Document Open Access Logo

Learning-Augmented Query Policies for Minimum Spanning Tree with Uncertainty

Authors Thomas Erlebach , Murilo Santos de Lima , Nicole Megow , Jens Schlöter

PDF

File

Thumbnail PDF
PDF
LIPIcs.ESA.2022.49.pdf
  • Filesize: 0.79 MB
  • 18 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • explorable uncertainty
  • queries
  • untrusted predictions

Metrics

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

Abstract

We study how to utilize (possibly erroneous) predictions in a model for computing under uncertainty in which an algorithm can query unknown data. Our aim is to minimize the number of queries needed to solve the minimum spanning tree problem, a fundamental combinatorial optimization problem that has been central also to the research area of explorable uncertainty. For all integral γ ≥ 2, we present algorithms that are γ-robust and (1+1/γ)-consistent, meaning that they use at most γOPT queries if the predictions are arbitrarily wrong and at most (1+1/γ)OPT queries if the predictions are correct, where OPT is the optimal number of queries for the given instance. Moreover, we show that this trade-off is best possible. Furthermore, we argue that a suitably defined hop distance is a useful measure for the amount of prediction error and design algorithms with performance guarantees that degrade smoothly with the hop distance. We also show that the predictions are PAC-learnable in our model. Our results demonstrate that untrusted predictions can circumvent the known lower bound of 2, without any degradation of the worst-case ratio. To obtain our results, we provide new structural insights for the minimum spanning tree problem that might be useful in the context of query-based algorithms regardless of predictions. In particular, we generalize the concept of witness sets - the key to lower-bounding the optimum - by proposing novel global witness set structures and completely new ways of adaptively using those.

Cite As Get BibTex

Thomas Erlebach, Murilo Santos de Lima, Nicole Megow, and Jens Schlöter. Learning-Augmented Query Policies for Minimum Spanning Tree with Uncertainty. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 49:1-49:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.ESA.2022.49

Author Details

Thomas Erlebach
  • Department of Computer Science, Durham University, UK
Murilo Santos de Lima
  • München, Germany
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

  • Erlebach, Thomas: Supported by EPSRC grant EP/S033483/2.
  • de Lima, Murilo Santos: Work done while employed at University of Leicester, United Kingdom, and 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.

