Improved ARV Rounding in Small-set Expanders and Graphs of Bounded Threshold Rank

We prove a structure theorem for the feasible solutions of the Arora-Rao-Vazirani SDP relaxation on low threshold rank graphs and on small-set expanders. We show that if G is a graph of bounded threshold rank or a small-set expander, then an optimal solution of the Arora-Rao-Vazirani relaxation (or of any stronger version of it) can be almost entirely covered by a small number of balls of bounded radius. Then, we show that, if k is the number of balls, a solution of this form can be rounded with an approximation factor of O( √ log k) in the case of the Arora-Rao-Vazirani relaxation, and with a constant-factor approximation in the case of the k-th round of the Sherali-Adams hierarchy starting at the Arora-Rao-Vazirani relaxation. The structure theorem and the rounding scheme combine to prove the following result, where G = (V,E) is a graph of expansion φ(G), λk is the k-th smallest eigenvalue of the normalized Laplacian of G, and φk(G) = mindisjoint S1,...,Sk max1≤i≤k φ(Si) is the largest expansion of any k disjoint subsets of V : if either λk & log 2.5 k ·φ(G) or φk(G) & log k · √ logn · log logn ·φ(G), then the Arora-Rao-Vazirani relaxation can be rounded in polynomial time with an approximation ratio O( √ log k). Stronger approximation guarantees are achievable in time exponential in k via relaxations in the Lasserre hierarchy. Guruswami and Sinop [GS13] and Arora, Ge and Sinop [AGS13] prove that 1 + ǫ approximation is achievable in time 2 poly(n) if either λk > φ(G)/ poly(ǫ), or if SSEn/k > √ log k logn · φ(G)/ poly(ǫ), where SSEs is the minimal expansion of sets of size at most s.