A Fine-Grained Perspective on Approximating Subset Sum and Partition

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 Min-Plus-Convolution, a problem that 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 is no approximation scheme 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 randomized approximation scheme. This adds the first approximation problem to the list of problems that are equivalent to Min-Plus-Convolution. For the related Partition problem, an important special case of Subset Sum, the state of the art is a randomized approximation scheme running in time $\tilde{O}(n+1/\varepsilon^{5/3})$ [Mucha~et~al.'19]. We adapt our reduction from Subset Sum to Min-Plus-Convolution to obtain a related reduction from Partition to Min-Plus-Convolution. This yields an improved approximation scheme for Partition running in time $\tilde{O}(n + 1/\varepsilon^{3/2})$. Our algorithm is the first deterministic approximation scheme for Partition that breaks the quadratic barrier.