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Exact Algorithms for Exact Satisfiability and Number of Perfect Matchings

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Abstract

We present exact algorithms with exponential running times for variants of n-element set cover problems, based on divide-and-conquer and on inclusion–exclusion characterizations.

We show that the Exact Satisfiability problem of size l with m clauses can be solved in time 2m l O(1) and polynomial space. The same bounds hold for counting the number of solutions. As a special case, we can count the number of perfect matchings in an n-vertex graph in time 2n n O(1) and polynomial space. We also show how to count the number of perfect matchings in time O(1.732n) and exponential space.

We give a number of examples where the running time can be further improved if the hypergraph corresponding to the set cover instance has low pathwidth. This yields exponential-time algorithms for counting k-dimensional matchings, Exact Uniform Set Cover, Clique Partition, and Minimum Dominating Set in graphs of degree at most three.

We extend the analysis to a number of related problems such as TSP and Chromatic Number.

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References

  1. Angelsmark, O., Thapper, J.: Partitioning based algorithms for some colouring problems. In: Recent Advances in Constraints. Lecture Notes in Artificial Intelligence, vol. 3978, pp. 44–58. Springer, Berlin (2005)

    Chapter  Google Scholar 

  2. Bax, E.T.: Inclusion and exclusion algorithm for the Hamiltonian Path problem. Inf. Process. Lett. 47(4), 203–207 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  3. Björklund, A., Husfeldt, T. Koivisto, M.: Set partitioning via inclusion–exclusion. SIAM J. Comput., to appear. Prelim. versions in Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science, Berkeley, CA, 21–24 October 2006, pp. 575–582, 583–590. IEEE Computer Society, Los Alamitos, CA (2006)

  4. Bodlaender, H., Kratsch, D.: An exact algorithm for graph coloring with polynomial memory. Technical report UU-CS-2006-015, Utrecht University (2006)

  5. Byskov, J.M.: Enumerating maximal independent sets with applications to graph colouring. Oper. Res. Lett. 32, 547–556 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  6. Byskov, J.M.: Exact algorithms for graph colouring and exact satisfiability. Ph.D. thesis, University of Aarhus (2004)

  7. Byskov, J.M., Madsen, B.A., Skjernaa, B.: New algorithms for Exact Satisfiability. Theor. Comput. Sci. 332(1–3), 515–541 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  8. Chien, S.: A determinant-based algorithm for counting perfect matchings in a general graph. In: Proc. 15th SODA, pp. 728–735 (2004)

  9. Christofides, N.: An algorithm for the chromatic number of a graph. Comput. J. 14, 38–39 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  10. Coppersmith, D., Winograd, S.: Matrix multiplication via arithmetic progressions. J. Symb. Comput. 9(3), 251–280 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  11. Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, New York (1999). ISBN 0-387-94883-X

    Google Scholar 

  12. Dahllöf, V., Jonsson, P., Beigel, R.: Algorithms for four variants of exact satisfiability. Theor. Comput. Sci. 320(2–3), 373–394 (2004)

    Article  MATH  Google Scholar 

  13. Davis, M., Logemann, G., Loveland, D.: A machine program for theorem proving. Commun. ACM 5(7), 394–397 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  14. Eppstein, D.: Small maximal independent sets and faster exact graph coloring. J. Graph Algorithms Appl. 7(2), 131–140 (2003)

    MATH  MathSciNet  Google Scholar 

  15. Fomin, F.V., Grandoni, F., Kratsch, D.: Some new techniques in design and analysis of exact (exponential) algorithms. In: Bull. EATCS, vol. 87 (2005)

  16. Fomin, F.V., Grandoni, F., Kratsch, D.: Measure and conquer: domination—a case study. In: Proceedings of the 32nd ICALP, pp. 191–203 (2005)

  17. Fomin, F.V., Höie, K.: Pathwidth of cubic graphs and exact algorithms. Inf. Process. Lett. 97(5), 191–196 (2006)

    Article  Google Scholar 

  18. Feder, T., Motwani, R.: Worst-case time bounds for coloring and satisfiability problems. J. Algorithms 45(2), 192–201 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  19. Feige, U., Kilian, J.: Exponential time algorithms for computing the bandwidth of a graph. Manuscript. Cited in [38]

