Consider a matroid where all elements are labeled with an element in $\mathbb{Z}$. We are interested in finding a base where the sum of the labels is congruent to $g \pmod m$. We show that this problem can be solved in $\tilde{O}(2^{4m} n r^{5/6})$ time for a matroid with $n$ elements and rank $r$, when $m$ is either the product of two primes or a prime power. The algorithm can be generalized to all moduli and, in fact, to all abelian groups if a classic additive combinatorics conjecture by Schrijver and Seymour holds true. We also discuss the optimization version of the problem.

翻译：考虑一个所有元素均被$\mathbb{Z}$中元素标记的拟阵。我们研究如何寻找一个基，使得其标签之和满足同余条件$g \pmod m$。我们证明，当$m$为两个素数的乘积或素数幂时，该问题可在$\tilde{O}(2^{4m} n r^{5/6})$时间内求解，其中拟阵包含$n$个元素且秩为$r$。该算法可推广至所有模数，事实上，若Schrijver与Seymour提出的经典加性组合学猜想成立，则该算法适用于所有阿贝尔群。此外，我们还讨论了该问题的优化版本。