We give an $O_{d,\varepsilon}(n+N^{d+\varepsilon})$-time algorithm for multidimensional subset sum on an explicitly listed multiset of $n$ vectors in $[-N,N]^d$, for every fixed $d$ and $\varepsilon>0$. This directly implies an $O_{d,\varepsilon}(n+N^{d+\varepsilon})$-time algorithm for feasibility test of integer programming. Prior to our work, the best-known algorithm has a running time of $O_{d}(n+N^{O(d^2)})$ [Eisenbrand, Weismantel'19]. Specifically, if $d=1$, our algorithm runs in $O_{\varepsilon}(n+N^{1+\varepsilon})$-time for the standard subset sum problem, which is near-linear in the largest input integer. Prior to our work, subset sum can be solved in time near-linear in the target $t$ [Bringmann'17], or near-linear in the mixed parameter $\sqrt{Nt}$ [Chen, Lian, Mao, Zhang'24]. It is a fundamental open problem whether the algorithm can be made near-linear in $N$, which is settled affirmatively in this paper up to an arbitrary small $N^{\varepsilon}$ term.

翻译：暂无翻译