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.

翻译：考虑一个拟阵，其基集被标记为阿贝尔群中的元素。我们将子集的标签定义为其元素标签之和。我们研究一类寻找具有标签限制的拟阵基和公共基的问题。对于零基和零公共基，结果大多是否定的。虽然寻找拟阵的非零基并不困难，但寻找非零公共基的复杂度取决于群的结构。具体而言，我们证明：若固定群包含二阶元素，则该问题是困难的；否则可在多项式时间内求解。作为零约束和非零约束的推广，我们进一步研究了$F$-避免约束，即寻找标签不在给定禁止标签集合$F$中的基或公共基。利用代数方法，我们给出随机化算法，用于在有限群（以运算表形式给出）且两个拟阵在相同域上表示时，寻找$F$-避免公共基。对于以预言机形式给出群的$F$-避免基问题，我们提出猜想：只要存在$F$-避免基，则可通过至多交换$|F|$个元素从任意基得到该基。当$|F|\le 2$或群为有序群时，我们证明了该猜想。基于固定有限域上可表示拟阵的结构观察，我们验证了这些拟阵的猜想松弛版本。作为推论，当$|F|$固定时，我们获得这些特殊情况下寻找$F$-避免基的多项式时间算法。