(Near) Optimal Adaptivity Gaps for Stochastic Multi-Value Probing

Authors Domagoj Bradac , Sahil Singla , Goran Zuzic



PDF
Thumbnail PDF

File

LIPIcs.APPROX-RANDOM.2019.49.pdf
  • Filesize: 0.63 MB
  • 21 pages

Document Identifiers

Author Details

Domagoj Bradac
  • Department of Mathematics, Faculty of Science, University of Zagreb, Croatia
Sahil Singla
  • Department of Computer Science, Princeton University, NJ, USA
Goran Zuzic
  • Department of Computer Science, Carnegie Mellon University, Pittsburgh, PA, USA

Acknowledgements

We would like to thank Anupam Gupta for helpful discussions.

Cite AsGet BibTex

Domagoj Bradac, Sahil Singla, and Goran Zuzic. (Near) Optimal Adaptivity Gaps for Stochastic Multi-Value Probing. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 49:1-49:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.49

Abstract

Consider a kidney-exchange application where we want to find a max-matching in a random graph. To find whether an edge e exists, we need to perform an expensive test, in which case the edge e appears independently with a known probability p_e. Given a budget on the total cost of the tests, our goal is to find a testing strategy that maximizes the expected maximum matching size. The above application is an example of the stochastic probing problem. In general the optimal stochastic probing strategy is difficult to find because it is adaptive - decides on the next edge to probe based on the outcomes of the probed edges. An alternate approach is to show the adaptivity gap is small, i.e., the best non-adaptive strategy always has a value close to the best adaptive strategy. This allows us to focus on designing non-adaptive strategies that are much simpler. Previous works, however, have focused on Bernoulli random variables that can only capture whether an edge appears or not. In this work we introduce a multi-value stochastic probing problem, which can also model situations where the weight of an edge has a probability distribution over multiple values. Our main technical contribution is to obtain (near) optimal bounds for the (worst-case) adaptivity gaps for multi-value stochastic probing over prefix-closed constraints. For a monotone submodular function, we show the adaptivity gap is at most 2 and provide a matching lower bound. For a weighted rank function of a k-extendible system (a generalization of intersection of k matroids), we show the adaptivity gap is between O(k log k) and k. None of these results were known even in the Bernoulli case where both our upper and lower bounds also apply, thereby resolving an open question of Gupta et al. [Gupta et al., 2017].

Subject Classification

ACM Subject Classification
  • Theory of computation → Stochastic control and optimization
  • Theory of computation → Submodular optimization and polymatroids
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Design and analysis of algorithms
Keywords
  • stochastic programming
  • adaptivity gaps
  • stochastic multi-value probing
  • submodular functions
  • k-extendible systems
  • adaptive strategy
  • matroid intersection

