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A Randomised Approximation Algorithm for the Partial Vertex Cover Problem in Hypergraphs

  • Mourad El Ouali
  • Helena Fohlin
  • Anand Srivastav
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7659)

Abstract

In this paper we present an approximation algorithm for the k-partial vertex cover problem in hypergraphs. Let \(\mathcal{H}=(V,\mathcal{E})\) be a hypergraph with set of vertices V, |V| = n and set of (hyper-)edges \(|\mathcal{E}|, |\mathcal{E}| =m\). The k-partial vertex cover problem in hypergraphs is the problem of finding a minimum cardinality subset of vertices in which at least k hyperedges are incident. It is a generalisation of the fundamental (partial) vertex cover problem in graphs and the hitting set problem in hypergraphs. Let l, l ≥ 2 be the maximum size of an edge, Δ be the maximum vertex degree and D be maximum edge degree. For a constant l, l ≥ 2 a non-approximabilty result is known: an approximation ratio better than l cannot be achieved in polynomial-time under the unique games conjecture (Khot and Rageev 2003, 2008). On the other hand, with the primal-dual method (Gandhi, Khuller, Srinivasan 2001) and the local-ratio method (Bar-Yehuda 2001), the l-approximation ratio can be proved. Thus approximations below the l-ratio for large classes of hypergraphs, for example those with constant D or Δ are interesting. In case of graphs (l = 2) such results are known. In this paper we break the l-approximation barrier for hypergraph classes with constant D resp. Δ for the partial vertex cover problem in hypergraphs. We propose a randomised algorithm of hybrid type which combines LP-based randomised rounding and greedy repairing. For hypergraphs with arbitrary l, l ≥ 3, and constant D the algorithm achieves an approximation ratio of l(1 − Ω(1/(D + 1))), and this can be improved to l (1 − Ω(1/Δ)) if Δ is constant and k ≥ m/4. For the class of l-uniform hypergraphs with both l and Δ being constants and l ≤ 4Δ, we get a further improvement to a ratio of \(l\left(1-\frac{l-1}{4\Delta}\right)\). The analysis relies on concentration inequalities and combinatorial arguments.

Keywords

Combinatorial optimization approximation algorithms hypergarphs vertex cover probabilistic methods 

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References

  1. 1.
    Alon, N., Moshkovitz, D., Safra, S.: Algorithmic construction of sets for k-restrictions. ACM Trans. Algorithms (ACM) 2, 153–177 (2006)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Alon, N., Spencer, J.: The probabilistic method, 2nd edn. Wiley Interscience (2000)Google Scholar
  3. 3.
    Bar-Yehuda, R.: Using homogeneous weights for approximating the partial cover problem. Journal of Algorithms 39(2), 137–144 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Berge, C.: Hypergraphs- combinatorics of finite sets. North Holland Mathematical Library (1989)Google Scholar
  5. 5.
    Chvátal, V.: A greedy heuristic for the set covering problem. Math. Oper. Res. 4(3), 233–235 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Duh, R., Fürer, M.: Approximating k-set cover by semi-local optimization. In: Proceedings of the 29th Annual ACM Symposium on Theory of Computing (STOC 1997), pp. 256–264 (May 1997)Google Scholar
  7. 7.
    Feige, U.: A treshold of ln n for approximating set cover. Journal of the ACM 45(4), 634–652 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Feige, U., Langberg, M.: Approximation algorithms for maximization problems arising in graph partitioning. Journal of Algorithms 41(2), 174–201 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Frieze, A., Jerrum, M.: Improved approximation algorithms for max k-cut and max bisection. Algorithmica 18, 67–81 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Gandhi, R., Khuller, S., Srinivasan, A.: Approximation Algorithms for Partial Covering Problems. J. Algorithms 53(1), 55–84 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Halperin, E.: Improved approximation algorithms for the vertex cover problem in graphs and hypergraphs. In: ACM-SIAM Symposium on Discrete Algorithms, vol. 11, pp. 329–337 (2000)Google Scholar
  12. 12.
    Halperin, E., Srinivasan, A.: Improved Approximation Algorithms for the Partial Vertex Cover Problem. In: Jansen, K., Leonardi, S., Vazirani, V.V. (eds.) APPROX 2002. LNCS, vol. 2462, pp. 161–174. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  13. 13.
    Hochbaum, D.S.: Approximation algorithms for the set covering and vertex cover problems. SIAM J. Computation 11(3), 555–556 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Jäger, G., Srivastav, A.: Improved approximation algorithms for maximum graph partitioning problems. Journal of Combinatorial Optimization 10(2), 133–167 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Khot, S., Regev, O.: Vertex cover might be hard to approximate to within 2-epsilon. J. Comput. Syst. Sci. 74(3), 335–349 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Krivelevich, J.: Approximate set covering in uniform hypergraphs. J. Algorithms 25(1), 118–143 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Lovász, L.: On the ratio of optimal integral and fractional covers. Discrete Math. 13, 383–390 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Lund, C., Yannakakis, M.: On the hardness of approximating minimization problems. J. Assoc. Comput. Mach. 41, 960–981 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Matousek, J.: Geometric Discrepancy. Algorithms and Combinatorics, vol. 18. Springer, Heidelberg (2010)zbMATHCrossRefGoogle Scholar
  20. 20.
    Matousek, J., Wagner, U.: New Constructions of Weak epsilon-Nets. Discrete & Computational Geometry 32(2), 195–206 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    McDiarmid, C.: On the method of bounded differences. Surveys in Combinatorics, Norwich, pp. 148–188. Cambridge Univ. Press, Cambridge (1989)Google Scholar
  22. 22.
    Peleg, D., Schechtman, G., Wool, A.: Randomized approximation of bounded multicovering problems. Algorithmica 18(1), 44–66 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Raz, R., Safra, S.: A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP. In: Proc. 29th ACM Symp. on Theory of Computing, pp. 475–484 (1997)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Mourad El Ouali
    • 1
  • Helena Fohlin
    • 2
  • Anand Srivastav
    • 1
  1. 1.Department of Computer ScienceUniversity of KielGermany
  2. 2.Department of Clinical and Experimental MedicineLinköping UniversitySweden

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