Bernstein, Aaron ; Disser, Yann ; Groß, Martin ; Himburg, Sandra (2024)
General bounds for incremental maximization.
In: Mathematical Programming: Series A, Series B, 2022, 191 (2)
doi: 10.26083/tuprints-00023878
Artikel, Zweitveröffentlichung, Verlagsversion
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Kurzbeschreibung (Abstract)
We propose a theoretical framework to capture incremental solutions to cardinality constrained maximization problems. The defining characteristic of our framework is that the cardinality/support of the solution is bounded by a value k∈N that grows over time, and we allow the solution to be extended one element at a time. We investigate the best-possible competitive ratio of such an incremental solution, i.e., the worst ratio over all k between the incremental solution after k steps and an optimum solution of cardinality k. We consider a large class of problems that contains many important cardinality constrained maximization problems like maximum matching, knapsack, and packing/covering problems. We provide a general 2.618-competitive incremental algorithm for this class of problems, and we show that no algorithm can have competitive ratio below 2.18 in general. In the second part of the paper, we focus on the inherently incremental greedy algorithm that increases the objective value as much as possible in each step. This algorithm is known to be 1.58-competitive for submodular objective functions, but it has unbounded competitive ratio for the class of incremental problems mentioned above. We define a relaxed submodularity condition for the objective function, capturing problems like maximum (weighted) d-dimensional matching, maximum (weighted) (b-)matching and a variant of the maximum flow problem. We show a general bound for the competitive ratio of the greedy algorithm on the class of problems that satisfy this relaxed submodularity condition. Our bound generalizes the (tight) bound of 1.58 slightly beyond sub-modular functions and yields a tight bound of 2.313 for maximum (weighted) (b-)matching. Our bound is also tight for a more general class of functions as the relevant parameter goes to infinity. Note that our upper bounds on the competitive ratios translate to approximation ratios for the underlying cardinality constrained problems, and our bounds for the greedy algorithm carry over both.
| Typ des Eintrags: | Artikel |
|---|---|
| Erschienen: | 2024 |
| Autor(en): | Bernstein, Aaron ; Disser, Yann ; Groß, Martin ; Himburg, Sandra |
| Art des Eintrags: | Zweitveröffentlichung |
| Titel: | General bounds for incremental maximization |
| Sprache: | Englisch |
| Publikationsjahr: | 23 April 2024 |
| Ort: | Darmstadt |
| Publikationsdatum der Erstveröffentlichung: | Februar 2022 |
| Ort der Erstveröffentlichung: | Berlin ; Heidelberg |
| Verlag: | Springer |
| Titel der Zeitschrift, Zeitung oder Schriftenreihe: | Mathematical Programming: Series A, Series B |
| Jahrgang/Volume einer Zeitschrift: | 191 |
| (Heft-)Nummer: | 2 |
| DOI: | 10.26083/tuprints-00023878 |
| URL / URN: | https://tuprints.ulb.tu-darmstadt.de/23878 |
| Zugehörige Links: | |
| Herkunft: | Zweitveröffentlichung DeepGreen |
| Kurzbeschreibung (Abstract): | We propose a theoretical framework to capture incremental solutions to cardinality constrained maximization problems. The defining characteristic of our framework is that the cardinality/support of the solution is bounded by a value k∈N that grows over time, and we allow the solution to be extended one element at a time. We investigate the best-possible competitive ratio of such an incremental solution, i.e., the worst ratio over all k between the incremental solution after k steps and an optimum solution of cardinality k. We consider a large class of problems that contains many important cardinality constrained maximization problems like maximum matching, knapsack, and packing/covering problems. We provide a general 2.618-competitive incremental algorithm for this class of problems, and we show that no algorithm can have competitive ratio below 2.18 in general. In the second part of the paper, we focus on the inherently incremental greedy algorithm that increases the objective value as much as possible in each step. This algorithm is known to be 1.58-competitive for submodular objective functions, but it has unbounded competitive ratio for the class of incremental problems mentioned above. We define a relaxed submodularity condition for the objective function, capturing problems like maximum (weighted) d-dimensional matching, maximum (weighted) (b-)matching and a variant of the maximum flow problem. We show a general bound for the competitive ratio of the greedy algorithm on the class of problems that satisfy this relaxed submodularity condition. Our bound generalizes the (tight) bound of 1.58 slightly beyond sub-modular functions and yields a tight bound of 2.313 for maximum (weighted) (b-)matching. Our bound is also tight for a more general class of functions as the relevant parameter goes to infinity. Note that our upper bounds on the competitive ratios translate to approximation ratios for the underlying cardinality constrained problems, and our bounds for the greedy algorithm carry over both. |
| Freie Schlagworte: | Incremental optimization, Maximization problems, Greedy algorithm, Competitive analysis, Cardinality constraint |
| Status: | Verlagsversion |
| URN: | urn:nbn:de:tuda-tuprints-238787 |
| Zusätzliche Informationen: | Mathematics Subject Classification: 68W27 · 68W25 · 90C27 · 68Q25 |
| Sachgruppe der Dewey Dezimalklassifikatin (DDC): | 500 Naturwissenschaften und Mathematik > 510 Mathematik |
| Fachbereich(e)/-gebiet(e): | 04 Fachbereich Mathematik 04 Fachbereich Mathematik > Optimierung |
| Hinterlegungsdatum: | 23 Apr 2024 12:45 |
| Letzte Änderung: | 25 Apr 2024 10:18 |
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| Suche nach Titel in: | TUfind oder in Google |
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