群标记拟阵基上的问题
Problems on Group-labeled Matroid Bases
AI总结：
本文研究群标记拟阵中带标签约束的基与公共基问题，刻画了非零公共基的复杂性条件，并提出F-回避基的随机算法及交换猜想，在特殊情形下给出多项式时间算法。
摘要：
考虑一个拟阵，其基础集被标记到一个阿贝尔群。我们将基础集的一个子集的标签定义为其元素的标签之和。我们研究了一类在标签受限条件下寻找拟阵的基和公共基的问题。对于零基和零公共基，结果大多是负面的。虽然寻找拟阵的非零基并不困难，但结果表明，寻找非零公共基的复杂性取决于群。具体来说，我们证明，对于一个固定群，如果它包含一个二阶元素，则该问题是困难的，否则它是多项式可解的。作为零约束和非零约束的推广，我们进一步研究了$F$-回避约束，即寻找一个基或公共基，使其标签不在给定的禁止标签集合$F$中。利用代数技术，对于以运算表形式给出的有限群，我们给出了一个随机算法，用于寻找在同一域上表示的两个拟阵的$F$-回避公共基。对以预言机形式给出的群的$F$-回避基的研究引出一个猜想：只要存在一个$F$-回避基，就可以通过交换至多$|F|$个元素从任意基得到一个$F$-回避基。我们证明了当$|F|\le 2$或群是有序群时的特殊情形的猜想。基于对可在固定有限域上表示的拟阵的结构性观察，我们验证了这些拟阵的该猜想的放宽版本。因此，当$|F|$固定时，我们在这些特殊情形下得到了一个寻找$F$-回避基的多项式时间算法。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
