Approximating Subset Sum is equivalent to Min-Plus-Convolution.
Karl Bringmann · arXiv (Cornell University) · 2019
Approximating Subset Sum is a classic and fundamental problem in computer science and mathematical optimization. The state-of-the-art approximation scheme for Subset Sum computes a $(1-\varepsilon)$-approximation in time $\tilde{O}(\min\{n/\varepsilon, n+1/\varepsilon^2\})$ [Gens, Levner'78, Kellerer et al.'97]. In particular, a $(1-1/n)$-approximation can be computed in time $O(n^2)$. We establish a connection to the Min-Plus-Convolution problem, which is of particular interest in fine-grained complexity theory and can be solved naively in time $O(n^2)$. Our main result is that computing a $(1-1/n)$-approximation for Subset Sum is subquadratically equivalent to Min-Plus-Convolution. Thus, assuming the Min-Plus-Convolution conjecture from fine-grained complexity theory, there are no approximation schemes for Subset Sum with strongly subquadratic dependence on $n$ and $1/\varepsilon$. In the other direction, our reduction allows us to transfer known lower order improvements from Min-Plus-Convolution to Subset Sum, which yields a mildly subquadratic approximation scheme. This adds the first approximation problem to the list of Min-Plus-Convolution-equivalent problems.