On the space complexity of linear programming with preprocessing

Yael Tauman Kalai, Ran Raz, Oded Regev

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

It is well known that Linear Programming is P-complete, with a log-space reduction. In this work we ask whether Linear Programming remains P-complete, even if the polyhedron (i.e., the set of linear inequality constraints) is a fixed polyhedron, for each input size, and only the objective function is given as input. More formally, we consider the following problem: maximize c x, subject to Ax ≥b; x €2 Rd, where A; b are fixed in advance and only c is given as an input. We start by showing that the problem remains P-complete with a log-space reduction, thus showing that no(1)-space algorithms are unlikely. This result is proved by a direct classical reduction. We then turn to study approximation algorithms and ask what is the best approximation factor that could be obtained by a small space algorithm. Since approximation factors are mostly meaningful when the objective function is nonnegative, we restrict ourselves to the case where x ≥0 and c ≥0. We show that (even in this possibly easier case) approximating the value of max c x (within any polynomial factor) is P-complete with a polylog space reduction, thus showing that 2(log n)o(1)-space approximation algorithms are unlikely. The last result is proved using a recent work of Kalai, Raz, and Rothblum, showing that every language in P has a nosignaling multi-prover interactive proof with poly-logarithmic communication complexity. To the best of our knowledge, our result gives the first space hardness of approximation result proved by a PCP-based argument.

Original languageEnglish (US)
Title of host publicationITCS 2016 - Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science
PublisherAssociation for Computing Machinery, Inc
Pages293-300
Number of pages8
ISBN (Print)9781450340571
DOIs
StatePublished - Jan 14 2016
Event7th ACM Conference on Innovations in Theoretical Computer Science, ITCS 2016 - Cambridge, United States
Duration: Jan 14 2016Jan 16 2016

Other

Other7th ACM Conference on Innovations in Theoretical Computer Science, ITCS 2016
CountryUnited States
CityCambridge
Period1/14/161/16/16

Fingerprint

Space Complexity
Linear programming
Preprocessing
Approximation algorithms
Polyhedron
Approximation Algorithms
Objective function
Hardness of Approximation
Interactive Proofs
Hardness
Polynomials
Communication Complexity
Linear Constraints
Inequality Constraints
Best Approximation
Communication
Linear Inequalities
Logarithmic
Maximise
Non-negative

Keywords

  • Linear programming
  • P-completeness
  • Preprocessing
  • Space complexity

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

Cite this

Kalai, Y. T., Raz, R., & Regev, O. (2016). On the space complexity of linear programming with preprocessing. In ITCS 2016 - Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science (pp. 293-300). Association for Computing Machinery, Inc. https://doi.org/10.1145/2840728.2840750

On the space complexity of linear programming with preprocessing. / Kalai, Yael Tauman; Raz, Ran; Regev, Oded.

ITCS 2016 - Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science. Association for Computing Machinery, Inc, 2016. p. 293-300.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Kalai, YT, Raz, R & Regev, O 2016, On the space complexity of linear programming with preprocessing. in ITCS 2016 - Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science. Association for Computing Machinery, Inc, pp. 293-300, 7th ACM Conference on Innovations in Theoretical Computer Science, ITCS 2016, Cambridge, United States, 1/14/16. https://doi.org/10.1145/2840728.2840750
Kalai YT, Raz R, Regev O. On the space complexity of linear programming with preprocessing. In ITCS 2016 - Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science. Association for Computing Machinery, Inc. 2016. p. 293-300 https://doi.org/10.1145/2840728.2840750
Kalai, Yael Tauman ; Raz, Ran ; Regev, Oded. / On the space complexity of linear programming with preprocessing. ITCS 2016 - Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science. Association for Computing Machinery, Inc, 2016. pp. 293-300
@inproceedings{36307825380549ef96d8f30ee6371645,
title = "On the space complexity of linear programming with preprocessing",
abstract = "It is well known that Linear Programming is P-complete, with a log-space reduction. In this work we ask whether Linear Programming remains P-complete, even if the polyhedron (i.e., the set of linear inequality constraints) is a fixed polyhedron, for each input size, and only the objective function is given as input. More formally, we consider the following problem: maximize c x, subject to Ax ≥b; x €2 Rd, where A; b are fixed in advance and only c is given as an input. We start by showing that the problem remains P-complete with a log-space reduction, thus showing that no(1)-space algorithms are unlikely. This result is proved by a direct classical reduction. We then turn to study approximation algorithms and ask what is the best approximation factor that could be obtained by a small space algorithm. Since approximation factors are mostly meaningful when the objective function is nonnegative, we restrict ourselves to the case where x ≥0 and c ≥0. We show that (even in this possibly easier case) approximating the value of max c x (within any polynomial factor) is P-complete with a polylog space reduction, thus showing that 2(log n)o(1)-space approximation algorithms are unlikely. The last result is proved using a recent work of Kalai, Raz, and Rothblum, showing that every language in P has a nosignaling multi-prover interactive proof with poly-logarithmic communication complexity. To the best of our knowledge, our result gives the first space hardness of approximation result proved by a PCP-based argument.",
keywords = "Linear programming, P-completeness, Preprocessing, Space complexity",
author = "Kalai, {Yael Tauman} and Ran Raz and Oded Regev",
year = "2016",
month = "1",
day = "14",
doi = "10.1145/2840728.2840750",
language = "English (US)",
isbn = "9781450340571",
pages = "293--300",
booktitle = "ITCS 2016 - Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science",
publisher = "Association for Computing Machinery, Inc",

