Hardness of robust graph isomorphism, lasserre gaps, and asymmetry of random graphs
Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms(2014)
摘要
Building on work of Cai, Fürer, 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 nonisomorphism requires degree Ω(n). In other words, we show an O(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 α-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 hypergraphs which may be of independent interest.
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关键词
algorithms,design,complexity measures and classes,theory,graph theory
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