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Towards the Proximity Conjecture on Group-Labeled Matroids

Dániel Garamvölgyi, Ryuhei Mizutani, Taihei Oki, Tamás Schwarcz, Yutaro Yamaguchi

Abstract

Consider a matroid $M$ whose ground set is equipped with a labeling to an abelian group. A basis of $M$ is called $F$-avoiding if the sum of the labels of its elements is not in a forbidden label set $F$. Hörsch, Imolay, Mizutani, Oki, and Schwarcz (2024) conjectured that if an $F$-avoiding basis exists, then any basis can be transformed into an $F$-avoiding basis by exchanging at most $|F|$ elements. This proximity conjecture is known to hold for certain specific groups; in the case where $|F| \le 2$; or when the matroid is subsequence-interchangeably base orderable (SIBO), which is a weakening of the so-called strongly base orderable (SBO) property. In this paper, we settle the proximity conjecture for sparse paving matroids or in the case where $|F| \le 4$. Related to the latter result, we present the first known example of a non-SIBO matroid. We further address the setting of multiple group-label constraints, showing proximity results for the cases of two labelings, SIBO matroids, matroids representable over a fixed, finite field, and sparse paving matroids.

Towards the Proximity Conjecture on Group-Labeled Matroids

Abstract

Consider a matroid whose ground set is equipped with a labeling to an abelian group. A basis of is called -avoiding if the sum of the labels of its elements is not in a forbidden label set . Hörsch, Imolay, Mizutani, Oki, and Schwarcz (2024) conjectured that if an -avoiding basis exists, then any basis can be transformed into an -avoiding basis by exchanging at most elements. This proximity conjecture is known to hold for certain specific groups; in the case where ; or when the matroid is subsequence-interchangeably base orderable (SIBO), which is a weakening of the so-called strongly base orderable (SBO) property. In this paper, we settle the proximity conjecture for sparse paving matroids or in the case where . Related to the latter result, we present the first known example of a non-SIBO matroid. We further address the setting of multiple group-label constraints, showing proximity results for the cases of two labelings, SIBO matroids, matroids representable over a fixed, finite field, and sparse paving matroids.

Paper Structure

This paper contains 11 sections, 15 theorems, 16 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Our contributions.
  3. Related work.
  4. Organization.
  5. Preliminaries
  6. Proximity Theorem for Sparse Paving Matroids
  7. Proximity Theorem for at Most 4 Forbidden Labels
  8. Proximity for Multiple Labelings
  9. Conclusion
  10. Acknowledgments
  11. CNF formulation of finding a non-SIBO matroid

Key Result

Lemma 2.1

Let $M$ be a matroid, $\psi\colon E(M)\to \Gamma$ a group labeling, and $F\subseteq \Gamma$ a finite set of forbidden labels. Assume that $(M, \psi, F)$ is a counterexample to conj:proximity, i.e., $M$ has an $F$-avoiding basis and it has a basis $A$ with $|A\setminus B|\ge |F|+1$ for any $F$-avoidi

Figures (1)

  • Figure 1: Labeling the elements of the matroid $R_{10}$ such that for indices $0 \le i < j \le 5$, the set $\hat{B}_{i,j}$ is a basis if and only if $(i,j)\ne (2,3)$. Recall that bases are the sets of size 5 containing no $4$-cycle of $K_5$.

Theorems & Definitions (38)

  • Conjecture 1.1: Proximity Conjecture horsch2024problems
  • Conjecture 1.2: Schrijver and Seymour schrijver1990spanning; see also devos2009generalization
  • Conjecture 1.3: Baumgart baumgart2009ranking
  • Conjecture 1.4: Gabow gabow1976decomposing, see also wiedemann1984cycliccordovil1993bases
  • Conjecture 1.5: Multi-Labeled Proximity Conjecture
  • Lemma 2.1: see horsch2024problems_arxiv
  • Lemma 2.2: see horsch2024problems_arxiv
  • proof
  • Theorem 3.1
  • proof
  • ...and 28 more