On Lagrangian relaxation for constrained maximization and reoptimization problems

Ariel Kulik, Hadas Shachnai, Gal Tamir

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

Abstract

We prove a general result demonstrating the power of Lagrangian relaxation in solving constrained maximization problems with arbitrary objective functions. This yields a unified approach for solving a wide class of subset selection problems with linear constraints. Given a problem in this class and some small ε∈(0,1), we show that if there exists an r-approximation algorithm for the Lagrangian relaxation of the problem, for some r∈(0,1), then our technique achieves a ratio of [Formula presented] to the optimal, and this ratio is tight. Using the technique we obtain (re)approximation algorithms for natural (reoptimization) variants of classic subset selection problems, including real-time scheduling, the maximum generalized assignment problem (GAP) and maximum weight independent set. For all of the problems studied in this paper, the number of calls to the r-approximation algorithm, used by our algorithms, is linear in the input size and in log(1∕ε) for inputs with a cardinality constraint, and polynomial in the input size and in log(1∕ε) for inputs with an arbitrary linear constraint.

Original languageEnglish
Pages (from-to)164-178
Number of pages15
JournalDiscrete Applied Mathematics
Volume296
DOIs
StatePublished - 15 Jun 2021
Externally publishedYes

Keywords

  • Approximation algorithms
  • Combinatorial reoptimization
  • Constrained maximization
  • Lagrangian relaxation
  • Set maximization problems
  • Subset selection

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

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