Families with Infants: A General Approach to Solve Hard Partition Problems
We introduce a general approach for solving partition problems where the goal is to represent a given set as a union (either disjoint or not) of subsets satisfying certain properties. Many NP-hard problems can be naturally stated as such partition problems. We show that if one can find a large enough system of so-called families with infants for a given problem, then this problem can be solved faster than by a straightforward algorithm. We use this approach to improve known bounds for several NP-hard problems (the traveling salesman problem, the graph coloring problem, the problem of counting perfect matchings) on graphs of bounded average degree, as well as to simplify the proofs of several known results.
Unable to display preview. Download preview PDF.
- 4.Björklund, A.: Determinant sums for undirected hamiltonicity. In: Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science, FOCS 2010, pp. 173–182 (2010)Google Scholar
- 5.Björklund, A.: Counting perfect matchings as fast as Ryser. In: Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, pp. 914–921. SIAM (2012)Google Scholar
- 6.Björklund, A., Husfeldt, T., Kaski, P., Koivisto, M.: The travelling salesman problem in bounded degree graphs. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 198–209. Springer, Heidelberg (2008)CrossRefGoogle Scholar
- 9.Cygan, M., Kratsch, S., Nederlof, J.: Fast hamiltonicity checking via bases of perfect matchings. In: Proceedings of the Forty-fifth Annual ACM Symposium on Theory of Computing, STOC 2013, pp. 301–310 (2013)Google Scholar
- 13.Izumi, T., Wadayama, T.: A new direction for counting perfect matchings. In: 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science (FOCS), pp. 591–598. IEEE (2012)Google Scholar
- 15.Lokshtanov, D., Nederlof, J.: Saving space by algebraization. In: Proceedings of the 42nd ACM symposium on Theory of computing, STOC 2010, pp. 321–330. ACM (2010)Google Scholar
- 20.Shoup, V.: A Computational Introduction to Number Theory and Algebra, 2nd edn. Cambridge University Press (2009)Google Scholar