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.
@InProceedings{disser_et_al:LIPIcs.ICALP.2023.47, author = {Disser, Yann and Klimm, Max and Schewior, Kevin and Weckbecker, David}, title = {{Incremental Maximization via Continuization}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {47:1--47:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-278-5}, ISSN = {1868-8969}, year = {2023}, volume = {261}, editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.47}, URN = {urn:nbn:de:0030-drops-180992}, doi = {10.4230/LIPIcs.ICALP.2023.47}, annote = {Keywords: incremental optimization, competitive analysis, robust matching, submodular function} }
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