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)


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.


Combinatorial optimization approximation algorithms hypergarphs vertex cover probabilistic methods 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  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

Personalised recommendations