A faster algorithm for solving general LPs

Shunhua Jiang; Zhao Song; Omri Weinstein; Hengjie Zhang

doi:10.1145/3406325.3451058

A faster algorithm for solving general LPs

Shunhua Jiang
, Zhao Song
, Omri Weinstein
, Hengjie Zhang

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

62 Scopus citations

Abstract

The fastest known LP solver for general (dense) linear programs is due to [Cohen, Lee and Song'19] and runs in O?(n? +n2.5-?/2 + n2+1/6) time. A number of follow-up works [Lee, Song and Zhang'19, Brand'20, Song and Yu'20] obtain the same complexity through different techniques, but none of them can go below n2+1/6, even if ?=2. This leaves a polynomial gap between the cost of solving linear systems (n?) and the cost of solving linear programs, and as such, improving the n2+1/6 term is crucial toward establishing an equivalence between these two fundamental problems. In this paper, we reduce the running time to O?(n? +n2.5-?/2 + n2+1/18) where ? and ? are the fast matrix multiplication exponent and its dual. Hence, under the common belief that ? ? 2 and ? ? 1, our LP solver runs in O?(n2.055) time instead of O?(n2.16).

Original language	English
Title of host publication	STOC 2021 - Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing
Editors	Samir Khuller, Virginia Vassilevska Williams
Publisher	Association for Computing Machinery
Pages	823-832
Number of pages	10
ISBN (Electronic)	9781450380539
DOIs	https://doi.org/10.1145/3406325.3451058
State	Published - 15 Jun 2021
Externally published	Yes
Event	53rd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2021 - Virtual, Online, Italy Duration: 21 Jun 2021 → 25 Jun 2021

Publication series

Name	Proceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)	0737-8017

Conference

Conference	53rd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2021
Country/Territory	Italy
City	Virtual, Online
Period	21/06/21 → 25/06/21

Bibliographical note

Publisher Copyright:
© 2021 ACM.

Keywords

Convex optimization
Dynamic data-structure
Linear programming

Access to Document

10.1145/3406325.3451058

Cite this

Jiang, S., Song, Z., Weinstein, O., & Zhang, H. (2021). A faster algorithm for solving general LPs. In S. Khuller, & V. V. Williams (Eds.), STOC 2021 - Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing (pp. 823-832). (Proceedings of the Annual ACM Symposium on Theory of Computing). Association for Computing Machinery. https://doi.org/10.1145/3406325.3451058

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title = "A faster algorithm for solving general LPs",

abstract = "The fastest known LP solver for general (dense) linear programs is due to [Cohen, Lee and Song'19] and runs in O?(n? +n2.5-?/2 + n2+1/6) time. A number of follow-up works [Lee, Song and Zhang'19, Brand'20, Song and Yu'20] obtain the same complexity through different techniques, but none of them can go below n2+1/6, even if ?=2. This leaves a polynomial gap between the cost of solving linear systems (n?) and the cost of solving linear programs, and as such, improving the n2+1/6 term is crucial toward establishing an equivalence between these two fundamental problems. In this paper, we reduce the running time to O?(n? +n2.5-?/2 + n2+1/18) where ? and ? are the fast matrix multiplication exponent and its dual. Hence, under the common belief that ? ? 2 and ? ? 1, our LP solver runs in O?(n2.055) time instead of O?(n2.16).",

keywords = "Convex optimization, Dynamic data-structure, Linear programming",

author = "Shunhua Jiang and Zhao Song and Omri Weinstein and Hengjie Zhang",

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Jiang, S, Song, Z, Weinstein, O & Zhang, H 2021, A faster algorithm for solving general LPs. in S Khuller & VV Williams (eds), STOC 2021 - Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing. Proceedings of the Annual ACM Symposium on Theory of Computing, Association for Computing Machinery, pp. 823-832, 53rd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2021, Virtual, Online, Italy, 21/06/21. https://doi.org/10.1145/3406325.3451058

A faster algorithm for solving general LPs. / Jiang, Shunhua; Song, Zhao; Weinstein, Omri et al.
STOC 2021 - Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing. ed. / Samir Khuller; Virginia Vassilevska Williams. Association for Computing Machinery, 2021. p. 823-832 (Proceedings of the Annual ACM Symposium on Theory of Computing).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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N2 - The fastest known LP solver for general (dense) linear programs is due to [Cohen, Lee and Song'19] and runs in O?(n? +n2.5-?/2 + n2+1/6) time. A number of follow-up works [Lee, Song and Zhang'19, Brand'20, Song and Yu'20] obtain the same complexity through different techniques, but none of them can go below n2+1/6, even if ?=2. This leaves a polynomial gap between the cost of solving linear systems (n?) and the cost of solving linear programs, and as such, improving the n2+1/6 term is crucial toward establishing an equivalence between these two fundamental problems. In this paper, we reduce the running time to O?(n? +n2.5-?/2 + n2+1/18) where ? and ? are the fast matrix multiplication exponent and its dual. Hence, under the common belief that ? ? 2 and ? ? 1, our LP solver runs in O?(n2.055) time instead of O?(n2.16).

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A faster algorithm for solving general LPs

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