Polynomial-Time Evaluation of Aardal-Lenstra Denumerants via Constant Term Method

Aardal and Lenstra systematically studied hard knapsack problems of the form $a_1x_1+\cdots+a_nx_n=b$, where $a_i=p_iM+r_iN$, $(M,N)$ is a coprime pair of positive integers, and the integers $|p_i|, |r_i|$ are small relative to $M$ and $N$. We investigate the corresponding challenging denumerant problem (i.e., counting the number of nonnegative integer solutions) and present a polynomial-time algorithm. This eliminates the computational bottlenecks caused by large values of $M$, $N$ and $b$. The proposed algorithm achieves a time complexity of $O(n^4\Delta^2\log n\log\Delta)$, which depends solely on the parameters $n$ and $\Delta=\max_{i,j}|r_i p_j - r_j p_i|$. Moreover, we consider the problem of expressing a general vector $(a_1,\dots,a_n)$ in the above form using the LLL algorithm.