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. Marek Adamczyk. Improved analysis of the greedy algorithm for stochastic matching. Inf. Process. Lett., 111(15):731-737, 2011. Google Scholar
  2. Marek Adamczyk, Fabrizio Grandoni, and Joydeep Mukherjee. Improved approximation algorithms for stochastic matching. In Algorithms-ESA 2015, pages 1-12. Springer, 2015. Google Scholar
  3. Marek Adamczyk, Maxim Sviridenko, and Justin Ward. Submodular Stochastic Probing on Matroids. In STACS, pages 29-40, 2014. Google Scholar
  4. Arash Asadpour and Hamid Nazerzadeh. Maximizing stochastic monotone submodular functions. Management Science, 62(8):2374-2391, 2016. Google Scholar
  5. Arash Asadpour, Hamid Nazerzadeh, and Amin Saberi. Stochastic submodular maximization. In International Workshop on Internet and Network Economics, pages 477-489. Springer, 2008. Full version appears as [Asadpour and Nazerzadeh, 2016]. Google Scholar
  6. Itai Ashlagi and Alvin E. Roth. New Challenges in Multihospital Kidney Exchange. American Economic Review, 102(3):354-59, 2012. Google Scholar
  7. Nikhil Bansal, Anupam Gupta, Jian Li, Julián Mestre, Viswanath Nagarajan, and Atri Rudra. When LP Is the Cure for Your Matching Woes: Improved Bounds for Stochastic Matchings. Algorithmica, 63(4):733-762, 2012. Google Scholar
  8. Nikhil Bansal and Viswanath Nagarajan. On the Adaptivity Gap of Stochastic Orienteering. In IPCO, pages 114-125, 2014. Google Scholar
  9. Alok Baveja, Amit Chavan, Andrei Nikiforov, Aravind Srinivasan, and Pan Xu. Improved Bounds in Stochastic Matching and Optimization. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2015, August 24-26, 2015, Princeton, NJ, USA, pages 124-134, 2015. Google Scholar
  10. Anand Bhalgat, Ashish Goel, and Sanjeev Khanna. Improved Approximation Results for Stochastic Knapsack Problems. In SODA, pages 1647-1665, 2011. Google Scholar
  11. Gruia Călinescu, Chandra Chekuri, Martin Pál, and Jan Vondrák. Maximizing a Monotone Submodular Function Subject to a Matroid Constraint. SIAM J. Comput., 40(6):1740-1766, 2011. Google Scholar
  12. Chandra Chekuri, Jan Vondrák, and Rico Zenklusen. Submodular Function Maximization via the Multilinear Relaxation and Contention Resolution Schemes. SIAM J. Comput., 43(6):1831-1879, 2014. URL: https://doi.org/10.1137/110839655.
  13. Ning Chen, Nicole Immorlica, Anna R. Karlin, Mohammad Mahdian, and Atri Rudra. Approximating Matches Made in Heaven. In Automata, Languages and Programming, 36th International Colloquium, ICALP 2009, Rhodes, Greece, July 5-12, 2009, Proceedings, Part I, pages 266-278, 2009. Google Scholar
  14. Brian C. Dean, Michel X. Goemans, and Jan Vondrák. Approximating the stochastic knapsack problem: The benefit of adaptivity. In Foundations of Computer Science, 2004. Proceedings. 45th Annual IEEE Symposium on, pages 208-217. IEEE, 2004. Google Scholar
  15. Brian C. Dean, Michel X. Goemans, and Jan Vondrák. Adaptivity and approximation for stochastic packing problems. In Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2005, Vancouver, British Columbia, Canada, January 23-25, 2005, pages 395-404, 2005. Google Scholar
  16. Amol Deshpande, Lisa Hellerstein, and Devorah Kletenik. Approximation Algorithms for Stochastic Boolean Function Evaluation and Stochastic Submodular Set Cover. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 1453-1466, 2014. URL: https://doi.org/10.1137/1.9781611973402.107.
  17. Sudipto Guha and Kamesh Munagala. Approximation algorithms for budgeted learning problems. In Proceedings of the thirty-ninth annual ACM symposium on Theory of computing, pages 104-113. ACM, 2007. Google Scholar
  18. Sudipto Guha and Kamesh Munagala. Multi-armed Bandits with Metric Switching Costs. In Automata, Languages and Programming, 36th Internatilonal Colloquium, ICALP 2009, Rhodes, Greece, July 5-12, 2009, Proceedings, Part II, pages 496-507, 2009. Google Scholar
  19. Anupam Gupta, Ravishankar Krishnaswamy, Marco Molinaro, and R. Ravi. Approximation Algorithms for Correlated Knapsacks and Non-martingale Bandits. In IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, Palm Springs, CA, USA, October 22-25, 2011, pages 827-836, 2011. Google Scholar
  20. Anupam Gupta, Ravishankar Krishnaswamy, Viswanath Nagarajan, and R. Ravi. Approximation algorithms for stochastic orienteering. In Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, Kyoto, Japan, January 17-19, 2012, pages 1522-1538, 2012. Google Scholar
  21. Anupam Gupta and Viswanath Nagarajan. A Stochastic Probing Problem with Applications. In Integer Programming and Combinatorial Optimization - 16th International Conference, IPCO 2013, Valparaíso, Chile, March 18-20, 2013. Proceedings, pages 205-216, 2013. Google Scholar
  22. Anupam Gupta, Viswanath Nagarajan, and Sahil Singla. Algorithms and adaptivity gaps for stochastic probing. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1731-1747. SIAM, 2016. Google Scholar
  23. Anupam Gupta, Viswanath Nagarajan, and Sahil Singla. Adaptivity Gaps for Stochastic Probing: Submodular and XOS Functions. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1688-1702. SIAM, 2017. Google Scholar
  24. Jian Li Hao Fu and Pan Xu. A PTAS for a Class of Stochastic Dynamic Programs. In Automata, Languages, and Programming - 39th International Colloquium, ICALP 2018, 2018. Google Scholar
  25. Lisa Hellerstein, Devorah Kletenik, and Patrick Lin. Discrete Stochastic Submodular Maximization: Adaptive vs. Non-adaptive vs. Offline. In Algorithms and Complexity - 9th International Conference, CIAC 2015, Paris, France, May 20-22, 2015. Proceedings, pages 235-248, 2015. URL: https://doi.org/10.1007/978-3-319-18173-8_17.
  26. Jian Li and Wen Yuan. Stochastic combinatorial optimization via poisson approximation. In Symposium on Theory of Computing Conference, STOC'13, Palo Alto, CA, USA, June 1-4, 2013, pages 971-980, 2013. URL: https://doi.org/10.1145/2488608.2488731.
  27. Zhen Liu, Srinivasan Parthasarathy, Anand Ranganathan, and Hao Yang. Near-optimal algorithms for shared filter evaluation in data stream systems. In Proceedings of the ACM SIGMOD International Conference on Management of Data, SIGMOD 2008, Vancouver, BC, Canada, June 10-12, 2008, pages 133-146, 2008. URL: https://doi.org/10.1145/1376616.1376633.
  28. Will Ma. Improvements and Generalizations of Stochastic Knapsack and Multi-Armed Bandit Approximation Algorithms: Extended Abstract. In SODA, pages 1154-1163, 2014. Google Scholar
  29. Julián Mestre. Greedy in approximation algorithms. In European Symposium on Algorithms, pages 528-539. Springer, 2006. Google Scholar
  30. Alvin E. Roth, Tayfun Sönmez, and M.Ũtku Ünver. Pairwise kidney exchange. J. Econom. Theory, 125(2):151-188, 2005. URL: https://doi.org/10.1016/j.jet.2005.04.004.
  31. Aviad Rubinstein and Sahil Singla. Combinatorial prophet inequalities. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1671-1687. SIAM, 2017. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail