Towards the Proximity Conjecture on Group-Labeled Matroids

TL;DR

This paper proves the proximity conjecture for group-labeled matroids in sparse paving cases and when the forbidden label set has size up to four, advancing understanding of basis transformations under group constraints.

Contribution

It settles the proximity conjecture for sparse paving matroids and small forbidden label sets, and introduces the first non-SIBO matroid example.

Findings

Proved the conjecture for sparse paving matroids.

Confirmed the conjecture for |F| ≤ 4.

Presented the first non-SIBO matroid example.

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 $F$. H\"orsch, 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.