Advertisement

Fixed-Parameter Algorithms for Maximum-Profit Facility Location Under Matroid Constraints

  • René van BevernEmail author
  • Oxana Yu. Tsidulko
  • Philipp Zschoche
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11485)

Abstract

We consider an uncapacitated discrete facility location problem where the task is to decide which facilities to open and which clients to serve for maximum profit so that the facilities form an independent set in given facility-side matroids and the clients form an independent set in given client-side matroids. We show that the problem is fixed-parameter tractable parameterized by the number of matroids and the minimum rank among the client-side matroids. To this end, we derive fixed-parameter algorithms for computing representative families for matroid intersections and maximum-weight set packings with multiple matroid constraints. To illustrate the modeling capabilities of the new problem, we use it to obtain algorithms for a problem in social network analysis. We complement our tractability results by lower bounds.

Keywords

Matroid set packing Matroid parity Matroid median Representative families Social network analysis Strong triadic closure 

Notes

Acknowledgments

This study was initiated at the 7th annual research retreat of the Algorithmics and Computational Complexity group of TU Berlin, Darlingerode, Germany, March 18th–23rd, 2018. R. van Bevern and O. Yu. Tsidulko were supported by the Russian Foundation for Basic Research, grants 16-31-60007 mol_a_dk and 18-31-00470 mol_a, respectively. Both were supported by Russian Science Foundation grant 16-11-10041 while working on Sect. 3.

References

  1. 1.
    Aardal, K., van den Berg, P.L., Gijswijt, D., Li, S.: Approximation algorithms for hard capacitated \(k\)-facility location problems. Eur. J. Oper. Res. 242(2), 358–368 (2015)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Ageev, A.A., Sviridenko, M.I.: An 0.828-approximation algorithm for the uncapacitated facility location problem. Discr. Appl. Math. 93(2), 149–156 (1999)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Alon, N., Yuster, R., Zwick, U.: Color-coding. J. ACM 42(4), 844–856 (1995)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bonnet, É., Paschos, V.T., Sikora, F.: Parameterized exact and approximation algorithms for maximum \(k\)-set cover and related satisfiability problems. RAIRO-Inf. Theor. Appl. 50(3), 227–240 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Calinescu, G., Chekuri, C., Pál, M., Vondrák, J.: Maximizing a monotone submodular function subject to a matroid constraint. SIAM J. Comput. 40(6), 1740–1766 (2011)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Cygan, M., et al.: Parameterized Algorithms. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  7. 7.
    Fellows, M.R., Fernau, H.: Facility location problems: a parameterized view. Discr. Appl. Math. 159(11), 1118–1130 (2011)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Filmus, Y., Ward, J.: The power of local search: maximum coverage over a matroid. In: Proceedings of 29th STACS, LIPIcs, vol. 14, pp. 601–612. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, Dagstuhl (2012)Google Scholar
  9. 9.
    Fomin, F.V., Lokshtanov, D., Panolan, F., Saurabh, S.: Efficient computation of representative families with applications in parameterized and exact algorithms. J. ACM 63(4), 29:1–29:60 (2016)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Fomin, F.V., Lokshtanov, D., Panolan, F., Saurabh, S.: Representative families of product families. ACM T. Algorithms 13(3), 36:1–36:29 (2017)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Golovach, P.A., Heggernes, P., Konstantinidis, A.L., Lima, P.T., Papadopoulos, C.: Parameterized aspects of strong subgraph closure. In: Proceedings of 16th SWAT, LIPIcs, vol. 101, pp. 23:1–23:13. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, Dagstuhl (2018)Google Scholar
  12. 12.
    Granovetter, M.S.: The strength of weak ties. Am. J. Sociol. 78(6), 1360–1380 (1973)CrossRefGoogle Scholar
  13. 13.
    Krishnaswamy, R., Kumar, A., Nagarajan, V., Sabharwal, Y., Saha, B.: Facility location with matroid or knapsack constraints. Math. Oper. Res. 40(2), 446–459 (2015)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Laporte, G., Nickel, S., Saldanha da Gama, F. (eds.): Location Science. Springer, Heidelberg (2015).  https://doi.org/10.1007/978-3-319-13111-5CrossRefzbMATHGoogle Scholar
  15. 15.
    Lawler, E.: Combinatorial Optimization-Networks and Matroids. Holt, Rinehart and Winston, New York (1976)zbMATHGoogle Scholar
  16. 16.
    Lee, J., Sviridenko, M., Vondrák, J.: Matroid matching: the power of local search. SIAM J. Comput. 42(1), 357–379 (2013)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Lokshtanov, D., Misra, P., Panolan, F., Saurabh, S.: Deterministic truncation of linear matroids. ACM Trans. Algorithms 14(2), 14:1–14:20 (2018)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Marx, D.: Parameterized complexity and approximation algorithms. Comput. J. 51(1), 60–78 (2008)CrossRefGoogle Scholar
  19. 19.
    Marx, D.: A parameterized view on matroid optimization problems. Theor. Comput. Sci. 410(44), 4471–4479 (2009)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Oxley, J.G.: Matroid Theory. Oxford University Press, Oxford (1992)zbMATHGoogle Scholar
  21. 21.
    Panolan, F., Saurabh, S.: Matroids in parameterized complexity and exact algorithms. In: Kao, M.Y. (ed.) Encyclopedia of Algorithms, pp. 1203–1205. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-1-4939-2864-4CrossRefGoogle Scholar
  22. 22.
    Rozenshtein, P., Tatti, N., Gionis, A.: Inferring the strength of social ties: a community-driven approach. In: Proceedings of 23rd ACM SIGKDD, pp. 1017–1025. ACM, New York (2017)Google Scholar
  23. 23.
    Schöbel, A., Hamacher, H.W., Liebers, A., Wagner, D.: The continuous stop location problem in public transportation networks. Asia Pac. J. Oper. Res. 26(1), 13–30 (2009)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency, Algorithms and Combinatorics, vol. 24. Springer, Heidelberg (2003).  https://doi.org/10.1007/s10288-004-0035-9CrossRefzbMATHGoogle Scholar
  25. 25.
    Swamy, C.: Improved approximation algorithms for matroid and knapsack median problems and applications. ACM Trans. Algorithms 12(4), 49:1–49:22 (2016)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • René van Bevern
    • 1
    • 2
    Email author
  • Oxana Yu. Tsidulko
    • 1
    • 2
  • Philipp Zschoche
    • 3
  1. 1.Department of Mechanics and MathematicsNovosibirsk State UniversityNovosibirskRussian Federation
  2. 2.Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of SciencesNovosibirskRussian Federation
  3. 3.Algorithmics and Computational Complexity, Fakultät IVTU BerlinBerlinGermany

Personalised recommendations