Algorithms for Subset Sum using linear sketching

Given n positive integers, the Modular Subset Sum problem asks if a subset adds up to a given target t modulo a given integer m. This is a natural generalization of the Subset Sum problem (where m = + [infinity symbol]) with ties to additive combinatorics and cryptography. The non-modular case was long known to be NP-complete but to admit pseudo-polynomial time algorithms and, recently, algorithms running in near-linear pseudo-polynomial time were developed [9, 211. For the modular case, however, the best known algorithm by Koiliaris and Xu [21] runs in time 0̃ (m⁵/⁴). In this thesis we tackle this problem by devising a faster algorithm for the Modular Subset Sum problem, running in 0̃(m) randomized time, which matches a recent conditional lower bound of [1] based on the Strong Exponential Time Hypothesis. Interestingly, in contrast to most previous results on Subset Sum, our algorithm does not use the Fast Fourier Transform. Instead, it is able to simulate the "textbook" Dynamic Programming algorithm much faster, using ideas from linear sketching. This is one of the first applications of sketching-based techniques to obtain fast algorithms for exact combinatorial problems in an offline setting.

Name thesis:degree_name

Master

Department dc:contributor.department

Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2019

Author dc:creator

• Axiotis, Kyriakos.

Advisor dc:contributor.advisor

• Aleksander Mądry.

• Electrical Engineering and Computer Science.

Statement dc:rights

• MIT theses are protected by copyright. They may be viewed, downloaded, or printed from this source but further reproduction or distribution in any format is prohibited without written permission.

Licence dc:rights.uri

• http://dspace.mit.edu/handle/1721.1/7582

Language dc:language.iso

eng

Handle dc:identifier.uri

https://hdl.handle.net/1721.1/122750

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/122750