Abstract
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.
The full version of this paper is available at http://arxiv.org/abs/1311.2456 .
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Arnborg, S., Lagergren, J., Seese, D.: Easy problems for tree-decomposable graphs. Journal of Algorithms 12(2), 308–340 (1991)
Bax, E., Franklin, J.: A permanent algorithm with \(\exp[{\Omega}(n ^{1/3}/2 \ln n )]\) expected speedup for 0-1 matrices. Algorithmica 32(1), 157–162 (2002)
Bellman, R.: Dynamic programming treatment of the travelling salesman problem. J. ACM 9, 61–63 (1962)
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)
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)
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)
Björklund, A., Husfeldt, T., Kaski, P., Koivisto, M.: Trimmed Moebius inversion and graphs of bounded degree. Theory of Computing Systems 47(3), 637–654 (2010)
Björklund, A., Husfeldt, T., Koivisto, M.: Set partitioning via inclusion-exclusion. SIAM Journal on Computing 39(2), 546–563 (2009)
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)
Cygan, M., Pilipczuk, M.: Exact and approximate bandwidth. Theoretical Computer Science 411(40-42), 3701–3713 (2010)
Cygan, M., Pilipczuk, M.: Faster exponential-time algorithms in graphs of bounded average degree. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds.) ICALP 2013, Part I. LNCS, vol. 7965, pp. 364–375. Springer, Heidelberg (2013)
Held, M., Karp, R.M.: A dynamic programming approach to sequencing problems. Journal of the Society for Industrial and Applied Mathematics 10(1), 196–210 (1962)
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)
Kronecker, L.: Grundzüge einer arithmetischen theorie der algebraischen grössen. J. Reine Angew. Math. 92, 1–122 (1882)
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)
van Rooij, J.M.M., Bodlaender, H.L., Rossmanith, P.: Dynamic programming on tree decompositions using generalised fast subset convolution. In: Fiat, A., Sanders, P. (eds.) ESA 2009. LNCS, vol. 5757, pp. 566–577. Springer, Heidelberg (2009)
Ryser, H.J.: Combinatorial mathematics. Mathematical Association of America, Washington, DC (1963)
Schonhage, A., Strassen, V.: Schnelle multiplikation grosser zahlen. Computing 7(3-4), 281–292 (1971)
Servedio, R.A., Wan, A.: Computing sparse permanents faster. Information Processing Letters 96(3), 89–92 (2005)
Shoup, V.: A Computational Introduction to Number Theory and Algebra, 2nd edn. Cambridge University Press (2009)
Turk, J.: Fast arithmetic operations on numbers and polynomials. Mathematisch Centrum Computational Methods in Number Theory 1, 43–54 (1982)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Golovnev, A., Kulikov, A.S., Mihajlin, I. (2014). Families with Infants: A General Approach to Solve Hard Partition Problems. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds) Automata, Languages, and Programming. ICALP 2014. Lecture Notes in Computer Science, vol 8572. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43948-7_46
Download citation
DOI: https://doi.org/10.1007/978-3-662-43948-7_46
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-43947-0
Online ISBN: 978-3-662-43948-7
eBook Packages: Computer ScienceComputer Science (R0)