Decomposing a graph into expanding subgraphs

A paradigm that was successfully applied in the study of both pure and algorithmic problems in graph theory can be colloquially summarized as stating that any graph is close to being the disjoint union of expanders. Our goal in this paper is to show that in several of the instantiations of the above approach, the quantitative bounds that were obtained are essentially best possible. Two examples of our results are the following: Motivated by the Unique Games Conjecture, Trevisan [FOCS '05] and Arora, Barak and Steurer [FOCS '10] showed that given a graph G, one can remove only 1% of G's edges and thus obtain a graph in which each connected component has good expansion properties. We show that in both of these decomposition results, the expansion properties they guarantee are (essentially) best possible even when one is allowed to remove 99% of G's edges. In particular, our results imply that the eigenspace enumeration approach of Arora-Barak-Steurer cannot give (even quasi-) polynomial time algorithms for unique games. A classical result of Lipton, Rose and Tarjan from 1979 states that if F is a hereditary family of graphs and every graph in F has a vertex separator of size n/ (1ogn)^{1+o (1)}, then every graph in F has O (n) edges. We construct a hereditary family of graphs with vertex separators of size n/ (1ogn)^{1-o (1)} such that not all graphs in the family have O (n) edges. The above results are obtained as corollaries of a new family of graphs, which we construct by picking random subgraphs of the hypercube, and analyze using (simple) arguments from the theory of metric embedding.