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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U2 - 10.1137/1.9781611973402.120

DO - 10.1137/1.9781611973402.120

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,

T2 - 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014

Y2 - 5 January 2014 through 7 January 2014

ER -