Minimum Latency Submodular Cover

  • Sungjin Im
  • Viswanath Nagarajan
  • Ruben van der Zwaan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7391)


We study the submodular ranking problem in the presence of metric costs. The input to the minimum latency submodular cover (MLSC ) problem consists of a metric (V,d) with source r ∈ V and m monotone submodular functions f 1, f 2, ..., f m : 2 V  → [0,1]. The goal is to find a path originating at r that minimizes the total cover time of all functions; the cover time of function f i is the smallest value t such that f i has value one for the vertices visited within distance t along the path. This generalizes many previously studied problems, such as submodular ranking [1] when the metric is uniform, and group Steiner tree [14] when m = 1 and f 1 is a coverage function. We give a polynomial time \(O(\log \frac{1}{\epsilon } \cdot \log^{2+\delta} |V|)\)-approximation algorithm for MLSC, where ε > 0 is the smallest non-zero marginal increase of any \(\{f_i\}_{i=1}^m\) and δ > 0 is any constant. This result is enabled by a simpler analysis of the submodular ranking algorithm from [1].

We also consider the stochastic submodular ranking problem where elements V have random instantiations, and obtain an adaptive algorithm with an O(log1/ ε) approximation ratio, which is best possible. This result also generalizes several previously studied stochastic problems, eg. adaptive set cover [15] and shared filter evaluation [24,23].


Approximation Algorithm Steiner Tree Steiner Tree Problem Cover Time Submodular Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Azar, Y., Gamzu, I.: Ranking with submodular valuations. In: SODA 2011, pp. 1070–1079 (2011)Google Scholar
  2. 2.
    Azar, Y., Gamzu, I., Yin, X.: Multiple intents re-ranking. In: STOC 2009, pp. 669–678 (2009)Google Scholar
  3. 3.
    Bansal, N., Gupta, A., Krishnaswamy, R.: A constant factor approximation algorithm for generalized min-sum set cover. In: SODA 2010, pp. 1539–1545 (2010)Google Scholar
  4. 4.
    Bar-Noy, A., Bellare, M., Halldórsson, M.M., Shachnai, H., Tamir, T.: On chromatic sums and distributed resource allocation. Inf. Comput. 140(2), 183–202 (1998)zbMATHCrossRefGoogle Scholar
  5. 5.
    Bhaskara, A., Charikar, M., Chlamtac, E., Feige, U., Vijayaraghavan, A.: Detecting high log-densities: an n 1/4 approximation for densest k-subgraph. In: STOC 2010, pp. 201–210 (2010)Google Scholar
  6. 6.
    Calinescu, G., Zelikovsky, A.: The polymatroid steiner problems. Journal of Combinatorial Optimization 9(3), 281–294 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Carr, R.D., Fleischer, L., Leung, V.J., Phillips, C.A.: Strengthening integrality gaps for capacitated network design and covering problems. In: SODA 2000, pp. 106–115 (2000)Google Scholar
  8. 8.
    Chakrabarty, D., Swamy, C.: Facility Location with Client Latencies: Linear Programming Based Techniques for Minimum Latency Problems. In: Günlük, O., Woeginger, G.J. (eds.) IPCO 2011. LNCS, vol. 6655, pp. 92–103. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  9. 9.
    Chaudhuri, K., Godfrey, B., Rao, S., Talwar, K.: Paths, trees, and minimum latency tours. In: FOCS 2003, pp. 36–45 (2003)Google Scholar
  10. 10.
    Chekuri, C., Even, G., Kortsarz, G.: A greedy approximation algorithm for the group steiner problem. Discrete Applied Mathematics 154(1), 15–34 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Chekuri, C., Pál, M.: A recursive greedy algorithm for walks in directed graphs. In: FOCS 2005, pp. 245–253 (2005)Google Scholar
  12. 12.
    Fakcharoenphol, J., Harrelson, C., Rao, S.: The k-traveling repairmen problem. ACM Transactions on Algorithms 3(4) (2007)Google Scholar
  13. 13.
    Feige, U., Lovász, L., Tetali, P.: Approximating min sum set cover. Algorithmica 40(4), 219–234 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Garg, N., Konjevod, G., Ravi, R.: A polylogarithmic approximation algorithm for the group steiner tree problem. J. Algorithms 37(1), 66–84 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Goemans, M.X., Vondrák, J.: Stochastic Covering and Adaptivity. In: Correa, J.R., Hevia, A., Kiwi, M. (eds.) LATIN 2006. LNCS, vol. 3887, pp. 532–543. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  16. 16.
    Golovin, D., Krause, A.: Adaptive submodularity: A new approach to active learning and stochastic optimization. In: COLT 2010, pp. 333–345 (2010)Google Scholar
  17. 17.
    Guillory, A., Bilmes, J.A.: Online submodular set cover, ranking, and repeated active learning. In: NIPS 2011 (2011)Google Scholar
  18. 18.
    Gupta, A., Nagarajan, V., Ravi, R.: Approximation Algorithms for Optimal Decision Trees and Adaptive TSP Problems. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010, Part I. LNCS, vol. 6198, pp. 690–701. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  19. 19.
    Gupta, A., Srinivasan, A.: An improved approximation ratio for the covering steiner problem. Theory of Computing 2(1), 53–64 (2006)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Halperin, E., Krauthgamer, R.: Polylogarithmic inapproximability. In: STOC 2003, pp. 585–594 (2003)Google Scholar
  21. 21.
    Im, S., Nagarajan, V., van der Zwaan, R.: Minimum latency submodular cover. CoRR, abs/1110.2207 (2011)Google Scholar
  22. 22.
    Konjevod, G., Ravi, R., Srinivasan, A.: Approximation algorithms for the covering steiner problem. Random Struct. Algorithms 20(3), 465–482 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Liu, Z., Parthasarathy, S., Ranganathan, A., Yang, H.: Near-optimal algorithms for shared filter evaluation in data stream systems. In: SIGMOD 2008, pp. 133–146 (2008)Google Scholar
  24. 24.
    Munagala, K., Srivastava, U., Widom, J.: Optimization of continuous queries with shared expensive filters. In: PODS 2007, pp. 215–224 (2007)Google Scholar
  25. 25.
    Nagarajan, V.: Approximation Algorithms for Sequencing Problems. PhD thesis. Tepper School of Business, Carnegie Mellon University (2009)Google Scholar
  26. 26.
    Schrijver, A.: Combinatorial optimization: polyhedra and efficiency. Springer, Berlin (2003)zbMATHGoogle Scholar
  27. 27.
    Wolsey, L.A.: An analysis of the greedy algorithm for the submodular set covering problem. Combinatorica 2(4), 385–393 (1982)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Sungjin Im
    • 1
  • Viswanath Nagarajan
    • 2
  • Ruben van der Zwaan
    • 3
  1. 1.Department of Computer ScienceUniversity of IllinoisUSA
  2. 2.IBM T. J. Watson Research CenterUSA
  3. 3.Maastricht UniversityThe Netherlands

Personalised recommendations