Fast Modular Subset Sum using Linear Sketching

TL;DR

This paper introduces a linear sketching-based algorithm for the Modular Subset Sum problem, achieving near-linear time complexity and matching theoretical lower bounds without relying on FFT, thus advancing combinatorial algorithm techniques.

Contribution

The paper presents the first linear sketching approach to solve Modular Subset Sum efficiently, improving runtime from previous algorithms and avoiding FFT usage.

Findings

Algorithm runs in near-linear time O~(m).

Matches the conditional lower bound based on ETH.

Avoids using Fast Fourier Transform, unlike previous methods.

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=+\infty) with ties to additive combinatorics and cryptography. Recently, in [Bringmann, SODA'17] and [Koiliaris and Xu, SODA'17], efficient algorithms have been developed for the non-modular case, running in near-linear pseudo-polynomial time. For the modular case, however, the best known algorithm by Koiliaris and Xu [Koiliaris and Xu, SODA'17] runs in time O~(m^{5/4}). In this paper, we present an algorithm running in time O~(m), which matches a recent conditional lower bound of [Abboud et al.'17] 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 combinatorial problems in an offline setting.