Consider a matroid where all elements are labeled with an element from $\mathbb{Z}_m$. 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 $O(m^{2m} n r \log r)$ 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 generalizes to all moduli, and in fact, to all abelian groups, if a classic additive combinatorics conjecture of Schrijver and Seymour is true. We also discuss the optimization version of the problem.

翻译：考虑一个所有元素均带有 $\mathbb{Z}_m$ 中标签的拟阵。我们感兴趣的是找到一个基，使得其标签之和与 $g \pmod m$ 同余。我们证明，当 $m$ 为两个素数的乘积或素数幂时，对于具有 $n$ 个元素和秩 $r$ 的拟阵，该问题可在 $O(m^{2m} n r \log r)$ 时间内解决。该算法可推广至所有模数，实际上，如果 Schrijver 和 Seymour 的一个经典加性组合学猜想成立，则可推广至所有阿贝尔群。我们还讨论了该问题的优化版本。