A Subquadratic Approximation Scheme for Partition

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 \(\widetilde{O}(n+1/\varepsilon^{5/3})\) time randomized FPTAS for Partition, which is derived from a certain relaxed form of a randomized FPTAS f...

Full description

Saved in:
Bibliographic Details
Published in:arXiv.org
Main Authors: Mucha, Marcin, Węgrzycki, Karol, Włodarczyk, Michał
Format: Paper
Language:English
Published: Ithaca Cornell University Library, arXiv.org 06.05.2019
Subjects:
Algorithms
Approximation
Combinatorial analysis
Complexity
Convolution
Lower bounds
Mathematical analysis
Partitions
Polynomials
Randomization
ISSN:2331-8422
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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 \(\widetilde{O}(n+1/\varepsilon^{5/3})\) 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 \(O((n+1/\varepsilon)^{2-\delta})\) for some \(\delta>0\). 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 Tree Sparsity. Finally, we argue why breaking the quadratic barrier for approximate Knapsack is unlikely by giving an \(\Omega((n+1/\varepsilon)^{2-o(1)})\) conditional lower bound.
Bibliography:SourceType-Working Papers-1
ObjectType-Working Paper/Pre-Print-1
content type line 50
ISSN:2331-8422
DOI:10.48550/arxiv.1804.02269