Families with Infants: A General Approach to Solve Hard Partition Problems

  • Alexander Golovnev
  • Alexander S. Kulikov
  • Ivan Mihajlin
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8572)


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.


Fast Fourier Transform Perfect Matchings Average Degree Travel Salesman Problem Hamiltonian Path 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Arnborg, S., Lagergren, J., Seese, D.: Easy problems for tree-decomposable graphs. Journal of Algorithms 12(2), 308–340 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    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)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Bellman, R.: Dynamic programming treatment of the travelling salesman problem. J. ACM 9, 61–63 (1962)CrossRefzbMATHGoogle Scholar
  4. 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. 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. 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
  7. 7.
    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)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Björklund, A., Husfeldt, T., Koivisto, M.: Set partitioning via inclusion-exclusion. SIAM Journal on Computing 39(2), 546–563 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 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
  10. 10.
    Cygan, M., Pilipczuk, M.: Exact and approximate bandwidth. Theoretical Computer Science 411(40-42), 3701–3713 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    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)CrossRefGoogle Scholar
  12. 12.
    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)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 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
  14. 14.
    Kronecker, L.: Grundzüge einer arithmetischen theorie der algebraischen grössen. J. Reine Angew. Math. 92, 1–122 (1882)zbMATHGoogle Scholar
  15. 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
  16. 16.
    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)CrossRefGoogle Scholar
  17. 17.
    Ryser, H.J.: Combinatorial mathematics. Mathematical Association of America, Washington, DC (1963)zbMATHGoogle Scholar
  18. 18.
    Schonhage, A., Strassen, V.: Schnelle multiplikation grosser zahlen. Computing 7(3-4), 281–292 (1971)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Servedio, R.A., Wan, A.: Computing sparse permanents faster. Information Processing Letters 96(3), 89–92 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Shoup, V.: A Computational Introduction to Number Theory and Algebra, 2nd edn. Cambridge University Press (2009)Google Scholar
  21. 21.
    Turk, J.: Fast arithmetic operations on numbers and polynomials. Mathematisch Centrum Computational Methods in Number Theory 1, 43–54 (1982)MathSciNetGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Alexander Golovnev
    • 1
  • Alexander S. Kulikov
    • 2
  • Ivan Mihajlin
    • 2
    • 3
  1. 1.New York UniversityUSA
  2. 2.St. Petersburg Department of SteklovInstitute of MathematicsUSA
  3. 3.St. Petersburg Academic UniversityUSA

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