### 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 language | English (US) |
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Title of host publication | ITCS 2016 - Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science |

Publisher | Association for Computing Machinery, Inc |

Pages | 293-300 |

Number of pages | 8 |

ISBN (Print) | 9781450340571 |

DOIs | |

State | Published - Jan 14 2016 |

Event | 7th ACM Conference on Innovations in Theoretical Computer Science, ITCS 2016 - Cambridge, United States Duration: Jan 14 2016 → Jan 16 2016 |

### Other

Other | 7th ACM Conference on Innovations in Theoretical Computer Science, ITCS 2016 |
---|---|

Country | United States |

City | Cambridge |

Period | 1/14/16 → 1/16/16 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*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.

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

*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

}

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 -