General bounds for incremental maximization

Aaron Bernstein, Yann Disser, Martin Groß

Research output: Chapter in Book/Report/Conference proceedingConference contribution

1 Scopus citations

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 define 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 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) (b-)matching and a variant of the maximum flow problem. We show that the greedy algorithm has competitive ratio (exactly) 2.313 for the class of problems that satisfy this relaxed submodularity condition. Note that our upper bounds on the competitive ratios translate to approximation ratios for the underlying cardinality constrained problems.

Original languageEnglish (US)
Title of host publication44th International Colloquium on Automata, Languages, and Programming, ICALP 2017
EditorsAnca Muscholl, Piotr Indyk, Fabian Kuhn, Ioannis Chatzigiannakis
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959770415
DOIs
StatePublished - Jul 1 2017
Externally publishedYes
Event44th International Colloquium on Automata, Languages, and Programming, ICALP 2017 - Warsaw, Poland
Duration: Jul 10 2017Jul 14 2017

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume80
ISSN (Print)1868-8969

Other

Other44th International Colloquium on Automata, Languages, and Programming, ICALP 2017
Country/TerritoryPoland
CityWarsaw
Period7/10/177/14/17

All Science Journal Classification (ASJC) codes

  • Software

Keywords

  • Cardinality constraint
  • Competitive analysis
  • Greedy algorithm
  • Incremental optimization
  • Maximization problems

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