2026-09-20T09:09:22Zhttps://dspace.mit.edu/server/oai/requestoai:dspace.mit.edu:1721.1/1227502021-07-05T14:03:20Zcom_1721.1_7582com_1721.1_7581col_1721.1_131023
Aleksander Mądry.
Axiotis, Kyriakos.
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science.
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
2019-11-04T20:22:26Z
2019-11-04T20:22:26Z
2019
2019
https://hdl.handle.net/1721.1/122750
1124855087
Thesis: S.M., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2019
Cataloged from PDF version of thesis.
Includes bibliographical references (pages 41-43).
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.
by Kyriakos Axiotis.
S.M.
S.M. Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science
43 pages ;
eng
Massachusetts Institute of Technology
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.
http://dspace.mit.edu/handle/1721.1/7582
Electrical Engineering and Computer Science.
Algorithms for Subset Sum using linear sketching
Thesis
application/pdf
2019-11-04T20:22:25Z
Publication
Master
EECS
unknown
unknown
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<Abstract>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.</Abstract>
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