Given a set S with n elements and a collection \(\mathcal {F}\) of multisets over S, the multiset multicover with multiplicity constraints problem (MSMC-MCP) entails finding the minimum number of multisets from \(\mathcal {F}\) required to cover each element \(s_i \in S\) at least specified \(b_i\) times, while ensuring that each multiset \(F_i \in \mathcal {F}\) can be selected no more than specified \(d_i\) times. Here, \(b = \max _{1\le i\le n} b_i\) . MSMC-MCP is a generalization of classic set cover problem. In this paper, we employ a novel algebraic approach that combines generating functions and the Discrete Fourier Transform over a ring. Through this method, we introduce an exact algorithm for MSMC-MCP, which exhibits \(O^*(b^4)\) space complexity and \(O^*(b^6(b+1)^n|\mathcal {F}|)\) time complexity (where the \(O^*\) notation suppresses polynomial factors involving n and \(\ln {b}\) ). Compared to the results presented in Hua et al.’s works (TCS, 2009) and (TCS, 2010), our approach significantly reduces the space complexity.