On the Congruency-Constrained Matroid Base

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 $g(\mathrm{m}\mathrm{o}\mathrm{d}m)$. We show that this problem can be solved in $\tilde{O}(2^{4m}n{r}^{5\mathrm{/}6})$ 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.