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Incremental Maximization via Continuization

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

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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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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.

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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

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.

References

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