### Abstract

This work is concerned with the proof-complexity of certifying that optimization problems do not have good solutions. Specifically we consider bounded-degree "Sum of Squares" (SOS) proofs, a powerful algebraic proof system introduced in 1999 by Grigoriev and Vorobjov. Work of Shor, Lasserre, and Parrilo shows that this proof is automatizable using semidefinite programming (SDP), meaning that any n-variable degree-d proof can be found in time n^{O(d)}. Furthermore, the SDP is dual to the well-known Lasserre SDP hierarchy, meaning that the "d/2-round Lasserre value" of an optimization problem is equal to the best bound provable using a degree-d SOS proof. These ideas were exploited in a recent paper by Barak et al. (STOC 2012) which shows that the known "hard instances" for the Unique-Games problem are in fact optimally solved by a constant level of the Lasserre SDP hierarchy. We continue the study of the power of SOS proofs in the context of difficult optimization problems. In particular, we show that the Balanced-Separator integrality gap instances proposed by Devanur et al. can have their optimal value certified by a degree-4 SOS proof. The key ingredient is an SOS proof of the KKL Theorem. We also investigate the extent to which the Khot-Vishnoi Max-Cut integrality gap instances can have their optimum value certified by an SOS proof. We show they can be certified to within a factor .952 (> .878) using a constant-degree proof. These investigations also raise an interesting mathematical question: is there a constant-degree SOS proof of the Central Limit Theorem?

Original language | English (US) |
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Title of host publication | Proceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013 |

Pages | 1537-1556 |

Number of pages | 20 |

State | Published - Apr 16 2013 |

Externally published | Yes |

Event | 24th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013 - New Orleans, LA, United States Duration: Jan 6 2013 → Jan 8 2013 |

### Publication series

Name | Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms |
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### Other

Other | 24th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013 |
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Country | United States |

City | New Orleans, LA |

Period | 1/6/13 → 1/8/13 |

### Fingerprint

### ASJC Scopus subject areas

- Software
- Mathematics(all)

### Cite this

*Proceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013*(pp. 1537-1556). (Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms).

**Approximability and proof complexity.** / O'Donnell, Ryan; Zhou, Yuan.

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

*Proceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013.*Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1537-1556, 24th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013, New Orleans, LA, United States, 1/6/13.

}

TY - GEN

T1 - Approximability and proof complexity

AU - O'Donnell, Ryan

AU - Zhou, Yuan

PY - 2013/4/16

Y1 - 2013/4/16

N2 - This work is concerned with the proof-complexity of certifying that optimization problems do not have good solutions. Specifically we consider bounded-degree "Sum of Squares" (SOS) proofs, a powerful algebraic proof system introduced in 1999 by Grigoriev and Vorobjov. Work of Shor, Lasserre, and Parrilo shows that this proof is automatizable using semidefinite programming (SDP), meaning that any n-variable degree-d proof can be found in time nO(d). Furthermore, the SDP is dual to the well-known Lasserre SDP hierarchy, meaning that the "d/2-round Lasserre value" of an optimization problem is equal to the best bound provable using a degree-d SOS proof. These ideas were exploited in a recent paper by Barak et al. (STOC 2012) which shows that the known "hard instances" for the Unique-Games problem are in fact optimally solved by a constant level of the Lasserre SDP hierarchy. We continue the study of the power of SOS proofs in the context of difficult optimization problems. In particular, we show that the Balanced-Separator integrality gap instances proposed by Devanur et al. can have their optimal value certified by a degree-4 SOS proof. The key ingredient is an SOS proof of the KKL Theorem. We also investigate the extent to which the Khot-Vishnoi Max-Cut integrality gap instances can have their optimum value certified by an SOS proof. We show they can be certified to within a factor .952 (> .878) using a constant-degree proof. These investigations also raise an interesting mathematical question: is there a constant-degree SOS proof of the Central Limit Theorem?

AB - This work is concerned with the proof-complexity of certifying that optimization problems do not have good solutions. Specifically we consider bounded-degree "Sum of Squares" (SOS) proofs, a powerful algebraic proof system introduced in 1999 by Grigoriev and Vorobjov. Work of Shor, Lasserre, and Parrilo shows that this proof is automatizable using semidefinite programming (SDP), meaning that any n-variable degree-d proof can be found in time nO(d). Furthermore, the SDP is dual to the well-known Lasserre SDP hierarchy, meaning that the "d/2-round Lasserre value" of an optimization problem is equal to the best bound provable using a degree-d SOS proof. These ideas were exploited in a recent paper by Barak et al. (STOC 2012) which shows that the known "hard instances" for the Unique-Games problem are in fact optimally solved by a constant level of the Lasserre SDP hierarchy. We continue the study of the power of SOS proofs in the context of difficult optimization problems. In particular, we show that the Balanced-Separator integrality gap instances proposed by Devanur et al. can have their optimal value certified by a degree-4 SOS proof. The key ingredient is an SOS proof of the KKL Theorem. We also investigate the extent to which the Khot-Vishnoi Max-Cut integrality gap instances can have their optimum value certified by an SOS proof. We show they can be certified to within a factor .952 (> .878) using a constant-degree proof. These investigations also raise an interesting mathematical question: is there a constant-degree SOS proof of the Central Limit Theorem?

UR - http://www.scopus.com/inward/record.url?scp=84876025683&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84876025683&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:84876025683

SN - 9781611972511

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 1537

EP - 1556

BT - Proceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013

ER -