Adaptive Submodular Ranking

  • Prabhanjan Kambadur
  • Viswanath Nagarajan
  • Fatemeh NavidiEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10328)


We study a general stochastic ranking problem where an algorithm needs to adaptively select a sequence of elements so as to “cover” a random scenario (drawn from a known distribution) at minimum expected cost. The coverage of each scenario is captured by an individual submodular function, where the scenario is said to be covered when its function value goes above some threshold. We obtain a logarithmic factor approximation algorithm for this adaptive ranking problem, which is the best possible (unless \(P=NP\)). This problem unifies and generalizes many previously studied problems with applications in search ranking and active learning. The approximation ratio of our algorithm either matches or improves the best result known in each of these special cases. Moreover, our algorithm is simple to state and implement. We also present preliminary experimental results on a real data set.


Approximation Algorithm Greedy Algorithm Approximation Ratio Submodular Function Stochastic Optimization Problem 
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.



Part of V. Nagarajan’s work was done while visiting the Simons institute for theoretical computer science (UC Berkeley). The authors thank Lisa Hellerstein for a clarification on [16] regarding the OR construction of submodular functions.


  1. 1.
    Adler, M., Heeringa, B.: Approximating optimal binary decision trees. Algorithmica 62(3–4), 1112–1121 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Azar, Y., Gamzu, I.: Ranking with submodular valuations. In: SODA, pp. 1070–1079 (2011)Google Scholar
  3. 3.
    Azar, Y., Gamzu, I., Yin, X.: Multiple intents re-ranking. In: STOC, pp. 669–678 (2009)Google Scholar
  4. 4.
    Bansal, N., Gupta, A., Krishnaswamy, R.: A constant factor approximation algorithm for generalized min-sum set cover. In: SODA, pp. 1539–1545 (2010)Google Scholar
  5. 5.
    Bansal, N., Gupta, A., Li, J., Mestre, J., Nagarajan, V., Rudra, A.: When LP is the cure for your matching woes: improved bounds for stochastic matchings. Algorithmica 63(4), 733–762 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bellala, G., Bhavnani, S.K., Scott, C.: Group-based active query selection for rapid diagnosis in time-critical situations. IEEE Trans. Inf. Theor. 58(1), 459–478 (2012)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Blum, A., Chalasani, P., Coppersmith, D., Pulleyblank, W.R., Raghavan, P., Sudan, M.: The minimum latency problem. In: STOC, pp. 163–171 (1994)Google Scholar
  8. 8.
    Chakaravarthy, V.T., Pandit, V., Roy, S., Awasthi, P., Mohania, M.K.: Decision trees for entity identification: approximation algorithms and hardness results. ACM Trans. Algorithms 7(2), 15 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chaudhuri, K., Godfrey, B., Rao, S., Talwar, K.: Paths, trees, and minimum latency tours. In: FOCS, pp. 36–45 (2003)Google Scholar
  10. 10.
    Cicalese, F., Laber, E.S., Saettler, A.M.: Diagnosis determination: decision trees optimizing simultaneously worst and expected testing cost. In: ICML, pp. 414–422 (2014)Google Scholar
  11. 11.
    Dasgupta, S.: Analysis of a greedy active learning strategy. In: NIPS (2004)Google Scholar
  12. 12.
    Dean, B.C., Goemans, M.X., Vondrák, J.: Approximating the stochastic knapsack problem: the benefit of adaptivity. Math. Oper. Res. 33(4), 945–964 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Feige, U.: A threshold of ln n for approximating set cover. J. ACM 45(4), 634–652 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Golovin, D., Krause, A.: Adaptive submodularity: theory and applications in active learning and stochastic optimization. J. Artif. Intell. Res. 42, 427–486 (2011)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Golovin, D., Krause, A., Ray, D.: Near-optimal Bayesian active learning with noisy observations. In: NIPS, pp. 766–774 (2010)Google Scholar
  16. 16.
    Grammel, N., Hellerstein, L., Kletenik, D., Lin, P.: Scenario submodular cover. CoRR abs/1603.03158 (2016). (to appear in WAOA 2016)Google Scholar
  17. 17.
    Guillory, A., Bilmes, J.: Average-case active learning with costs. In: Gavaldà, R., Lugosi, G., Zeugmann, T., Zilles, S. (eds.) ALT 2009. LNCS, vol. 5809, pp. 141–155. Springer, Heidelberg (2009). doi: 10.1007/978-3-642-04414-4_15 CrossRefGoogle 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. LNCS, vol. 6198, pp. 690–701. Springer, Heidelberg (2010). doi: 10.1007/978-3-642-14165-2_58 CrossRefGoogle Scholar
  19. 19.
    Harper, F.M., Konstan, J.A.: The movielens datasets: history and context. ACM Trans. Interact. Intell. Syst. (TiiS) 5(4), 19 (2015)Google Scholar
  20. 20.
    Hyafil, L., Rivest, R.L.: Constructing optimal binary decision trees is \(NP\)-complete. Inf. Process. Lett. 5(1), 15–17 (1976/1977)Google Scholar
  21. 21.
    Im, S., Nagarajan, V., Zwaan, R.: Minimum latency submodular cover. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds.) ICALP 2012. LNCS, vol. 7391, pp. 485–497. Springer, Heidelberg (2012). doi: 10.1007/978-3-642-31594-7_41 CrossRefGoogle Scholar
  22. 22.
    Im, S., Sviridenko, M., van der Zwaan, R.: Preemptive and non-preemptive generalized min sum set cover. Math. Program. 145(1–2), 377–401 (2014)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Javdani, S., Chen, Y., Karbasi, A., Krause, A., Bagnell, D., Srinivasa, S.S.: Near optimal bayesian active learning for decision making. In: AISTATS, pp. 430–438 (2014)Google Scholar
  24. 24.
    Kosaraju, S.R., Przytycka, T.M., Borgstrom, R.: On an optimal split tree problem. In: Dehne, F., Sack, J.-R., Gupta, A., Tamassia, R. (eds.) WADS 1999. LNCS, vol. 1663, pp. 157–168. Springer, Heidelberg (1999). doi: 10.1007/3-540-48447-7_17 CrossRefGoogle Scholar
  25. 25.
    Navidi, F., Kambadur, P., Nagarajan, V.: Adaptive submodular ranking. arXiv preprint arXiv:1606.01530 (2016)
  26. 26.
    Skutella, M., Williamson, D.P.: A note on the generalized min-sum set cover problem. Oper. Res. Lett. 39(6), 433–436 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Wolsey, L.: An analysis of the greedy algorithm for the submodular set covering problem. Combinatorica 2(4), 385–393 (1982)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Prabhanjan Kambadur
    • 1
  • Viswanath Nagarajan
    • 2
  • Fatemeh Navidi
    • 2
    Email author
  1. 1.Bloomberg LPNew York CityUSA
  2. 2.Department of Industrial and Operations EngineeringUniversity of MichiganAnn ArborUSA

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