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Incremental Maximization via Continuization

Authors Yann Disser , Max Klimm , Kevin Schewior , David Weckbecker

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Author Details

Yann Disser
  • TU Darmstadt, Germany
Max Klimm
  • TU Berlin, Germany
Kevin Schewior
  • University of Southern Denmark, Odense, Denmark
David Weckbecker
  • TU Darmstadt, Germany

Funding

  • Klimm, Max: Supported by Deutsche Forschungsgemeinschaft under Germany’s Excellence Strategy, Berlin Mathematics Research Center (grant EXC-2046/1, Project 390685689).
  • Schewior, Kevin: Supported in part by the Independent Research Fund Denmark, Natural Sciences, grant DFF-0135-00018B.
  • Weckbecker, David: Supported by DFG grant DI 2041/2.

Cite As Get BibTex

Yann Disser, Max Klimm, Kevin Schewior, and David Weckbecker. Incremental Maximization via Continuization. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 47:1-47:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.ICALP.2023.47

Abstract

We consider the problem of finding an incremental solution to a cardinality-constrained maximization problem that not only captures the solution for a fixed cardinality, but also describes how to gradually grow the solution as the cardinality bound increases. The goal is to find an incremental solution that guarantees a good competitive ratio against the optimum solution for all cardinalities simultaneously. The central challenge is to characterize maximization problems where this is possible, and to determine the best-possible competitive ratio that can be attained. A lower bound of 2.18 and an upper bound of φ + 1 ≈ 2.618 are known on the competitive ratio for monotone and accountable objectives [Bernstein et al., Math. Prog., 2022], which capture a wide range of maximization problems. We introduce a continuization technique and identify an optimal incremental algorithm that provides strong evidence that φ + 1 is the best-possible competitive ratio. Using this continuization, we obtain an improved lower bound of 2.246 by studying a particular recurrence relation whose characteristic polynomial has complex roots exactly beyond the lower bound. Based on the optimal continuous algorithm combined with a scaling approach, we also provide a 1.772-competitive randomized algorithm. We complement this by a randomized lower bound of 1.447 via Yao’s principle.

Subject Classification

ACM Subject Classification
  • Theory of computation → Online algorithms
  • Mathematics of computing → Combinatorial algorithms
  • Mathematics of computing → Combinatorial optimization
Keywords
  • incremental optimization
  • competitive analysis
  • robust matching
  • submodular function

