Centrala begrepp
The authors propose an FPTAS (Fully Polynomial-Time Approximation Scheme) that can approximate the Partition problem in near-linear time, matching the best possible running time up to a polylogarithmic factor.
Sammanfattning
The paper presents an efficient algorithm for approximating the Partition problem, which is a special case of the Subset Sum problem. The key highlights and insights are:
The authors develop a randomized weak approximation scheme for the Subset Sum problem that runs in near-linear time (Theorem 1). This immediately implies an FPTAS for the Partition problem (Theorem 2).
The algorithm utilizes a combination of sparse convolution and an additive combinatorics result to efficiently approximate the set of subset sums. It reduces the number of tree nodes via two-layer color coding and estimates the density of each level to decide whether to compute it or use the additive combinatorics result.
The authors show that their algorithm matches the conditional lower bound of (n + 1/ε)^(1-o(1)) assuming the Strong Exponential Time Hypothesis, making Partition the first NP-hard problem that admits an FPTAS that is near-linear in both n and 1/ε.
The technical approach, especially the usage of arithmetic progressions from additive combinatorics, is inspired by prior work on dense Subset Sum, but the authors obtain the arithmetic progression in a different way.