}

TY - GEN

T1 - On the space complexity of linear programming with preprocessing

AU - Kalai, Yael Tauman

AU - Raz, Ran

AU - Regev, Oded

PY - 2016/1/14

Y1 - 2016/1/14

N2 - It is well known that Linear Programming is P-complete, with a log-space reduction. In this work we ask whether Linear Programming remains P-complete, even if the polyhedron (i.e., the set of linear inequality constraints) is a fixed polyhedron, for each input size, and only the objective function is given as input. More formally, we consider the following problem: maximize c x, subject to Ax ≥b; x €2 Rd, where A; b are fixed in advance and only c is given as an input. We start by showing that the problem remains P-complete with a log-space reduction, thus showing that no(1)-space algorithms are unlikely. This result is proved by a direct classical reduction. We then turn to study approximation algorithms and ask what is the best approximation factor that could be obtained by a small space algorithm. Since approximation factors are mostly meaningful when the objective function is nonnegative, we restrict ourselves to the case where x ≥0 and c ≥0. We show that (even in this possibly easier case) approximating the value of max c x (within any polynomial factor) is P-complete with a polylog space reduction, thus showing that 2(log n)o(1)-space approximation algorithms are unlikely. The last result is proved using a recent work of Kalai, Raz, and Rothblum, showing that every language in P has a nosignaling multi-prover interactive proof with poly-logarithmic communication complexity. To the best of our knowledge, our result gives the first space hardness of approximation result proved by a PCP-based argument.

AB - It is well known that Linear Programming is P-complete, with a log-space reduction. In this work we ask whether Linear Programming remains P-complete, even if the polyhedron (i.e., the set of linear inequality constraints) is a fixed polyhedron, for each input size, and only the objective function is given as input. More formally, we consider the following problem: maximize c x, subject to Ax ≥b; x €2 Rd, where A; b are fixed in advance and only c is given as an input. We start by showing that the problem remains P-complete with a log-space reduction, thus showing that no(1)-space algorithms are unlikely. This result is proved by a direct classical reduction. We then turn to study approximation algorithms and ask what is the best approximation factor that could be obtained by a small space algorithm. Since approximation factors are mostly meaningful when the objective function is nonnegative, we restrict ourselves to the case where x ≥0 and c ≥0. We show that (even in this possibly easier case) approximating the value of max c x (within any polynomial factor) is P-complete with a polylog space reduction, thus showing that 2(log n)o(1)-space approximation algorithms are unlikely. The last result is proved using a recent work of Kalai, Raz, and Rothblum, showing that every language in P has a nosignaling multi-prover interactive proof with poly-logarithmic communication complexity. To the best of our knowledge, our result gives the first space hardness of approximation result proved by a PCP-based argument.

KW - Linear programming

KW - P-completeness

KW - Preprocessing

KW - Space complexity

UR - http://www.scopus.com/inward/record.url?scp=84966524887&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84966524887&partnerID=8YFLogxK

U2 - 10.1145/2840728.2840750

DO - 10.1145/2840728.2840750

M3 - Conference contribution

SN - 9781450340571

SP - 293

EP - 300

BT - ITCS 2016 - Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science

PB - Association for Computing Machinery, Inc

ER -