Approximation algorithms for the covering-type k-violation linear program

  • Yotaro TakazawaEmail author
  • Shinji Mizuno
  • Tomonari Kitahara
Original Paper


We study the covering-type k-violation linear program where at most k of the constraints can be violated. This problem is formulated as a mixed integer program and known to be strongly NP-hard. In this paper, we present a simple \((k+1)\)-approximation algorithm using a natural LP relaxation. We also show that the integrality gap of the LP relaxation is \(k+1\). This implies we can not get better approximation algorithms when we use the LP-relaxation as a lower bound of the optimal value.


Approximation algorithms Mixed integer program k-violation linear program Linear relaxation Rounding algorithm 



We thank the anonymous referees for their helpful comments and suggestions. This research is supported in part by Grant-in-Aid for Science Research (A) 26242027 of Japan Society for the Promotion of Science and Grant-in-Aid for Young Scientist (B) 15K15941.


  1. 1.
    Ahmed, S., Xie, W.: Relaxations and approximations of chance constraints under finite distributions. Math. Program. 170(1), 43–65 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Chan, T.M.: Low-dimensional linear programming with violations. SIAM J. Comput. 34(4), 879–893 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Charikar, M., Khuller, S., Mount, D.M., Narasimhan, G.: Algorithms for facility location problems with outliers. In: Proceedings of the Twelfth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 642–651. Society for Industrial and Applied Mathematics (2001)Google Scholar
  4. 4.
    Chen, K.: A constant factor approximation algorithm for k-median clustering with outliers. In: Proceedings of the Twelfth Annual ACM-SIAM Symposium on Discrete Algorithms, vol. 8, pp. 826–835 (2008)Google Scholar
  5. 5.
    Dinitz, M., Gupta, A.: Packing Interdiction and Partial Covering Problems, pp. 157–168. Springer, Berlin (2013)zbMATHGoogle Scholar
  6. 6.
    Gandhi, R., Khuller, S., Srinivasan, A.: Approximation algorithms for partial covering problems. J. Algorithms 53(1), 55–84 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Qiu, F., Ahmed, S., Dey, S.S., Wolsey, L.A.: Covering linear programming with violations. INFORMS J. Comput. 26(3), 531–546 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ran, Y., Shi, Y., Zhang, Z.: Local ratio method on partial set multi-cover. J. Comb. Optim. 34(1), 302–313 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Roos, T., Widmayer, P.: k-violation linear programming. Inf. Process. Lett. 52(2), 109–114 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Takazawa, Y., Mizuno, S., Kitahara, T.: An approximation algorithm for the partial covering 0–1 integer program. Discrete Appl. Math. (2017).
  11. 11.
    Vazirani, V.V.: Introduction to LP-duality. In: Vazirani, V.V. (ed.) Approximation Algorithms, pp. 93–107. Springer, Berlin (2003)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Industrial Engineering and EconomicsTokyo Institute of TechnologyTokyoJapan

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