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References

  1. Aaron Bernstein, Yann Disser, Martin Groß, and Sandra Himburg. General bounds for incremental maximization. Math. Program., 191(2):953-979, 2022. URL: https://doi.org/10.1007/s10107-020-01576-0.
  2. Avrim Blum, Prasad Chalasani, Don Coppersmith, William R. Pulleyblank, Prabhakar Raghavan, and Madhu Sudan. The minimum latency problem. In Proceedings of the Twenty-Sixth Annual ACM Symposium on Theory of Computing (STOC), pages 163-171. ACM, 1994. URL: https://doi.org/10.1145/195058.195125.
  3. Marek Chrobak, Claire Kenyon, John Noga, and Neal E. Young. Incremental medians via online bidding. Algorithmica, 50(4):455-478, 2008. URL: https://doi.org/10.1007/s00453-007-9005-x.
  4. Yann Disser, Max Klimm, Nicole Megow, and Sebastian Stiller. Packing a knapsack of unknown capacity. SIAM J. Discret. Math., 31(3):1477-1497, 2017. URL: https://doi.org/10.1137/16M1070049.
  5. Yann Disser, Max Klimm, Kevin Schewior, and David Weckbecker. Incremental maximization via continuization. URL: https://arxiv.org/abs/2305.01310v1.
  6. Yann Disser, Max Klimm, and David Weckbecker. Fractionally subadditive maximization under an incremental knapsack constraint. In Proceedings of the 19th International Workshop on Approximation and Online Algorithms (WAOA), pages 206-223. Springer, 2021. URL: https://doi.org/10.1007/978-3-030-92702-8_13.
  7. Ryo Fujita, Yusuke Kobayashi, and Kazuhisa Makino. Robust matchings and matroid intersections. SIAM J. Discret. Math., 27(3):1234-1256, 2013. URL: https://doi.org/10.1137/100808800.
  8. Michel X. Goemans and Jon M. Kleinberg. An improved approximation ratio for the minimum latency problem. Math. Program., 82:111-124, 1998. URL: https://doi.org/10.1007/BF01585867.
  9. Michel X. Goemans and Francisco Unda. Approximating incremental combinatorial optimization problems. In Proceedings of the 20th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX), pages 6:1-6:14, 2017. URL: https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2017.6.
  10. Teofilo F. Gonzalez. Clustering to minimize the maximum intercluster distance. Theor. Comput. Sci., 38:293-306, 1985. URL: https://doi.org/10.1016/0304-3975(85)90224-5.
  11. Jeff Hartline and Alexa Sharp. An incremental model for combinatorial maximization problems. In Proceedings of the 5th International Workshop on Experimental Algorithms (WEA), pages 36-48, 2006. URL: https://doi.org/10.1007/11764298_4.
  12. Jeff Hartline and Alexa Sharp. Incremental flow. Networks, 50(1):77-85, 2007. URL: https://doi.org/10.1002/net.20168.
  13. Refael Hassin and Shlomi Rubinstein. Robust matchings. SIAM J. Discret. Math., 15(4):530-537, 2002. URL: https://doi.org/10.1137/S0895480198332156.
  14. Refael Hassin and Danny Segev. Robust subgraphs for trees and paths. ACM Trans. Algorithms, 2(2):263-281, 2006. URL: https://doi.org/10.1145/1150334.1150341.
  15. Naonori Kakimura and Kazuhisa Makino. Robust independence systems. SIAM J. Discret. Math., 27(3):1257-1273, 2013. URL: https://doi.org/10.1137/120899480.
  16. Ming-Yang Kao, John H Reif, and Stephen R Tate. Searching in an unknown environment: An optimal randomized algorithm for the cow-path problem. Information and Computation, 131(1):63-79, 1996. Google Scholar
  17. Max Klimm and Martin Knaack. Maximizing a submodular function with bounded curvature under an unknown knapsack constraint. In Proceedings of the 25th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX), pages 49:1-49:19, 2022. URL: https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2022.49.
  18. Guolong Lin, Chandrashekhar Nagarajan, Rajmohan Rajaraman, and David P. Williamson. A general approach for incremental approximation and hierarchical clustering. SIAM J. Comput., 39(8):3633-3669, 2010. URL: https://doi.org/10.1137/070698257.
  19. Jannik Matuschke, Martin Skutella, and José A. Soto. Robust randomized matchings. Math. Oper. Res., 43(2):675-692, 2018. URL: https://doi.org/10.1287/moor.2017.0878.
  20. Nicole Megow and Julián Mestre. Instance-sensitive robustness guarantees for sequencing with unknown packing and covering constraints. In Proceedings of the 4th Innovations in Theoretical Computer Science Conference (ITCS), pages 495-504, 2013. URL: https://doi.org/10.1145/2422436.2422490.
  21. Julián Mestre. Greedy in approximation algorithms. In Proceedings of the 14th Annual European Symposium on Algorithms (ESA), pages 528-539, 2006. URL: https://doi.org/10.1007/11841036_48.
  22. Ramgopal R. Mettu and C. Greg Plaxton. The online median problem. SIAM J. Comput., 32(3):816-832, 2003. URL: https://doi.org/10.1137/S0097539701383443.
  23. C. Greg Plaxton. Approximation algorithms for hierarchical location problems. J. Comput. Syst. Sci., 72(3):425-443, 2006. URL: https://doi.org/10.1016/j.jcss.2005.09.004.
  24. Andrew Chi-Chin Yao. Probabilistic computations: Toward a unified measure of complexity. In 18th Annual Symposium on Foundations of Computer Science (sfcs 1977), pages 222-227, 1977. URL: https://doi.org/10.1109/SFCS.1977.24.
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