  In this paper we show lower bounds for a certain large class of algorithms
solving the Graph Isomorphism problem, even on expander graph instances.
Spielman [25] shows an algorithm for isomorphism of strongly regular expander
graphs that runs in time exp(O(n^(1/3)) (this bound was recently improved to
expf O(n^(1/5) [5]). It has since been an open question to remove the
requirement that the graph be strongly regular. Recent algorithmic results show
that for many problems the Lasserre hierarchy works surprisingly well when the
underlying graph has expansion properties. Moreover, recent work of Atserias
and Maneva [3] shows that k rounds of the Lasserre hierarchy is a
generalization of the k-dimensional Weisfeiler-Lehman algorithm for Graph
Isomorphism. These two facts combined make the Lasserre hierarchy a good
candidate for solving graph isomorphism on expander graphs. Our main result
rules out this promising direction by showing that even Omega(n) rounds of the
Lasserre semidefinite program hierarchy fail to solve the Graph Isomorphism
problem even on expander graphs.
