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.
For a finite sequence of positive integers $\boldsymbol{a}=(a_1,\dots,a_n)$, the restricted partition function $q_{\boldsymbol{a}}(k)$ denote the number of nonnegative integer solutions to the equation $a_1x_1+a_2x_2+\cdots +a_nx_n=k$. It is proved to be a quasi-polynomial of degree $n-1$. Write $q_{\boldsymbol{a}}(k)=\sum_{j=0}^{n-1}c_j(k)k^j$ with periodic coefficient functions $c_j$, and set $b_m=\#\{i:m\mid a_i\}$. In 2008, Beck, Sam, and Woods conjectured that the minimum period of $c_j(k)$ is $\mathrm{lcm}\{m:b_m>j\}$. In this paper, we derive an exact root-of-unity formula for every coefficient function $c_j(k)$. The formula proves the conjectured divisibility upper bound, but it also reveals a lower bound for the period of $c_j(k)$. Both divisibility bounds are sharp. This leads us to construct a family of counterexamples to this conjecture.
Feihu Liu, Jinlong Tang, Guoce Xin et al.· 0 citations
