### Abstract

Building on work of Cai, Fiirer, and Immerman [18], we show two hardness results for the Graph Isomorphism problem. First, we show that there are pairs of nonisomorphic n- vertex graphs G and H such that any sum-of-squares (SOS) proof of nonisomor- phism requires degree Ω(n). In other words, we show an Ω (n)- round integrality gap for the Lasserre SDP relaxation. In fact, we show this for pairs G and H which are not even (1 - 10^{-14})- isomorphic. (Here we say that two n-vertex, m-edge graphs G and H are a-isomorphic if there is a bijection between their vertices which preserves at least αm edges.) Our second result is that under the R3XOR Hypothesis [23] (and also any of a class of hypotheses which generalize the R3XOR Hypothesis), the robust Graph Isomorphism is hard. I.e. for every ε > 0, there is no efficient algorithm which can distinguish graph pairs which are (1 - ε) isomorphic from pairs which are not even (1 - ε_{0})-)- isomorphic for some universal constant ε_{0})- Along the way we prove a robust asymmetry result for random graphs and hyper- graphs which may be of independent interest.

Original language | English (US) |
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Title of host publication | Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014 |

Publisher | Association for Computing Machinery, |

Pages | 1659-1677 |

Number of pages | 19 |

ISBN (Print) | 9781611973389 |

State | Published - Jan 1 2014 |

Externally published | Yes |

Event | 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014 - Portland, OR, United States Duration: Jan 5 2014 → Jan 7 2014 |

### Publication series

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

Other | 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014 |
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Country | United States |

City | Portland, OR |

Period | 1/5/14 → 1/7/14 |

### Fingerprint

### ASJC Scopus subject areas

- Software
- Mathematics(all)

### Cite this

*Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014*(pp. 1659-1677). (Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms). Association for Computing Machinery,.

**Hardness of robust graph isomorphism, lasserre gaps, and asymmetry of random graphs.** / O'Donnell, Ryan; Wright, John; Wu, Chenggang; Zhou, Yuan.

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

*Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014.*Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms, Association for Computing Machinery, pp. 1659-1677, 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, OR, United States, 1/5/14.

}

TY - GEN

T1 - Hardness of robust graph isomorphism, lasserre gaps, and asymmetry of random graphs

AU - O'Donnell, Ryan

AU - Wright, John

AU - Wu, Chenggang

AU - Zhou, Yuan

PY - 2014/1/1

Y1 - 2014/1/1

N2 - Building on work of Cai, Fiirer, and Immerman [18], we show two hardness results for the Graph Isomorphism problem. First, we show that there are pairs of nonisomorphic n- vertex graphs G and H such that any sum-of-squares (SOS) proof of nonisomor- phism requires degree Ω(n). In other words, we show an Ω (n)- round integrality gap for the Lasserre SDP relaxation. In fact, we show this for pairs G and H which are not even (1 - 10-14)- isomorphic. (Here we say that two n-vertex, m-edge graphs G and H are a-isomorphic if there is a bijection between their vertices which preserves at least αm edges.) Our second result is that under the R3XOR Hypothesis [23] (and also any of a class of hypotheses which generalize the R3XOR Hypothesis), the robust Graph Isomorphism is hard. I.e. for every ε > 0, there is no efficient algorithm which can distinguish graph pairs which are (1 - ε) isomorphic from pairs which are not even (1 - ε0)-)- isomorphic for some universal constant ε0)- Along the way we prove a robust asymmetry result for random graphs and hyper- graphs which may be of independent interest.

AB - Building on work of Cai, Fiirer, and Immerman [18], we show two hardness results for the Graph Isomorphism problem. First, we show that there are pairs of nonisomorphic n- vertex graphs G and H such that any sum-of-squares (SOS) proof of nonisomor- phism requires degree Ω(n). In other words, we show an Ω (n)- round integrality gap for the Lasserre SDP relaxation. In fact, we show this for pairs G and H which are not even (1 - 10-14)- isomorphic. (Here we say that two n-vertex, m-edge graphs G and H are a-isomorphic if there is a bijection between their vertices which preserves at least αm edges.) Our second result is that under the R3XOR Hypothesis [23] (and also any of a class of hypotheses which generalize the R3XOR Hypothesis), the robust Graph Isomorphism is hard. I.e. for every ε > 0, there is no efficient algorithm which can distinguish graph pairs which are (1 - ε) isomorphic from pairs which are not even (1 - ε0)-)- isomorphic for some universal constant ε0)- Along the way we prove a robust asymmetry result for random graphs and hyper- graphs which may be of independent interest.

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UR - http://www.scopus.com/inward/citedby.url?scp=84902089891&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:84902089891

SN - 9781611973389

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

SP - 1659

EP - 1677

BT - Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014

PB - Association for Computing Machinery,

ER -