Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs

Given two graphs which are almost isomorphic, is it possible to find a bijection which preserves most of the edges between the two? This is the algorithmic task of Robust Graph Isomorphism, which is a natural approximation variation of the Graph Isomorphism problem. In this talk, we show that no polynomial-time algorithm solves this problem, conditioned on Feige’s Random 3XOR Hypothesis. In addition, we show that the Lasserre/SOS SDP hierarchy, the most powerful SDP hierarchy known, fails quite spectacularly on this problem: it needs a linear number of rounds to distinguish two isomorphic graphs from two far-from-isomorphic graphs. Along the way, we venture into the theory of random graphs by showing that a random graph is robustly asymmetric whp, meaning that any permutation which is close to an automorphism is itself close to the identity permutation.

Joint work with Ryan O’Donnell, John Wright, and Chenggang Wu.

Speaker Details

Yuan Zhou received his bachelor’s degree in Computer Science from Tsinghua University in 2009, where he attended the Special Pilot CS Class supervised by Turing Award laureate Professor Andrew Yao. Yuan is currently a graduate student at Carnegie Mellon University, working on theoretical computer science with Professors Venkatesan Guruswami and Ryan O’Donnell. His research interests focus on approximability of fundamental optimization problems. In his past research projects, he designed approximation algorithms and proved inapproximability results for solving sparse linear equations, graph cut and graph bisection problems, unique games, several constraint satisfaction and ordering problems, and the densest subgraph problem. He also publishes work on other topics of theoretical computer science, including analysis of Boolean functions, algebraic dichotomy theory, algorithmic game theory and quantum information theory.

Yuan Zhou
Carnegie Mellon University

Series: Microsoft Research Talks