Advertisement

Approximating the \(k\)-Set Packing Problem by Local Improvements

  • Martin Fürer
  • Huiwen YuEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8596)

Abstract

We study algorithms based on local improvements for the \(k\)-Set Packing problem. The well-known local improvement algorithm by Hurkens and Schrijver [14] has been improved by Sviridenko and Ward [15] from \(\frac{k}{2}+\epsilon \) to \(\frac{k+2}{3}\), and by Cygan [7] to \(\frac{k+1}{3}+\epsilon \) for any \(\epsilon >0\). In this paper, we achieve the approximation ratio \(\frac{k+1}{3}+\epsilon \) for the \(k\)-Set Packing problem using a simple polynomial-time algorithm based on the method by Sviridenko and Ward [15]. With the same approximation guarantee, our algorithm runs in time singly exponential in \(\frac{1}{\epsilon ^2}\), while the running time of Cygan’s algorithm [7] is doubly exponential in \(\frac{1}{\epsilon }\). On the other hand, we construct an instance with locality gap \(\frac{k+1}{3}\) for any algorithm using local improvements of size \(O(n^{1/5})\), where \(n\) is the total number of sets. Thus, our approximation guarantee is optimal with respect to results achievable by algorithms based on local improvements.

Keywords

\(k\)-set packing Tail change Local improvement Color coding 

References

  1. 1.
    Alon, N., Naor, M.: Derandomization, witnesses for boolean matrix multiplication and construction of perfect hash functions. Algorithmica 16(4/5), 434–449 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Alon, N., Yuster, R., Zwick, U.: Color-coding. J. ACM 42(4), 844–856 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Berman, P.: A d/2 approximation for maximum weight independent set in d-claw free graphs. Nord. J. Comput. 7(3), 178–184 (2000)zbMATHGoogle Scholar
  4. 4.
    Berman, P., Fürer, M.: Approximating maximum independent set in bounded degree graphs. In: SODA, pp. 365–371 (1994)Google Scholar
  5. 5.
    Chan, Y., Lau, L.: On linear and semidefinite programming relaxations for hypergraph matching. In: SODA, pp. 1500–1511 (2010)Google Scholar
  6. 6.
    Chandra, B., Halldórsson, M.: Greedy local improvement and weighted set packing approximation. In: SODA, pp. 169–176 (1999)Google Scholar
  7. 7.
    Cygan, M.: Improved approximation for 3-dimensional matching via bounded pathwidth local search. In: FOCS, pp. 509–518 (2013)Google Scholar
  8. 8.
    Cygan, M., Grandoni, F., Mastrolilli, M.: How to sell hyperedges: the hypermatching assignment problem. In: SODA, pp. 342–351 (2013)Google Scholar
  9. 9.
    Fellows, M.R., Knauer, C., Nishimura, N., Ragde, P., Rosamond, F., Stege, U., Thilikos, D.M., Whitesides, S.: Faster fixed-parameter tractable algorithms for matching and packing problems. Algorithmica 52(2), 167–176 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Fürer, M., Yu, H.: Approximating the k-set packing problem by local improvements. http://arxiv.org/abs/1307.2262
  11. 11.
    Halldórsson, M.: Approximating discrete collections via local improvements. In: SODA, pp. 160–169 (1995)Google Scholar
  12. 12.
    Håstad, J.: Clique is hard to approximate within \(n^{1-\epsilon }\). In: FOCS, pp. 627–636 (1996)Google Scholar
  13. 13.
    Hazan, E., Safra, S., Schwartz, O.: On the complexity of approximating k-set packing. Comput. Complex. 15(1), 20–39 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Hurkens, C.A., Shrijver, J.: On the size of systems of sets every \(t\) of which have an SDR, with an application to the worst-case ratio of heuristics for packing problems. SIAM J. Discrete Math. 2(1), 68–72 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Sviridenko, M., Ward, J.: Large neighborhood local search for the maximum set packing problem. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds.) ICALP 2013, Part I. LNCS, vol. 7965, pp. 792–803. Springer, Heidelberg (2013) CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Department of Computer Science and EngineeringThe Pennsylvania State UniversityUniversity ParkUSA

Personalised recommendations