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

Abstract

Building on work of Cai, Fürer, and Immerman [CFI92], 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 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 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 [Fei02] (and also any of a class of hypotheses which generalize the R3XOR Hypothesis), the robust Graph Isomorphism problem is hard. I.e. for every > 0, there is no efficient algorithm which can distinguish graph pairs which are (1 − )-isomorphic from pairs which are not even (1 − 0 )-isomorphic for some universal constant 0 . Along the way we prove a robust asymmetry result for random graphs and hypergraphs which may be of independent interest. the heuristic is powerful enough to work correctly for all trees and for almost all n-vertex graphs in the Erdős-Rényi G(n, 1/2) model [BES80, BK79]. (We say the heuristic "works correctly" on a graph G if the stabilized coloring for G is distinct from the stabilized coloring of any graph not isomorphic to G.) To overcome WL's failure for regular graphs, several researchers independently introduced the "k-dimensional generalization" WL k (see [Wei76, CFI92] for discussion). Briefly, in the WL k heuristic, each k-tuple of vertices has a color, and color refinement involves looking at all "neighbors" of each k-tuple of vertices (v 1 , . . . , v k ) (where the neighbors are all tuples of the formThe WL k heuristic can be performed in time n k+O(1) and is thus a polynomial-time algorithm for any constant k.The WL k heuristic is very powerful. For example, it is known to work correctly in polynomial time for all graphs which exclude a fixed minor [Gro12], a class which includes all graphs of bounded treewidth or bounded genus. Spielman's 2 O(n 1/3 ) -time graph isomorphism algorithm [Spi96] for strongly regular graphs is achieved by WL k with k = O(n 1/3 ). The WL k algorithm with k = O( √ n) is also a key component in the 2 O( √ n log n) -time GISO algorithm [BL83]. Throughout the '80s there was some speculation that GISO might be solvable on all graphs by running the WL k algorithm with k = O(log n) or even k = O(1). However this was disproved in the notable work of Cai, Fürer, and Immerman [CFI92], which showed the existence of nonisomorphic n-vertex graphs G and H which are not distinguished by WL k unless k = Ω(n). 1 The GRAPHISOMORPHISM problem can be thought of as kind of constraint satisfaction problem (CSP), and readers familiar with LP/SDP hierarchies for CSPs might see an analogy between k-dimensional WL and level-k LP/SDP relaxations. A very interesting recent work of Atserias and Maneva [AM13] (see also [GO12]) shows tha...

Limitations of Algebraic Approaches to Graph Isomorphism Testing

“…As our main lower bounds, we prove that for every field F of characteristic = 2, there is a family of nonisomorphic graphs G k , H k of size O(k) that cannot be distinguished by the polynomial calculus in degree k. Furthermore, we prove that there is a family of nonisomorphic graphs G k , H k of size O(k) that cannot be distinguished by the Positivstellensatz calculus in degree k. The Positivstellensatz calculus [13] is an extension of the polynomial calculus over the reals and subsumes semi-definite programming hierarchies. Thus, our results slightly generalise the results of O'Donnell et al [20] on the Lasserre hierarchy (described above). Technically, our contribution is a low-degree reduction from systems of equations describing so-called Tseitin tautologies to the systems for graph isomorphism.…”

“…Our intuition is supported by Theorem 6.2, which implies that low-degree PC is not able to distinguish Cai-Fürer-Immerman graphs. Thus, polynomial calculus has similar limitations as the Weisfeiler-Lehman algorithm [8], Resolution [24], the Sherali-Adams hierarchy [1,15] and the Positivstellensatz [20]. Proof.…”

“…Complementing recent research on linear and semidefinite programming approaches to GI [1,10,15,19,20], we investigate the power of GI-algorithms based on algebraic reasoning techniques like Gröbner basis computation. The idea of all these approaches is to encode isomorphisms between two graphs as solutions to a system of equations and possibly inequalities and then try to solve this system or relaxations of it.…”

“…Already in 1992, Cai, Fürer, and Immerman [8] had proved that for every k there are non-isomorphic graphs G k , H k (called CFI-graphs in the following) of size O(k) that are not distinguished by k-WL, and combined with the results of Atserias-Maneva and Malkin, this implies that no sublinear level of the Sherali-Adams hierarchy suffices to decide isomorphism. O'Donnell, Wright, Wu, and Zhou [20] and Codenotti, Schoenbeck, and Snook [10] studied the Lasserre hierarchy [18] of semi-definite relaxations of the integer linear program for GI. They proved that the same CFI-graphs cannot even be distinguished by sublinear levels of the Lasserre hierarchy.…”

Linear Diophantine Equations, Group CSPs, and Graph Isomorphism

“…al. [17] and Codenotti et al [11] proved that that even the more powerful semi-definite Lasserre hierarchy fails to distinguish CFI-graphs.…”