References

  1. Mohamed Abdel-Nasser, Karar Mahmoud, Osama A. Omer, Matti Lehtonen, and Domenec Puig. Link quality prediction in wireless community networks using deep recurrent neural networks. Alexandria Engineering Journal, 59(5):3531-3543, 2020. URL: https://doi.org/10.1016/j.aej.2020.05.037.
  2. Susanne Albers and Alexander Eckl. Explorable uncertainty in scheduling with non-uniform testing times. In WAOA, volume 12806 of Lecture Notes in Computer Science, pages 127-142. Springer, 2020. URL: https://doi.org/10.1007/978-3-030-80879-2_9.
  3. Susanne Albers and Alexander Eckl. Scheduling with testing on multiple identical parallel machines. In WADS, volume 12808 of Lecture Notes in Computer Science, pages 29-42. Springer, 2021. URL: https://doi.org/10.1007/978-3-030-83508-8_3.
  4. Spyros Angelopoulos, Christoph Dürr, Shendan Jin, Shahin Kamali, and Marc P. Renault. Online computation with untrusted advice. In ITCS, volume 151 of LIPIcs, pages 52:1-52:15, 2020. URL: https://doi.org/10.4230/LIPIcs.ITCS.2020.52.
  5. Antonios Antoniadis, Christian Coester, Marek Eliás, Adam Polak, and Bertrand Simon. Online metric algorithms with untrusted predictions. In ICML, volume 119 of Proceedings of Machine Learning Research, pages 345-355. PMLR, 2020. URL: http://proceedings.mlr.press/v119/antoniadis20a.html.
  6. Antonios Antoniadis, Themis Gouleakis, Pieter Kleer, and Pavel Kolev. Secretary and online matching problems with machine learned advice. In NeurIPS, 2020. Google Scholar
  7. Luciana Arantes, Evripidis Bampis, Alexander V. Kononov, Manthos Letsios, Giorgio Lucarelli, and Pierre Sens. Scheduling under uncertainty: A query-based approach. In IJCAI, pages 4646-4652. ijcai.org, 2018. URL: https://doi.org/10.24963/ijcai.2018/646.
  8. Sepehr Assadi, Deeparnab Chakrabarty, and Sanjeev Khanna. Graph connectivity and single element recovery via linear and OR queries. In ESA, volume 204 of LIPIcs, pages 7:1-7:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.ESA.2021.7.
  9. Yossi Azar, Stefano Leonardi, and Noam Touitou. Flow time scheduling with uncertain processing time. In STOC, pages 1070-1080. ACM, 2021. URL: https://doi.org/10.1145/3406325.3451023.
  10. Evripidis Bampis, Konstantinos Dogeas, Alexander V. Kononov, Giorgio Lucarelli, and Fanny Pascual. Speed scaling with explorable uncertainty. In SPAA, pages 83-93. ACM, 2021. URL: https://doi.org/10.1145/3409964.3461812.
  11. Evripidis Bampis, Christoph Dürr, Thomas Erlebach, Murilo Santos de Lima, Nicole Megow, and Jens Schlöter. Orienting (hyper)graphs under explorable stochastic uncertainty. In ESA, volume 204 of LIPIcs, pages 10:1-10:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.ESA.2021.10.
  12. Paul Beame, Sariel Har-Peled, Sivaramakrishnan Natarajan Ramamoorthy, Cyrus Rashtchian, and Makrand Sinha. Edge estimation with independent set oracles. ACM Trans. Algorithms, 16(4):52:1-52:27, 2020. Google Scholar
  13. Paul Beame, Sariel Har-Peled, Sivaramakrishnan Natarajan Ramamoorthy, Cyrus Rashtchian, and Makrand Sinha. Edge estimation with independent set oracles. ACM Trans. Algorithms, 16(4):52:1-52:27, 2020. URL: https://doi.org/10.1145/3404867.
  14. Norman Biggs, E Keith Lloyd, and Robin J Wilson. Graph Theory, 1736-1936. Oxford University Press, 1986. Google Scholar
  15. Richard Bruce, Michael Hoffmann, Danny Krizanc, and Rajeev Raman. Efficient update strategies for geometric computing with uncertainty. Theory Comput. Syst., 38(4):411-423, 2005. URL: https://doi.org/10.1007/s00224-004-1180-4.
  16. 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. URL: https://doi.org/10.1561/2200000024.
  17. Steven Chaplick, Magnús M. Halldórsson, Murilo S. de Lima, and Tigran Tonoyan. Query minimization under stochastic uncertainty. Theor. Comput. Sci., 895:75-95, 2021. URL: https://doi.org/10.1016/j.tcs.2021.09.032.
  18. Christoph Dürr, Thomas Erlebach, Nicole Megow, and Julie Meißner. An adversarial model for scheduling with testing. Algorithmica, 82(12):3630-3675, 2020. URL: https://doi.org/10.1007/s00453-020-00742-2.
  19. Franziska Eberle, Alexander Lindermayr, Nicole Megow, Lukas Nölke, and Jens Schlöter. Robustification of online graph exploration methods. In AAAI, 2022. Google Scholar
  20. Thomas Erlebach, Murilo Santos de Lima, Nicole Megow, and Jens Schlöter. Learning-augmented query policies for minimum spanning tree with uncertainty. CoRR, abs/2206.15201, 2022. Google Scholar
  21. Thomas Erlebach and Michael Hoffmann. Minimum spanning tree verification under uncertainty. In WG 2014: International Workshop on Graph-Theoretic Concepts in Computer Science, volume 8747 of Lecture Notes in Computer Science, pages 164-175. Springer, 2014. URL: https://doi.org/10.1007/978-3-319-12340-0_14.
  22. Thomas Erlebach and Michael Hoffmann. Query-competitive algorithms for computing with uncertainty. Bull. EATCS, 116, 2015. URL: https://doi.org/10.1016/j.tcs.2015.11.025.
  23. Thomas Erlebach, Michael Hoffmann, and Frank Kammer. Query-competitive algorithms for cheapest set problems under uncertainty. Theor. Comput. Sci., 613:51-64, 2016. Google Scholar
  24. Thomas Erlebach, Michael Hoffmann, Danny Krizanc, Matús Mihalák, and Rajeev Raman. Computing minimum spanning trees with uncertainty. In STACS, volume 1 of LIPIcs, pages 277-288. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, Germany, 2008. URL: https://doi.org/10.4230/LIPIcs.STACS.2008.1358.