  20. Gurevich, Y., Shelah, S.: Expected computation time for Hamiltonian path problem. SIAM J. Comput. 16(3), 486–502 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  21. Hujter, M., Tuza, Z.: On the number of maximal independent sets in triangle-free graphs. SIAM J. Discrete Math. 6(2), 284–288 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  22. Jerrum, M., Sinclair, A., Vigoda, E.: A polynomial-time approximation algorithm for the permanent of a matrix with non-negative entries. In: Proc. 33rd STOC, pp. 712–721 (2001)

  23. Karp, R.M.: Dynamic programming meets the principle of inclusion-exclusion. Oper. Res. Lett. 1, 49–51 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  24. Kohn, S., Gottlieb, A., Kohn, M.: A generating function approach to the Traveling Salesman Problem. In: ACM ’77: Proceedings of the 1977 annual conference, pp. 294–300. ACM Press, New York (1977)

    Chapter  Google Scholar 

  25. Kneis, J., Mölle, D., Richter, S., Rossmanith, P.: Algorithms based on the treewidth of sparse graphs. In: Proceedings of the 31st International Workshop on Graph-Theoretic Concepts in Computer Science, pp. 385–396 (2005)

  26. Lawler, E.L.: A note on the complexity of the chromatic number problem. Inf. Process. Lett. 5(3), 66–67 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  27. Lovász, L., Plummer, M.D.: Matching Theory. North-Holland, Amsterdam (1986)

    MATH  Google Scholar 

  28. Madsen, B.A.: An algorithm for exact satisfiability analysed with the number of clauses as parameter. Inf. Process. Lett. 97(1), 28–30 (2006)

    Article  MathSciNet  Google Scholar 

  29. Moon, J.W., Moser, L.: On cliques in graphs. Isr. J. Math. (1965)

  30. Monien, B., Speckenmeyer, E., Vornberger, O.: Upper bounds for covering problems. Methods Oper. Res. 43, 419–431 (1981)

    MATH  MathSciNet  Google Scholar 

  31. Porschen, S.: On some weighted satisfiability and graph problems. In: Proc. 31st SOFSEM. Lecture Notes in Computer Science, vol. 3381, pp. 278–287 (2005)

  32. Ryser, H.J.: Combinatorial Mathematics. Carus Math. Monographs, no. 14. Math. Assoc. of America, Washington, DC (1963)

    MATH  Google Scholar 

  33. Tsukiyama, S., Ide, M., Ariyoshi, H., Shirakawa, I.: A new algorithm for generating all the maximal independent sets. SIAM J. Comput. 6(3), 505–517 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  34. Vadhan, S.P.: The complexity of counting in sparse, regular, and planar graps. SIAM J. Comput. 32(2), 398–427 (2001)

    Article  MathSciNet  Google Scholar 

  35. Valiant, L.G.: The complexity of computing the permanent. Theor. Comput. Sci. 8, 189–201 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  36. Williams, R.: A new algorithm for optimal 2-constraint satisfaction and its implications. Theor. Comput. Sci. 348(1–2), 357–365 (2005)

    Article  MATH  Google Scholar 

  37. Woeginger, G.J.: Exact algorithms for NP-hard problems: a survey. In: Combinatorial Optimization: Eureka, You Shrink!, pp. 185–207. Springer, Berlin (2003)

    Chapter  Google Scholar 

  38. Woeginger, G.J.: Space and time complexity of exact algorithms: Some open problems. In: Proc. 1st IWPEC. Lecture Notes in Computer Science, vol. 3162, pp. 281–290 (2004)

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Correspondence to Thore Husfeldt.

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Björklund, A., Husfeldt, T. Exact Algorithms for Exact Satisfiability and Number of Perfect Matchings. Algorithmica 52, 226–249 (2008). https://doi.org/10.1007/s00453-007-9149-8

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