–49 (1997)
Coppersmith, D., Winograd, S.: Matrix multiplication via arithmetic progressions. Journal of Symbolic Computation 9, 251–280 (1990)
Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 2nd edn. MIT Press and McGraw-Hill Book Company (2001)
Dorn, F., Penninkx, E., Bodlaender, H., Fomin, F.V.: Efficient exact algorithms on planar graphs: Exploiting sphere cut branch decompositions. In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, pp. 95–106. Springer, Heidelberg (2005)
Dorn, F., Penninkx, E., Bodlaender, H., Fomin, F.V.: Efficient exact algorithms on planar graphs: Exploiting sphere cut decompositions (manuscript, 2006), http://archive.cs.uu.nl/pub/RUU/CS/techreps/CS-2006/2006-006.pdf
Dorn, F., Telle, J.A.: Two birds with one stone: the best of branchwidth and treewidth with one algorithm. In: Correa, J.R., Hevia, A., Kiwi, M. (eds.) LATIN 2006. LNCS, vol. 3887, pp. 386–397. Springer, Heidelberg (2006)
Fomin, F.V., Thilikos, D.M.: Dominating sets in planar graphs: branch-width and exponential speed-up. In: SODA 2003: Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, Baltimore, MD, pp. 168–177. ACM, New York (2003)
Fomin, F.V., Thilikos, D.M.: Fast parameterized algorithms for graphs on surfaces: Linear kernel and exponential speed-up. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 581–592. Springer, Heidelberg (2004)
Fomin, F.V., Thilikos, D.M.: A simple and fast approach for solving problems on planar graphs. In: Diekert, V., Habib, M. (eds.) STACS 2004. LNCS, vol. 2996, pp. 56–67. Springer, Heidelberg (2004)
Gu, Q.-P., Tamaki, H.: Optimal branch-decomposition of planar graphs in O(n 3) time. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 373–384. Springer, Heidelberg (2005)
Heggernes, P., Telle, J.A., Villanger, Y.: Computing minimal triangulations in time O(n α log n) = o(n 2.376). SIAM Journal on Discrete Mathematics 19, 900–913 (2005)
Itai, A., Rodeh, M.: Finding a minimum circuit in a graph. SIAM Journal on Computing 7, 413–423 (1978)
Kratsch, D., Spinrad, J.: Between O(nm) and O(n α). In: SODA 2003: Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, Baltimore, MD, 2003, pp. 158–167. ACM, New York (2003)
Seidel, R.: On the all-pairs-shortest-path problem in unweighted undirected graphs. Journal of Computer and System Sciences 51, 400–403 (1995)
Seymour, P.D., Thomas, R.: Call routing and the ratcatcher. Combinatorica 14, 217–241 (1994)
Shoshan, A., Zwick, U.: All pairs shortest paths in undirected graphs with integer weights. In: 40th Annual Symposium on Foundations of Computer Science (FOCS 1999). LNCS, pp. 605–615. Springer, Heidelberg (1999)
Telle, J.A., Proskurowski, A.: Algorithms for vertex partitioning problems on partial k-trees. SIAM J. Discrete Math 10, 529–550 (1997)
Vaidya, P.M.: Speeding-up linear programming using fast matrix multiplication. In: 30th Annual Symposium on Foundations of Computer Science (FOCS 1989), pp. 332–337 (1989)
Vassilevska, V., Williams, R.: Finding a maximum weight triangle in n (3 − δ) time, with applications. In: ACM Symposium on Theory of Computing (STOC 2006) (to appear, 2006), http://www.cs.cmu.edu/~ryanw/max-weight-triangle.pdf
Williams, R.: A new algorithm for optimal constraint satisfaction and its implications. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 1227–1237. Springer, Heidelberg (2004)
Zwick, U.: All pairs shortest paths using bridging sets and rectangular matrix multiplication. Journal of the ACM 49, 289–317 (2002)
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Dorn, F. (2006). Dynamic Programming and Fast Matrix Multiplication. In: Azar, Y., Erlebach, T. (eds) Algorithms – ESA 2006. ESA 2006. Lecture Notes in Computer Science, vol 4168. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11841036_27
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DOI: https://doi.org/10.1007/11841036_27
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