Advertisement

Primal Dual Algorithm for Partial Set Multi-cover

  • Yingli Ran
  • Yishuo Shi
  • Zhao Zhang
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11346)

Abstract

In a minimum partial set multi-cover problem (MinPSMC), given an element set E, a collection of subsets \(\mathcal S \subseteq 2^E\), a cost \(w_S\) on each set \(S\in \mathcal S\), a covering requirement \(r_e\) for each element \(e\in E\), and an integer k, the goal is to find a sub-collection \(\mathcal F \subseteq \mathcal S\) to fully cover at least k elements such that the cost of \(\mathcal F\) is as small as possible, where element e is fully covered by \(\mathcal F\) if it belongs to at least \(r_e\) sets of \(\mathcal F\). On the application side, the problem has its background in the seed selection problem in a social network. On the theoretical side, it is a natural combination of the minimum partial (single) set cover problem (MinPSC) and the minimum set multi-cover problem (MinSMC). Although both MinPSC and MinSMC admit good approximations whose performance ratios match those lower bounds for the classic set cover problem, previous studies show that theoretical study on MinPSMC is quite challenging. In this paper, we prove that MinPSMC cannot be approximated within factor \(O(n^\frac{1}{2(\log \log n)^c})\) under the ETH assumption. A primal dual algorithm for MinPSMC is presented with a guaranteed performance ratio \(O(\sqrt{n})\) when \(r_{\max }\) and f are constants, where \(r_{\max } =\max _{e\in E} r_e\) is the maximum covering requirement and f is the maximum frequency of elements (that is the maximum number of sets containing a common element). We also improve the ratio for a restricted version of MinPSMC which possesses a graph-type structure.

Keywords

Positive influence seeding problem Partial set multi-cover problem Densest l-subgraph problem Approximation algorithm 

Notes

Acknowledgements

This research is supported by NSFC (11771013, 11531011).

References

  1. 1.
    Kempe, D., Kleinberg, J., Tardos, E.: Maximizing the spread of influence through a social network. In: Proceedings of the Ninth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 137–146 (2003)Google Scholar
  2. 2.
    Kempe, D., Kleinberg, J., Tardos, É.: Influential nodes in a diffusion model for social networks. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 1127–1138. Springer, Heidelberg (2005).  https://doi.org/10.1007/11523468_91CrossRefGoogle Scholar
  3. 3.
    Chen, N.: On the approximability of influence in social networks. SIAM J. Discrete Math. 23(3), 1400–1415 (2008). A preliminary version appears in SODA’08, pp. 1029–1037Google Scholar
  4. 4.
    Ran, Y., Zhang, Z., Du, H., Zhu, Y.: Approximation algorithm for partial positive influence problem in social network. J. Comb. Optim. 33(2), 791–802 (2017)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Ran, Y., Shi, Y., Zhang, Z.: Local ratio method on partial set multi-cover. J. Comb. Optim. 34, 302–313 (2017)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Dinh, T.N., Shen, Y., Nguyen, D.T., Thai, M.T.: On the approximability of positive influence dominating set in social networks. J. Comb. Optim. 27, 487–503 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Wang, F., Camacho, E., Xu, K.: Positive influence dominating set in online social networks. In: Du, D.-Z., Hu, X., Pardalos, P.M. (eds.) COCOA 2009. LNCS, vol. 5573, pp. 313–321. Springer, Heidelberg (2009).  https://doi.org/10.1007/978-3-642-02026-1_29CrossRefGoogle Scholar
  8. 8.
    Wang, F., et al.: On positive influence dominating sets in social networks. Theoret. Comput. Sci. 412, 265–269 (2011)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W. (eds.) Complexity of Computer Computations, pp. 85–103. Plenum Press, New York (1972)CrossRefGoogle Scholar
  10. 10.
    Feige, U.: A threshold of \(\ln n\) for approximating set cover. In: Proceedings of 28th ACM Symposium on the Theory of Computing, pp. 312–318 (1996)Google Scholar
  11. 11.
    Dinur, I., Steurer, D.: Analytical approach to parallel repetition. In: STOC, pp. 624–633 (2014)Google Scholar
  12. 12.
    Johnson, D.S.: Approximation algorithms for combinatorial problems. J. Comput 9, 256–278 (1974)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Lovász, L.: On the ratio of optimal integral and fractional covers. Discrete Math. 13, 383–390 (1975)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Chvatal, V.: A greedy heuristic for the set-covering problem. Math. Oper. Res. 233–235 (1979)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Vazirani, V.V.: Approximation Algorithms. Springer, Heidelberg (2001)Google Scholar
  16. 16.
    Hochbaum, D.S.: Approximation algorithms for the set covering and vertex cover problems. SIAM J. Comput. 11, 555–556 (1982)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Bar-Yehuda, R., Even, S.: A local-ratio theorem for approximating the weighted vertex cover problem. North-Holland Math. Stud. 109, 27–45 (1985)Google Scholar
  18. 18.
    Khot, S., Regev, O.: Vertex cover might be hard to approximate to within \(2-\varepsilon \). J. Comput. Syst. Sci. 74(3), 335–349 (2008)Google Scholar
  19. 19.
    Rajagopalan, S., Vazirani, V.V.: Primal-dual RNC approximation algorithms for (multi)-set (multi)-cover and covering integer programs. In: Proceedings of the 34th Annual IEEE Symposium on Foundations of Computer Science, pp. 322–331 (1993)Google Scholar
  20. 20.
    Kearns, M.: The Computational Complexity of Machine Learning. MIT Press, Cambridge (1990)Google Scholar
  21. 21.
    Slavík, P.: Improved performance of the greedy algorithm for partial cover. Inf. Process. Lett. 64(5), 251–254 (1997)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Gandhi, R., Khuller, S., Srinivasan, A.: Approximation algorithms for partial covering problems. J. Algorithms 53(1), 55–84 (2004)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Bar-Yehuda, R.: Using homogeneous weights for approximating the partial cover problem. J. Algorithms 39, 137–144 (2001)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Manurangsi, P.: Almost-polynomial ratio ETH-hardness of approximating densest \(k\)-subgraph. In: STOC, pp. 19–23 (2017)Google Scholar
  25. 25.
    Lovász, L.: Submodular functions and convexity. In: Bachem, A., Korte, B., Grötschel, M. (eds.) Mathematical Programming the State of the Art, pp. 235–257. Springer, Heidelberg (1983).  https://doi.org/10.1007/978-3-642-68874-4_10CrossRefGoogle Scholar
  26. 26.
    Fleisher, L., Iwata, S.: A push-relabel framework for submodular function minimization and applications to parametric optimization. Discrete Appl. Math. 131, 311–322 (2003)Google Scholar
  27. 27.
    Bhaskara, A., Charikar, M., Chlamtac, E., Feige, U., Vijayaraghavan, A.: Detecting high log-densities: an \(O(n^{1/4})\) approximation for densest \(k\)-subgraph. In: STOC, pp. 201–210 (2010)Google Scholar
  28. 28.
    Edmonds, J.: Submodular functions, matroids, and certain polyhedra. In: Guy, R., Hanani, H., Sauer, N., Schönheim, J. (eds.) Combinatorial Structures and Their Applications, pp. 69–87. Gordon and Breach, New York (1970)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.College of Mathematics, Physics, and Information EngineeringZhejiang Normal UniversityJinhuaChina
  2. 2.Institute of Information Science, Academia SinicaTaibeiTaiwan

Personalised recommendations