2018/04/30 by Marcin Mucha, Karol Węgrzycki, Michał Włodarczyk · 16 citations
Computer Science · Mathematics · #Algorithm #Approximation algorithm #Combinatorics #Complexity and Algorithms in Graphs #Computational Geometry and Mesh Generation #Context (archaeology) #Discrete mathematics #Knapsack problem #Mathematics #Omega #Optimization and Search Problems #Partition (number theory) #Partition problem #Polynomial-time approximation scheme #Quadratic equation #Quadratic growth #Scheme (mathematics) #Time complexity #cs.DS
paper · pdf · doi:10.1137/1.9781611975482.5
published in Society for Industrial and Applied Mathematics eBooks, 70-88 (Society for Industrial and Applied Mathematics) · Extended abstract published in the proceedings of SODA 2019
openalex publication_date 2019/01/01 · arxiv created 2019/05/06 · arxiv updated 2019/05/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
The subject of this paper is the time complexity of approximating Knapsack, Subset Sum, Partition, and some other related problems. The main result is an Õ(n + 1/ε5/3) time randomized FPTAS for Partition, which is derived from a certain relaxed form of a randomized FPTAS for Subset Sum. To the best of our knowledge, this is the first NP-hard problem that has been shown to admit a subquadratic time approximation scheme, i.e., one with time complexity of O((n + 1/ε2–δ) for some δ > 0. To put these developments in context, note that a quadratic FPTAS for Partition has been known for 40 years. Our main contribution lies in designing a mechanism that reduces an instance of Subset Sum to several simpler instances, each with some special structure, and keeps track of interactions between them. This allows us to combine techniques from approximation algorithms, pseudo-polynomial algorithms, and additive combinatorics. We also prove several related results. Notably, we improve approximation schemes for 3SUM, (min, +)-convolution, and TreeSparsity. Finally, we argue why breaking the quadratic barrier for approximate Knapsack is unlikely by giving an Ω((n + 1/ε)2–o(1)) conditional lower bound.