Approximating Subset Sum is equivalent to Min-Plus-Convolution
share
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-ε)-approximation in time Õ(min{n/ε, n+1/ε^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/ε. 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.
READ FULL TEXT
