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On Optimization and Counting of Non-Broken Bases of Matroids

Authors Dorna Abdolazimi, Kasper Lindberg, Shayan Oveis Gharan

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Author Details

Dorna Abdolazimi
  • University of Washington, Seattle, WA, USA
Kasper Lindberg
  • University of Washington, Seattle, WA, USA
Shayan Oveis Gharan
  • University of Washington, Seattle, WA, USA

Funding

  • Abdolazimi, Dorna: Research supported by NSF grant CCF-2203541, and Air Force Office of Scientific Research grant FA9550-20-1-0212.
  • Lindberg, Kasper: Research supported by NSF grant CCF-2203541.
  • Gharan, Shayan Oveis: Research supported by NSF grant CCF-2203541, Air Force Office of Scientific Research grant FA9550-20-1-0212, and Simons Investigator grant.

Cite AsGet BibTex

Dorna Abdolazimi, Kasper Lindberg, and Shayan Oveis Gharan. On Optimization and Counting of Non-Broken Bases of Matroids. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 40:1-40:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2023.40

Abstract

Given a matroid M = (E,I), and a total ordering over the elements E, a broken circuit is a circuit where the smallest element is removed and an NBC independent set is an independent set in I with no broken circuit. The set of NBC independent sets of any matroid M define a simplicial complex called the broken circuit complex which has been the subject of intense study in combinatorics. Recently, Adiprasito, Huh and Katz showed that the face of numbers of any broken circuit complex form a log-concave sequence, proving a long-standing conjecture of Rota. We study counting and optimization problems on NBC bases of a generic matroid. We find several fundamental differences with the independent set complex: for example, we show that it is NP-hard to find the max-weight NBC base of a matroid or that the convex hull of NBC bases of a matroid has edges of arbitrary large length. We also give evidence that the natural down-up walk on the space of NBC bases of a matroid may not mix rapidly by showing that for some family of matroids it is NP-hard to count the number of NBC bases after certain conditionings.

Subject Classification

ACM Subject Classification
  • Theory of computation → Random walks and Markov chains
  • Theory of computation → Computational complexity and cryptography
Keywords
  • Complexity
  • Hardness
  • Optimization
  • Counting
  • Random walk
  • Local to Global
  • Matroids

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