Algorithms for Subset Sum using linear sketching
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science.
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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  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  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.
Thesis: S.M., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2019Cataloged from PDF version of thesis.Includes bibliographical references (pages 41-43).
DepartmentMassachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology
Electrical Engineering and Computer Science.