Problems on Group-labeled Matroid Bases
Florian Hörsch, András Imolay, Ryuhei Mizutani, Taihei Oki, Tamás Schwarcz
TL;DR
This work studies bases of group-labeled matroids, where a ground-set labeling $\\psi: E \to \\Gamma$ restricts feasible bases $B$ by requiring $\\psi(B) = \sum_{e\in B} \\psi(e) \notin F$. It establishes a dichotomy: Non-Zero Common Basis is polynomial-time solvable when the group $\\Gamma$ has no order-2 element (i.e., $\\Z_2 \not\le \\Gamma$), while hardness arises when $\\Z_2 \le \\Gamma$, using sparse paving matroids for information-theoretic arguments. The paper also develops randomized algebraic algorithms for $F$-avoiding bases in representable matroids (via group rings and Cauchy–Binet) and proves polynomial-time solvability in several structured cases (partition/graphic matroids) and for fixed $|F|$ under various representations. Additionally, it introduces the concept of $(\\alpha,k)$-weakly base-orderable matroids to connect exchange properties to $F$-avoiding basesp and explores conjectures about exchange-distance bounds, providing partial derandomization results and a counterexample to a conjecture of Liu and Xu. Overall, the results advance algorithmic understanding of label-constrained matroid optimization, offering both tractable cases and clear hardness boundaries, with implications for reconfiguration problems and lattice-theoretic perspectives on non-zero constraints.
Abstract
Consider a matroid equipped with a labeling of its ground set to an abelian group. We define the label of a subset of the ground set as the sum of the labels of its elements. We study a collection of problems on finding bases and common bases of matroids with restrictions on their labels. For zero bases and zero common bases, the results are mostly negative. While finding a non-zero basis of a matroid is not difficult, it turns out that the complexity of finding a non-zero common basis depends on the group. Namely, we show that the problem is hard for a fixed group if it contains an element of order two, otherwise it is polynomially solvable. As a generalization of both zero and non-zero constraints, we further study $F$-avoiding constraints where we seek a basis or common basis whose label is not in a given set $F$ of forbidden labels. Using algebraic techniques, we give a randomized algorithm for finding an $F$-avoiding common basis of two matroids represented over the same field for finite groups given as operation tables. The study of $F$-avoiding bases with groups given as oracles leads to a conjecture stating that whenever an $F$-avoiding basis exists, an $F$-avoiding basis can be obtained from an arbitrary basis by exchanging at most $|F|$ elements. We prove the conjecture for the special cases when $|F|\le 2$ or the group is ordered. By relying on structural observations on matroids representable over fixed, finite fields, we verify a relaxed version of the conjecture for these matroids. As a consequence, we obtain a polynomial-time algorithm in these special cases for finding an $F$-avoiding basis when $|F|$ is fixed.