A Heuristic Algorithm for the Set k-Cover Problem

  • Amir SalehipourEmail author
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 1173)


The set k-cover problem (SkCP) is an extension of the classical set cover problem (SCP), in which each row needs to be covered by at least k columns while the coverage cost is minimized. The case of \(k=1\) refers to the classical SCP. SkCP has many applications including in computational biology. We develop a simple and effective heuristic for both weighted and unweighted SkCP. In the weighted SkCP, there is a cost associated with a column and in the unweighted variant, all columns have the identical cost. The proposed heuristic first generates a lower bound and then builds a feasible solution from the lower bound. We improve the feasible solution through several procedures including a removal local search. We consider three different values for k and test the heuristic on 45 benchmark instances of SCP from OR library. Therefore, we solve 135 instances. Over the solved instances, we show that our proposed heuristic obtains quality solutions.


Set cover problem Multiple coverage Heuristic 



Amir Salehipour is the recipient of an Australian Research Council Discovery Early Career Researcher Award (project number DE170100234) funded by the Australian Government.


  1. 1.
    Baker, E.K.: Efficient heuristic algorithms for the weighted set covering problem. Comput. Oper. Res. 8(4), 303–310 (1981)CrossRefGoogle Scholar
  2. 2.
    Balas, E., Carrera, M.C.: A dynamic subgradient-based branch-and-bound procedure for set covering. Oper. Res. 44(6), 875–890 (1996)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Balas, E., Ho, A.: Set covering algorithms using cutting planes, heuristics, and subgradient optimization: a computational study. In: Padberg, M.W. (ed.) Combinatorial Optimization. Mathematical Programming Studies, vol. 12, pp. 37–60. Springer, Heidelberg (1980). Scholar
  4. 4.
    Bautista, J., Pereira, J.: A GRASP algorithm to solve the unicost set covering problem. Comput. Oper. Res. 34(10), 3162–3173 (2007)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Beasley, J.E.: An algorithm for set covering problem. Eur. J. Oper. Res. 31(1), 85–93 (1987)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Beasley, J.E.: A Lagrangian heuristic for set-covering problems. Nav. Res. Logist. 37(1), 151–164 (1990)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Beasley, J.E., Chu, P.C.: A genetic algorithm for the set covering problem. Eur. J. Oper. Res. 94(2), 392–404 (1996)CrossRefGoogle Scholar
  8. 8.
    Beasley, J., Jørnsten, K.: Practical combinatorial optimization enhancing an algorithm for set covering problems. Eur. J. Oper. Res. 58(2), 293–300 (1992)CrossRefGoogle Scholar
  9. 9.
    Caprara, A., Fischetti, M., Toth, P.: A heuristic method for the set covering problem. Oper. Res. 47(5), 730–743 (1999)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Caprara, A., Toth, P., Fischetti, M.: Algorithms for the set covering problem. Ann. Oper. Res. 98(1–4), 353–371 (2000)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Caserta, M.: Tabu search-based metaheuristic algorithm for large-scale set covering problems. In: Doerner, K.F., Gendreau, M., Greistorfer, P., Gutjahr, W., Hartl, R.F., Reimann, M. (eds.) Metaheuristics. ORSIS, vol. 39, pp. 43–63. Springer, Boston (2007). Scholar
  12. 12.
    Ceria, S., Nobili, P., Sassano, A.: A Lagrangian-based heuristic for large-scale set covering problems. Math. Program. 81(2), 215–228 (1998)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Chvatal, V.: A greedy heuristic for the set covering problem. Math. Oper. Res. 4(3), 233–235 (1979)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Feo, T.A., Resende, M.G.C.: A probabilistic heuristic for a computationally difficult set covering problem. Oper. Res. Lett. 8(2), 67–71 (1989)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Fisher, M.L., Kedia, P.: Optimal solution of set covering/partitioning problems using dual heuristics. Manag. Sci. 36(6), 674–688 (1990)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Garfinkel, R.S., Nemhauser, G.L.: Integer Programming. Wiley, New York (1972)zbMATHGoogle Scholar
  17. 17.
    Haddadi, S.: Simple Lagrangian heuristic for the set covering problem. Eur. J. Oper. Res. 97(1), 200–204 (1997)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Hua, Q.-S., Wang, Y., Yu, D., Lau, F.C.: Dynamic programming based algorithms for set multicover and multiset multicover problems. Theoret. Comput. Sci. 411(26-28), 2467–2474 (2010)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Lan, G., DePuy, G.W., Whitehouse, G.E.: An effiective and simple heuristic for the set covering problem. Eur. J. Oper. Res. 176(3), 1387–1403 (2007)CrossRefGoogle Scholar
  20. 20.
    Lovász, L.: On the ratio of optimal integral and fractional covers. Discret. Math. 13(4), 383–390 (1975)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Naji-Azimi, Z., Toth, P., Galli, L.: An electromagnetism metaheuristic for the unicost set covering problem. Eur. J. Oper. Res. 205(2), 290–300 (2010)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Pessoa, L.S., Resende, M.G.C., Ribeiro, C.C.: Experiments with LAGRASP heuristic for set k-covering. Optim. Lett. 5(3), 407–419 (2011)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Pessoa, L.S., Resende, M.G.C., Ribeiro, C.C.: A hybrid Lagrangean heuristic with GRASP and path-relinking for set K-covering. Comput. Oper. Res. 40(12), 3132–3146 (2013)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Vasko, F.J.: An efficient heuristic for large set covering problems. Nav. Res. Logist. Q. 31(1), 163–171 (1984)CrossRefGoogle Scholar
  25. 25.
    Vasko, F.J., Wilson, G.R.: Hybrid heuristics for minimum cardinality set covering problems. Nav. Res. Logist. Q. 33(2), 241–249 (1986)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Yagiura, M., Kishida, M., Ibaraki, T.: A 3-flip neighborhood local search for the set covering problem. Eur. J. Oper. Res. 172(2), 472–499 (2006)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.School of Mathematical and Physical SciencesUniversity of Technology SydneySydneyAustralia

Personalised recommendations