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Matroid bases with cardinality constraints on the intersection.

Stefan Lendl1,2, Britta Peis3, Veerle Timmermans3

  • 1Institute of Operations and Information Systems, University of Graz, Graz, Austria.

Mathematical Programming
|July 5, 2022
PubMed
Summary
This summary is machine-generated.

This study introduces efficient algorithms for matroid intersection problems with constraints on base set intersections. These methods generalize existing problems and offer strongly polynomial-time solutions for robust matroid and polymatroid base problems.

Keywords:
Intersection constraintsMatroidsPolymatroidsRobust optimization

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Area of Science:

  • Combinatorial Optimization
  • Discrete Mathematics
  • Operations Research

Background:

  • Matroid theory provides a framework for studying independence properties in various mathematical structures.
  • Optimization problems involving matroid bases with intersection constraints are computationally challenging.
  • The Recoverable Robust Matroid problem with interval uncertainty remained an open question regarding efficient algorithms.

Purpose of the Study:

  • To develop strongly polynomial-time algorithms for finding pairs of bases in two matroids that minimize total cost under intersection size constraints.
  • To generalize existing matroid optimization problems to polymatroids and more than two matroids.
  • To address the open question of efficiently solving the Recoverable Robust Matroid problem.

Main Methods:

  • Reduction of constrained matroid intersection problems to weighted matroid intersection.
  • Application of primal-dual algorithms for solving the weighted matroid intersection problems.
  • Analysis of asymptotic running times compared to established algorithms like Frank's matroid intersection algorithm.

Main Results:

  • The problems with lower and upper bound constraints on the intersection size are reducible to weighted matroid intersection.
  • A strongly polynomial-time primal-dual algorithm is presented for solving these constrained problems.
  • The algorithm computes minimum cost solutions for all feasible intersection sizes efficiently.
  • Generalizations to polymatroids and multiple matroids are achieved, including a strongly polynomial algorithm for the recoverable robust polymatroid base problem.

Conclusions:

  • Constrained matroid intersection problems can be efficiently solved using weighted matroid intersection techniques.
  • The developed primal-dual algorithms offer significant improvements in solving robust matroid and polymatroid optimization problems.
  • This work provides a comprehensive framework for a class of combinatorial optimization problems with broad applicability.