  25. Tomás Feder, Rajeev Motwani, Liadan O'Callaghan, Chris Olston, and Rina Panigrahy. Computing shortest paths with uncertainty. J. Algorithms, 62(1):1-18, 2007. URL: https://doi.org/10.1016/j.jalgor.2004.07.005.
  26. Tomás Feder, Rajeev Motwani, Rina Panigrahy, Chris Olston, and Jennifer Widom. Computing the median with uncertainty. SIAM J. Comput., 32(2):538-547, 2003. URL: https://doi.org/10.1137/S0097539701395668.
  27. 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.
  28. John Gittins, Kevin Glazebrook, and Richard Weber. Multi-armed Bandit Allocation Indices. Wiley, 2nd edition, 2011. Google Scholar
  29. Marc Goerigk, Manoj Gupta, Jonas Ide, Anita Schöbel, and Sandeep Sen. The robust knapsack problem with queries. Comput. Oper. Res., 55:12-22, 2015. URL: https://doi.org/10.1016/j.cor.2014.09.010.
  30. Oded Goldreich. Introduction to Property Testing. Cambridge University Press, 2017. URL: https://doi.org/10.1017/9781108135252.
  31. Sreenivas Gollapudi and Debmalya Panigrahi. Online algorithms for rent-or-buy with expert advice. In ICML, volume 97 of Proceedings of Machine Learning Research, pages 2319-2327. PMLR, 2019. Google Scholar
  32. Anupam Gupta, Haotian Jiang, Ziv Scully, and Sahil Singla. The markovian price of information. In IPCO, volume 11480 of Lecture Notes in Computer Science, pages 233-246. Springer, 2019. URL: https://doi.org/10.1007/978-3-030-17953-3_18.
  33. Manoj Gupta, Yogish Sabharwal, and Sandeep Sen. The update complexity of selection and related problems. Theory Comput. Syst., 59(1):112-132, 2016. URL: https://doi.org/10.1007/s00224-015-9664-y.
  34. Magnús M. Halldórsson and Murilo Santos de Lima. Query-competitive sorting with uncertainty. Theor. Comput. Sci., 867:50-67, 2022. URL: https://doi.org/10.1016/j.tcs.2021.03.021.
  35. Simon Kahan. A model for data in motion. In STOC, pages 267-277. ACM, 1991. URL: https://doi.org/10.1145/103418.103449.
  36. Sanjeev Khanna and Wang Chiew Tan. On computing functions with uncertainty. In PODS, pages 171-182. ACM, 2001. URL: https://doi.org/10.1145/375551.375577.
  37. Ravi Kumar, Manish Purohit, Aaron Schild, Zoya Svitkina, and Erik Vee. Semi-online bipartite matching. In ITCS, volume 124 of LIPIcs, pages 50:1-50:20. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.ITCS.2019.50.
  38. Silvio Lattanzi, Thomas Lavastida, Benjamin Moseley, and Sergei Vassilvitskii. Online scheduling via learned weights. In SODA, pages 1859-1877. SIAM, 2020. URL: https://doi.org/10.1137/1.9781611975994.114.
  39. Alexander Lindermayr, Nicole Megow, and Bertrand Simon. Double Coverage with Machine-Learned Advice. In ITCS, volume 215 of LIPIcs, pages 99:1-99:18, 2022. URL: https://doi.org/10.4230/LIPIcs.ITCS.2022.99.
  40. Thodoris Lykouris and Sergei Vassilvitskii. Competitive caching with machine learned advice. J. ACM, 68(4):24:1-24:25, 2021. URL: https://doi.org/10.1145/3447579.
  41. Takanori Maehara and Yutaro Yamaguchi. Stochastic packing integer programs with few queries. Math. Program., 182(1):141-174, 2020. URL: https://doi.org/10.1007/s10107-019-01388-x.
  42. Hanna Mazzawi. Optimally reconstructing weighted graphs using queries. In SODA, pages 608-615. SIAM, 2010. URL: https://doi.org/10.1137/1.9781611973075.51.
  43. Nicole Megow, Julie Meißner, and Martin Skutella. Randomization helps computing a minimum spanning tree under uncertainty. SIAM J. Comput., 46(4):1217-1240, 2017. URL: https://doi.org/10.1137/16M1088375.
  44. Arturo I. Merino and José A. Soto. The minimum cost query problem on matroids with uncertainty areas. In ICALP, volume 132 of LIPIcs, pages 83:1-83:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.ICALP.2019.83.
  45. Michael Mitzenmacher. Scheduling with predictions and the price of misprediction. In ITCS, volume 151 of LIPIcs, pages 14:1-14:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.ITCS.2020.14.
  46. Noam Nisan. The demand query model for bipartite matching. In SODA, pages 592-599. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976465.36.
  47. Chris Olston and Jennifer Widom. Offering a precision-performance tradeoff for aggregation queries over replicated data. In VLDB, pages 144-155. Morgan Kaufmann, 2000. Google Scholar
  48. Manish Purohit, Zoya Svitkina, and Ravi Kumar. Improving online algorithms via ML predictions. In NeurIPS, pages 9684-9693, 2018. Google Scholar
  49. Dhruv Rohatgi. Near-optimal bounds for online caching with machine learned advice. In SODA, pages 1834-1845. SIAM, 2020. URL: https://doi.org/10.1137/1.9781611975994.112.
  50. Aviad Rubinstein, Tselil Schramm, and S. Matthew Weinberg. Computing exact minimum cuts without knowing the graph. In ITCS, 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.
  51. Sahil Singla. The price of information in combinatorial optimization. In SODA, pages 2523-2532. SIAM, 2018. URL: https://doi.org/10.1137/1.9781611975031.161.
  52. 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. URL: https://doi.org/10.2307/2332286.
  53. Alexander Wei. Better and simpler learning-augmented online caching. In APPROX-RANDOM, volume 176 of LIPIcs, pages 60:1-60:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2020.60.
  54. Alexander Wei and Fred Zhang. Optimal robustness-consistency trade-offs for learning-augmented online algorithms. In NeurIPS, 2020. Google Scholar
  55. Martin Weitzman. Optimal search for the best alternative. Econometrica, 47(3):641-54, 1979. URL: https://doi.org/10.2307/1910412.
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
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact