A Fine-Grained Perspective on Approximating Subset Sum and Partition
Karl Bringmann, Vasileios Nakos

TL;DR
This paper connects the complexity of approximating Subset Sum and Partition to Min-Plus-Convolution, establishing equivalences and deriving improved algorithms, including the first deterministic scheme for Partition breaking quadratic time barriers.
Contribution
It establishes a subquadratic equivalence between Subset Sum approximation and Min-Plus-Convolution, and improves approximation schemes for Partition, including the first deterministic subquadratic algorithm.
Findings
Subset Sum approximation is subquadratically equivalent to Min-Plus-Convolution.
Assuming Min-Plus-Convolution conjecture, no strongly subquadratic approximation scheme exists.
New deterministic approximation scheme for Partition breaks quadratic time barrier.
Abstract
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 -approximation in time [Gens, Levner'78, Kellerer et al.'97]. In particular, a -approximation can be computed in time . 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 . Our main result is that computing a -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 and . In…
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