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Orienting (Hyper)graphs Under Explorable Stochastic Uncertainty

Authors Evripidis Bampis , Christoph Dürr , Thomas Erlebach , Murilo Santos de Lima , Nicole Megow , Jens Schlöter

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Author Details

Evripidis Bampis
  • Sorbonne Université, CNRS, LIP6, Paris, France
Christoph Dürr
  • Sorbonne Université, CNRS, LIP6, Paris, France
Thomas Erlebach
  • School of Informatics, University of Leicester, UK
Murilo Santos de Lima
  • School of Informatics, University of Leicester, UK
Nicole Megow
  • Faculty of Mathematics and Computer Science, University of Bremen, Germany
Jens Schlöter
  • Faculty of Mathematics and Computer Science, University of Bremen, Germany

Funding

  • Bampis, Evripidis: Supported by the grant ANR-19-CE48-0016 from the French National Research Agency (ANR).
  • Dürr, Christoph: Supported by the grant ANR-19-CE48-0016 from the French National Research Agency (ANR).
  • Erlebach, Thomas: Supported by EPSRC grant EP/S033483/1.
  • de Lima, Murilo Santos: Funded by EPSRC grant EP/S033483/1.
  • Megow, Nicole: Supported by the German Science Foundation (DFG) under contract ME 3825/1.
  • Schlöter, Jens: Funded by the German Science Foundation (DFG) under contract ME 3825/1.

Acknowledgements

The authors would like to thank the anonymous referees for their careful reading and helpful suggestions.

Cite AsGet BibTex

Evripidis Bampis, Christoph Dürr, Thomas Erlebach, Murilo Santos de Lima, Nicole Megow, and Jens Schlöter. Orienting (Hyper)graphs Under Explorable Stochastic Uncertainty. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 10:1-10:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.ESA.2021.10

Abstract

Given a hypergraph with uncertain node weights following known probability distributions, we study the problem of querying as few nodes as possible until the identity of a node with minimum weight can be determined for each hyperedge. Querying a node has a cost and reveals the precise weight of the node, drawn from the given probability distribution. Using competitive analysis, we compare the expected query cost of an algorithm with the expected cost of an optimal query set for the given instance. For the general case, we give a polynomial-time f(α)-competitive algorithm, where f(α) ∈ [1.618+ε,2] depends on the approximation ratio α for an underlying vertex cover problem. We also show that no algorithm using a similar approach can be better than 1.5-competitive. Furthermore, we give polynomial-time 4/3-competitive algorithms for bipartite graphs with arbitrary query costs and for hypergraphs with a single hyperedge and uniform query costs, with matching lower bounds.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Mathematics of computing → Discrete mathematics
Keywords
  • Explorable uncertainty
  • queries
  • stochastic optimization
  • graph orientation
  • selection problems

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References

  1. Marek Adamczyk, Maxim Sviridenko, and Justin Ward. Submodular stochastic probing on matroids. Math. Oper. Res., 41(3):1022-1038, 2016. Google Scholar
  2. Susanne Albers and Alexander Eckl. Explorable uncertainty in scheduling with non-uniform testing times. In WAOA 2020: 18th International Workshop on Approximation and Online Algorithms, 2021. Google Scholar
  3. Luciana Arantes, Evripidis Bampis, Alexander V. Kononov, Manthos Letsios, Giorgio Lucarelli, and Pierre Sens. Scheduling under uncertainty: A query-based approach. In IJCAI 2018: 27th International Joint Conference on Artificial Intelligence, pages 4646-4652, 2018. URL: https://doi.org/10.24963/ijcai.2018/646.
  4. Sepehr Assadi, Deeparnab Chakrabarty, and Sanjeev Khanna. Graph connectivity and single element recovery via linear and OR queries. In ESA 2021: 29th Annual European Symposium on Algorithms, volume 204 of LIPIcs. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. Google Scholar
  5. Sepehr Assadi, Sanjeev Khanna, and Yang Li. The stochastic matching problem with (very) few queries. ACM Trans. Economics and Comput., 7(3):16:1-16:19, 2019. Google Scholar
  6. Evripidis Bampis, Christoph Dürr, Thomas Erlebach, Murilo S. de Lima, Nicole Megow, and Jens Schlöter. Orienting (hyper)graphs under explorable stochastic uncertainty. CoRR, abs/2107.00572, 2021. URL: https://arxiv.org/abs/2107.00572.
  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. Math. Program., 154(1-2):145-172, 2015. Google Scholar
  9. Reuven Bar-Yehuda, Keren Bendel, Ari Freund, and Dror Rawitz. Local ratio: A unified framework for approxmation algrithms. In memoriam: Shimon Even 1935-2004. ACM Comput. Surv., 36(4):422-463, 2004. URL: https://doi.org/10.1145/1041680.1041683.
  10. Reuven Bar-Yehuda and Shimon Even. On approximating a vertex cover for planar graphs. In Harry R. Lewis, Barbara B. Simons, Walter A. Burkhard, and Lawrence H. Landweber, editors, STOC 1982: 14th Annual ACM Symposium on Theory of Computing, pages 303-309. ACM, 1982. URL: https://doi.org/10.1145/800070.802205.
  11. Paul Beame, Sariel Har-Peled, Sivaramakrishnan Natarajan Ramamoorthy, Cyrus Rashtchian, and Makrand Sinha. Edge estimation with independent set oracles. In Anna R. Karlin, editor, ITCS 2018: 9th Innovations in Theoretical Computer Science Conference, volume 94 of LIPIcs, pages 38:1-38:21. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. URL: https://doi.org/10.4230/LIPIcs.ITCS.2018.38.
  12. Soheil Behnezhad, Alireza Farhadi, MohammadTaghi Hajiaghayi, and Nima Reyhani. Stochastic matching with few queries: New algorithms and tools. In SODA 2019: 30th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 2855-2874. SIAM, 2019. Google Scholar
  13. Avrim Blum, John P. Dickerson, Nika Haghtalab, Ariel D. Procaccia, Tuomas Sandholm, and Ankit Sharma. Ignorance is almost bliss: Near-optimal stochastic matching with few queries. Oper. Res., 68(1):16-34, 2020. Google Scholar
  14. Richard Bruce, Michael Hoffmann, Danny Krizanc, and Rajeev Raman. Efficient update strategies for geometric computing with uncertainty. Theory of Computing Systems, 38(4):411-423, 2005. URL: https://doi.org/10.1007/s00224-004-1180-4.
  15. Sébastien Bubeck and Nicolò Cesa-Bianchi. Regret analysis of stochastic and nonstochastic multi-armed bandit problems. Foundations and Trends in Machine Learning, 5(1):1-122, 2012. Google Scholar
  16. Steven Chaplick, Magnuús M. Halldórsson, Murilo S. de Lima, and Tigran Tonoyan. Query minimization under stochastic uncertainty. In Y. Kohayakawa and F. K. Miyazawa, editors, LATIN 2020: 14th Latin American Theoretical Informatics Symposium, volume 12118 of Lecture Notes in Computer Science, pages 181-193. Springer Berlin Heidelberg, 2020. Google Scholar
  17. Ning Chen, Nicole Immorlica, Anna R. Karlin, Mohammad Mahdian, and Atri Rudra. Approximating matches made in heaven. In ICALP 2009: 36th International Colloquium on Automata, Languages and Programming, Part I, volume 5555 of Lecture Notes in Computer Science, pages 266-278. Springer, 2009. URL: https://doi.org/10.1007/978-3-642-02927-1_23.
  18. Xi Chen, Amit Levi, and Erik Waingarten. Nearly optimal edge estimation with independent set queries. In Shuchi Chawla, editor, SODA 2020: Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms, pages 2916-2935. SIAM, 2020. URL: https://doi.org/10.1137/1.9781611975994.177.
  19. Miroslav Chlebik and Janka Chlebíková. The complexity of combinatorial optimization problems on d-dimensional boxes. SIAM Journal on Discrete Mathematics, 21(1):158-169, 2007. Google Scholar
  20. Brian C. Dean, Michel X. Goemans, and Jan Vondrák. Approximating the stochastic knapsack problem: The benefit of adaptivity. Math. Oper. Res., 33(4):945-964, 2008. Google Scholar
  21. Christoph Dürr, Thomas Erlebach, Nicole Megow, and Julie Meißner. An adversarial model for scheduling with testing. Algorithmica, 82(12):3630-3675, 2020. Google Scholar
  22. Thomas Erlebach and Michael Hoffmann. Minimum spanning tree verification under uncertainty. In D. Kratsch and I. Todinca, editors, WG 2014: International Workshop on Graph-Theoretic Concepts in Computer Science, volume 8747 of Lecture Notes in Computer Science, pages 164-175. Springer Berlin Heidelberg, 2014. URL: https://doi.org/10.1007/978-3-319-12340-0_14.
  23. Thomas Erlebach, Michael Hoffmann, and Murilo S. de Lima. Round-competitive algorithms for uncertainty problems with parallel queries. In Markus Bläser and Benjamin Monmege, editors, STACS 2021: 38th International Symposium on Theoretical Aspects of Computer Science, volume 187 of LIPIcs, pages 27:1-27:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.STACS.2021.27.
  24. Thomas Erlebach, Michael Hoffmann, Murilo S. de Lima, Nicole Megow, and Jens Schlöter. Untrusted predictions improve trustable query policies. CoRR, abs/2011.07385, 2020. URL: https://arxiv.org/abs/2011.07385.
  25. Thomas Erlebach, Michael Hoffmann, Danny Krizanc, Matús Mihalák, and Rajeev Raman. Computing minimum spanning trees with uncertainty. In STACS'08: 25th International Symposium on Theoretical Aspects of Computer Science, volume 1 of LIPIcs, pages 277-288. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Germany, 2008. URL: https://doi.org/10.4230/LIPIcs.STACS.2008.1358.
  26. Tomás Feder, Rajeev Motwani, Liadan O'Callaghan, Chris Olston, and Rina Panigrahy. Computing shortest paths with uncertainty. Journal of Algorithms, 62(1):1-18, 2007. URL: https://doi.org/10.1016/j.jalgor.2004.07.005.
  27. Tomás Feder, Rajeev Motwani, Rina Panigrahy, Chris Olston, and Jennifer Widom. Computing the median with uncertainty. SIAM Journal on Computing, 32(2):538-547, 2003. URL: https://doi.org/10.1137/S0097539701395668.
  28. Jacob Focke, Nicole Megow, and Julie Meißner. Minimum spanning tree under explorable uncertainty in theory and experiments. ACM J. Exp. Algorithmics, 25:1-20, 2020. URL: https://doi.org/10.1145/3422371.
  29. András Frank. Connections in Combinatorial Optimization. Oxford University Press, USA, 2011. Google Scholar
  30. John Gittins, Kevin Glazebrook, and Richard Weber. Multi-armed Bandit Allocation Indices. Wiley, 2nd edition, 2011. Google Scholar
  31. Marc Goerigk, Manoj Gupta, Jonas Ide, Anita Schöbel, and Sandeep Sen. The robust knapsack problem with queries. Computers & Operations Research, 55:12-22, 2015. URL: https://doi.org/10.1016/j.cor.2014.09.010.
  32. Oded Goldreich. Introduction to Property Testing. Cambridge University Press, 2017. URL: https://doi.org/10.1017/9781108135252.
  33. Anupam Gupta, Haotian Jiang, Ziv Scully, and Sahil Singla. The Markovian price of information. In IPCO 2019: 20th International Conference on Integer Programming and Combinatorial Optimization, volume 11480 of Lecture Notes in Computer Science, pages 233-246. Springer, 2019. Google Scholar
  34. Anupam Gupta, Ravishankar Krishnaswamy, Viswanath Nagarajan, and R. Ravi. Running errands in time: Approximation algorithms for stochastic orienteering. Math. Oper. Res., 40(1):56-79, 2015. Google Scholar
  35. Anupam Gupta and Viswanath Nagarajan. A stochastic probing problem with applications. In IPCO 2013: 16th International Conference on Integer Programming and Combinatorial Optimization, volume 7801 of Lecture Notes in Computer Science, pages 205-216. Springer, 2013. Google Scholar
