research-article

Faster space-efficient algorithms for subset sum and k-sum

Authors:

Jesper Nederlof,

Nikhil VyasAuthors Info & Claims

STOC 2017: Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing

Pages 198 - 209

https://doi.org/10.1145/3055399.3055467

Published: 19 June 2017 Publication History

Abstract

We present randomized algorithms that solve Subset Sum and Knapsack instances with n items in O^*(2^0.86n) time, where the O^*(·) notation suppresses factors polynomial in the input size, and polynomial space, assuming random read-only access to exponentially many random bits. These results can be extended to solve Binary Linear Programming on n variables with few constraints in a similar running time. We also show that for any constant k≥ 2, random instances of k-Sum can be solved using O(n^k-0.5(n)) time and O(logn) space, without the assumption of random access to random bits.

Underlying these results is an algorithm that determines whether two given lists of length n with integers bounded by a polynomial in n share a common value. Assuming random read-only access to random bits, we show that this problem can be solved using O(logn) space significantly faster than the trivial O(n²) time algorithm if no value occurs too often in the same list.

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Cited By

Abboud ABringmann KHermelin DShabtay D(2022)SETH-based Lower Bounds for Subset Sum and Bicriteria PathACM Transactions on Algorithms10.1145/345052418:1(1-22)Online publication date: 23-Jan-2022
https://dl.acm.org/doi/10.1145/3450524
Bringmann KWellnitz PMarx D(2021)On near-linear-time algorithms for dense subset sumProceedings of the Thirty-Second Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3458064.3458171(1777-1796)Online publication date: 10-Jan-2021
https://dl.acm.org/doi/10.5555/3458064.3458171
Lagerkvist VWahlström M(2021)The (Coarse) Fine-Grained Structure of NP-Hard SAT and CSP ProblemsACM Transactions on Computation Theory10.1145/349233614:1(1-54)Online publication date: 15-Dec-2021
https://dl.acm.org/doi/10.1145/3492336
Show More Cited By

Index Terms

Faster space-efficient algorithms for subset sum and k-sum
1. Theory of computation
  1. Design and analysis of algorithms

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cover image ACM Conferences

STOC 2017: Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing

June 2017

1268 pages

ISBN:9781450345286

DOI:10.1145/3055399

General Chairs:
Hamed Hatami
McGill University
,
Pierre McKenzie
Université de Montréal
,
Program Chair:
Valerie King
University of Victoria

Copyright © 2017 ACM.

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Published: 19 June 2017

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Cited By

Abboud ABringmann KHermelin DShabtay D(2022)SETH-based Lower Bounds for Subset Sum and Bicriteria PathACM Transactions on Algorithms10.1145/345052418:1(1-22)Online publication date: 23-Jan-2022
https://dl.acm.org/doi/10.1145/3450524
Bringmann KWellnitz PMarx D(2021)On near-linear-time algorithms for dense subset sumProceedings of the Thirty-Second Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3458064.3458171(1777-1796)Online publication date: 10-Jan-2021
https://dl.acm.org/doi/10.5555/3458064.3458171
Lagerkvist VWahlström M(2021)The (Coarse) Fine-Grained Structure of NP-Hard SAT and CSP ProblemsACM Transactions on Computation Theory10.1145/349233614:1(1-54)Online publication date: 15-Dec-2021
https://dl.acm.org/doi/10.1145/3492336
Bringmann KNakos VMakarychev KMakarychev YTulsiani MKamath GChuzhoy J(2020)Top-𝑘-convolution and the quest for near-linear output-sensitive subset sumProceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing10.1145/3357713.3384308(982-995)Online publication date: 22-Jun-2020
https://dl.acm.org/doi/10.1145/3357713.3384308
Esser AMay A(2020)Low Weight Discrete Logarithm and Subset Sum in with Polynomial MemoryAdvances in Cryptology – EUROCRYPT 202010.1007/978-3-030-45727-3_4(94-122)Online publication date: 10-May-2020
https://dl.acm.org/doi/10.1007/978-3-030-45727-3_4
Mucha MWęgrzycki KWłodarczyk MChan T(2019)A subquadratic approximation scheme for partitionProceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3310435.3310440(70-88)Online publication date: 6-Jan-2019
https://dl.acm.org/doi/10.5555/3310435.3310440
Abboud ABringmann KHermelin DShabtay DChan T(2019)SETH-based lower bounds for subset sum and bicriteria pathProceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3310435.3310438(41-57)Online publication date: 6-Jan-2019
https://dl.acm.org/doi/10.5555/3310435.3310438
Bai SGalbraith SLi LSheffield D(2019)Improved Combinatorial Algorithms for the Inhomogeneous Short Integer Solution ProblemJournal of Cryptology10.1007/s00145-018-9304-132:1(35-83)Online publication date: 1-Jan-2019
https://dl.acm.org/doi/10.1007/s00145-018-9304-1
Delaplace CEsser AMay A(2019)Improved Low-Memory Subset Sum and LPN Algorithms via Multiple CollisionsCryptography and Coding10.1007/978-3-030-35199-1_9(178-199)Online publication date: 18-Nov-2019
https://doi.org/10.1007/978-3-030-35199-1_9
Seliverstov A(2018)Binary Solutions to Some Systems of Linear EquationsOptimization Problems and Their Applications10.1007/978-3-319-93800-4_15(183-192)Online publication date: 17-Jun-2018
https://doi.org/10.1007/978-3-319-93800-4_15

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