We present an efficient algorithm to find a good solution to the Unique Games problem when the constraint graph is an expander.We introduce a new analysis of the standard SDP in this case that involves correlations among distant vertices. It also leads to a parallel repetition theorem for unique games when the graph is an expander.

We consider the problem of identifying underlying community-like structures in graphs. Towards this end we study the Stochastic Block Model (SBM) on k-clusters: a random model on n = km vertices, partitioned in k equal sized clusters, with edges sampled independently across clusters with probability q and within clusters with probability p, p > q. The goal is to recover the initial "hidden" partition of [n]. We study semidefinite programming (SDP) based algorithms in this context. In the regime p = α log(m)

We consider a variation of the spectral sparsification problem where we are required to keep a subgraph of the original graph. Formally, given a union of two weighted graphs G and W and an integer k, we are asked to find a k-edge weighted graph W k such that G + W k is a good spectral sparsifer of G + W . We will refer to this problem as the subgraph (spectral) sparsification. We present a nontrivial condition on G and W such that a good sparsifier exists and give a polynomial-time algorithm to find the sparsifer.As a significant application of our technique, we show that for each positive integer k, every n-vertex weighted graph has an (n − 1 + k)-edge spectral sparsifier with relative condition number at most n k log nÕ(log log n) whereÕ() hides lower order terms. Our bound nearly settles a question left open by Spielman and Teng about ultrasparsifiers, which is a key component in their nearly linear-time algorithms for solving diagonally dominant symmetric linear systems.We also present another application of our technique to spectral optimization in which the goal is to maximize the algebraic connectivity of a graph (e.g. turn it into an expander) with a limited number of edges.

Abstract-High throughput is of particular interest in data center and HPC networks. Although myriad network topologies have been proposed, a broad head-to-head comparison across topologies and across traffic patterns is absent, and the right way to compare worst-case throughput performance is a subtle problem.In this paper, we develop a framework to benchmark the throughput of network topologies, using a two-pronged approach. First, we study performance on a variety of synthetic and experimentally-measured traffic matrices (TMs). Second, we show how to measure worst-case throughput by generating a near-worst-case TM for any given topology. We apply the framework to study the performance of these TMs in a wide range of network topologies, revealing insights into the performance of topologies with scaling, robustness of performance across TMs, and the effect of scattered workload placement. Our evaluation code is freely available.

In this paper, we study the average case complexity of the Unique Games problem. We propose a natural semi-random model, in which a unique game instance is generated in several steps. First an adversary selects a completely satisfiable instance of Unique Games, then she chooses an ε-fraction of all edges, and finally replaces ("corrupts") the constraints corresponding to these edges with new constraints. If all steps are adversarial, the adversary can obtain any (1 − ε) satisfiable instance, so then the problem is as hard as in the worst case. In our semi-random model, one of the steps is random, and all other steps are adversarial. We show that known algorithms for unique games (in particular, all algorithms that use the standard SDP relaxation) fail to solve semi-random instances of Unique Games.We present an algorithm that with high probability finds a solution satisfying a (1 − δ) fraction of all constraints in semi-random instances (we require that the average degree of the graph isΩ(log k)). To this end, we consider a new non-standard SDP program for Unique Games, which is not a relaxation for the problem, and show how to analyze it. We present a new rounding scheme that simultaneously uses SDP and LP solutions, which we believe is of independent interest.Our result holds only for ε less than some absolute constant. We prove that if ε ≥ 1/2, then the problem is hard in one of the models, that is, no polynomial-time algorithm can distinguish between the following two cases: (i) the instance is a (1 − ε) satisfiable semi-random instance and (ii) the instance is at most δ satisfiable (for every δ > 0); the result assumes the 2-to-2 conjecture.Finally, we study semi-random instances of Unique Games that are at most (1 − ε) satisfiable. We present an algorithm that with high probability, distinguishes between the case when the instance is a semi-random instance and the case when the instance is an (arbitrary) (1 − δ) satisfiable instance if ε > cδ (for some absolute constant c).