  36. Anupam Gupta, Viswanath Nagarajan, and Sahil Singla. Algorithms and adaptivity gaps for stochastic probing. In SODA 2016: 27th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1731-1747. SIAM, 2016. Google Scholar
  37. Manoj Gupta, Yogish Sabharwal, and Sandeep Sen. The update complexity of selection and related problems. Theory of Computing Systems, 59(1):112-132, 2016. URL: https://doi.org/10.1007/s00224-015-9664-y.
  38. Magnús M. Halldórsson and Murilo Santos de Lima. Query-competitive sorting with uncertainty. Theor. Comput. Sci., 867:50-67, 2021. URL: https://doi.org/10.1016/j.tcs.2021.03.021.
  39. Eran Halperin. Improved approximation algorithms for the vertex cover problem in graphs and hypergraphs. SIAM Journal on Computing, 31(5):1608-1623, 2002. Google Scholar
  40. Simon Kahan. A model for data in motion. In STOC'91: 23rd Annual ACM Symposium on Theory of Computing, pages 265-277, 1991. URL: https://doi.org/10.1145/103418.103449.
  41. Sanjeev Khanna and Wang-Chiew Tan. On computing functions with uncertainty. In PODS'01: 20th ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems, pages 171-182, 2001. URL: https://doi.org/10.1145/375551.375577.
  42. Retsef Levi, Thomas L. Magnanti, and Yaron Shaposhnik. Scheduling with testing. Manag. Sci., 65(2):776-793, 2019. Google Scholar
  43. Will Ma. Improvements and generalizations of stochastic knapsack and Markovian bandits approximation algorithms. Math. Oper. Res., 43(3):789-812, 2018. Google Scholar
  44. Hanna Mazzawi. Optimally reconstructing weighted graphs using queries. In Moses Charikar, editor, SODA 2010: 21st Annual ACM-SIAM Symposium on Discrete Algorithms, pages 608-615. SIAM, 2010. URL: https://doi.org/10.1137/1.9781611973075.51.
  45. Nicole Megow, Julie Meißner, and Martin Skutella. Randomization helps computing a minimum spanning tree under uncertainty. SIAM Journal on Computing, 46(4):1217-1240, 2017. URL: https://doi.org/10.1137/16M1088375.
  46. Arturo I. Merino and José A. Soto. The minimum cost query problem on matroids with uncertainty areas. In ICALP 2019: 46th International Colloquium on Automata, Languages, and Programming, volume 132 of LIPIcs, pages 83:1-83:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. Google Scholar
  47. George L. Nemhauser and Leslie E. Trotter Jr. Vertex packings: Structural properties and algorithms. Mathematical Programming, 8:232-248, 1975. Google Scholar
  48. Noam Nisan. The demand query model for bipartite matching. In SODA 2021: Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, pages 592-599. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976465.36.
  49. Chris Olston and Jennifer Widom. Offering a precision-performance tradeoff for aggregation queries over replicated data. In VLDB 2000: 26th International Conference on Very Large Data Bases, pages 144-155, 2000. URL: http://ilpubs.stanford.edu:8090/437/.
  50. Herbert E. Robbins. A theorem on graphs, with an application to a problem of traffic control. The American Mathematical Monthly, 46(5):281-283, 1939. URL: http://www.jstor.org/stable/2303897.
  51. Aviad Rubinstein, Tselil Schramm, and S. Matthew Weinberg. Computing exact minimum cuts without knowing the graph. In Anna R. Karlin, editor, ITCS 2018: 9th Innovations in Theoretical Computer Science Conference, volume 94 of LIPIcs, pages 39:1-39:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. URL: https://doi.org/10.4230/LIPIcs.ITCS.2018.39.
  52. Alexander Schrijver. Combinatorial Optimization - Polyhedra and Efficiency. Springer, 2003. Google Scholar
  53. Sahil Singla. The price of information in combinatorial optimization. In SODA 2018: 29th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 2523-2532. SIAM, 2018. URL: https://doi.org/10.1137/1.9781611975031.161.
  54. William R. Thompson. On the likelihood that one unknown probability exceeds another in view of the evidence of two samples. Biometrika, 25(3/4):285-294, 1933. Google Scholar
  55. Martin Weitzman. Optimal search for the best alternative. Econometrica, 47(3):641-54, 1979. Google Scholar
  56. Mihalis Yannakakis and Fanica Gavril. Edge dominating sets in graphs. SIAM Journal on Applied Mathematics, 38(3):364-372, 1980. URL: https://doi.org/10.1137/0138030.
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