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Below All Subsets for Some Permutational Counting Problems

Author Andreas Björklund

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Andreas Björklund

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Andreas Björklund. Below All Subsets for Some Permutational Counting Problems. In 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 53, pp. 17:1-17:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
https://doi.org/10.4230/LIPIcs.SWAT.2016.17

Abstract

We show that the two problems of computing the permanent of an n*n matrix of poly(n)-bit integers and counting the number of Hamiltonian cycles in a directed n-vertex multigraph with exp(poly(n)) edges can be reduced to relatively few smaller instances of themselves. In effect we derive the first deterministic algorithms for these two problems that run in o(2^n) time in the worst case. Classic poly(n)2^n time algorithms for the two problems have been known since the early 1960's. Our algorithms run in 2^{n-Omega(sqrt{n/log(n)})} time.
Keywords
  • Matrix Permanent
  • Hamiltonian Cycles
  • Asymmetric TSP

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References

  1. E. T. Bax. Inclusion and exclusion algorithm for the hamiltonian path problem. Inform. Process. Lett, 47:203-207, 1993. Google Scholar
  2. E. T. Bax and J. Franklin. A permanent algorithm with exp[ω(n ^1/3/2 ln(n))] expected speedup for 0-1 matrices. Algorithmica, 32:157-162, 2002. Google Scholar
  3. R. Bellman. Dynamic programming treatment of the travelling salesman problem. J. Assoc. Comput. Mach, 9:61-63, 1962. Google Scholar
  4. A. Björklund. Counting perfect matchings as fast as ryser. In Proceedings of the ACM-SIAM SODA, pages 914-921, 2012. Google Scholar
  5. A. Björklund. Determinant sums for undirected hamiltonicity. SIAM J. Comput., 43:280-299, 2014. Google Scholar
  6. A. Björklund, H. Dell, and T. Husfeldt. The parity of set systems under random restrictions with applications to exponential time problems. In Proceedings of ICALP,, pages 231-242, 2015. Google Scholar
  7. A. Björklund and T. Husfeldt. The parity of directed hamiltonian cycles. In Proceedings of the IEEE FOCS,, pages 724-735, 2013. Google Scholar
  8. M. Cygan, S. Kratsch, and J. Nederlof. Fast hamiltonicity checking via bases of perfect matchings. In Proceedings of the ACM STOC, pages 301-310, 2013. Google Scholar
  9. M. Held and R. M. Karp. A dynamic programming approach to sequencing problems. J. Soc. Indust. Appl. Math., 10:196-210, 1962. Google Scholar
  10. R. M. Karp. Dynamic programming meets the principle of inclusion and exclusion. Oper. Res. Lett., 1:49-51, 1982. Google Scholar
  11. D. E. Knuth. The Art of Computer Programming. Vol. 2: Seminumerical Algorithms. Addison-Wesley, third edition, 1997. Google Scholar
  12. S. Kohn, A. Gottlieb, and M. Kohn. A generating function approach to the traveling salesman problem. In Proceedings of the Annual Conference (ACM'77), Association for Computing Machinery, pages 294-300, 1977. Google Scholar
  13. B. Rosser. Explicit bounds for some functions of prime numbers. American Journal of Mathematics, 63:211-232, 1941. Google Scholar
  14. H. J. Ryser. Combinatorial Mathematics. The Carus mathematical monographs, The Mathematical Association of America, 1963. Google Scholar
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