Subset Sum and Integer Programming Feasibility Test in Near-Optimal Time

Abstract: Given an integer $t$ and a set of integers in $[N]$, Subset Sum asks whether there exists a subset that sums to $t$. A long open problem is whether Subset Sum can be solved in $O(N)$ time. We show that Subset Sum can be solved in $O(N^{1 + \varepsilon})$ time where $\varepsilon > 0$ is an arbitrarily small constant. Our result also generalizes to multi-sets in multi-dimension. That is, for any fixed $d$ and $\varepsilon >0$, the $d$-dimensional subset sum can be solved in $O(n + N^{d + \varepsilon})$ time. As an application, for any fixed $d$ and $\varepsilon$, feasibility test of integer programming with $d$ constraints can be done in $$O(N^{{d+\varepsilon}} \mathrm{poly}(\ell_{\mathrm{in}}))$$ time, where $N$ is the maximum absolute value in the constraint matrix and $\ell_{\mathrm{in}}$ is the input size. Although the algorithm is quite involved, its high-level idea is clear. It is already known that when the input set is dense and $t$ lies in the interior of the zonotope, then Subset Sum is easy to solve [Galil-Margalit '91, Bringmann-Wellnitz '21, Chen-Hu-Mao-Zhang '26]. Our key observation is that a sparse input can exhibit a dense structure when lifted to higher dimension. We show that such a lifting can be done efficiently. After lifting, the interior of the zonotope is easy to tackle, and one only needs to deal with the thin area near the boundary of the zonotope. Then we show that the boundary area can be decomposed into subproblems and solved recursively, and that there cannot be too many recursion levels since each level of recursive reduces the volume of the area significantly.