We give subexponential time approximation algorithms for the unique games and the small set expansion problems. Specifically, for some absolute constant c, we give: 1. An exp(kn ε)-time algorithm that, given as input a k-alphabet unique game on n variables that has an assignment satisfying 1 − ε c fraction of its constraints, outputs an assignment satisfying 1 − ε fraction of the constraints. 2. An exp(n ε /δ)-time algorithm that, given as input an n-vertex regular graph that has a set S of δn vertices with edge expansion at most ε c, outputs a set S ′ of at most δn vertices with edge expansion at most ε. We also obtain a subexponential algorithm with improved approximation for the Multi-Cut problem, as well as subexponential algorithms with improved approximations to Max-Cut, Sparsest-Cut and Vertex Cover on some interesting subclasses of instances. Khot’s Unique Games Conjecture (UGC) states that it is NP-hard to achieve approximation guarantees such as ours for unique games. While our results stop short of refusing the UGC, they do suggest that Unique Games is significantly easier than NP-hard problems such as 3SAT,3LIN, Label Cover and more, that are believed not to have a subexponential algorithm achieving a non-trivial approximation ratio. The main component in our algorithms is a new result on graph decomposition that may have other applications. Namely we show that for every δ> 0 and a regular n-vertex graph G, by changing at most δ fraction of G’s edges, one can break G into disjoint parts so that the induced graph on each part has at most n ε eigenvalues larger than 1 − η (where ε, η depend polynomially on δ). Our results are based on combining this decomposition with previous algorithms for unique games on graphs with few large eigenvalues (Kolla and Tulsiani 2007, Kolla 2010). 1

In this paper, we consider the following graph partitioning problem: The input is an undirected graph G = (V, E), a balance parameter b ∈ (0, 1/2] and a target conductance value γ ∈ (0, 1). The output is a cut which, if non-empty, is of conductance at most O ( f), for some function f (G, γ), and which is either balanced or well correlated with all cuts of conductance at most γ. In a seminal paper, Spielman and Teng γ log 3 |V| [16] gave an Õ(|E|/γ2)-time algorithm for f = and used it to decompose graphs into a collection of nearexpanders [18]. We present a new spectral algorithm for this problem which runs in time Õ(|E|/γ) for f = √ γ. Our result yields the first nearly-linear time algorithm for the classic Balanced Separator problem that achieves the asymptotically optimal approximation guarantee for spectral methods. Our method has the advantage of being conceptually simple and relies on a primal-dual semidefinite-programming (SDP) approach. We first consider a natural SDP relaxation for the Balanced Separator problem. While it is easy to obtain from this SDP a certificate of the fact that the graph has no balanced cut of conductance less than γ, somewhat surprisingly, we can obtain a certificate for the stronger correlation condition. This is achieved via a novel separation oracle for our SDP and by appealing to Arora and Kale’s [3] framework to bound the running time. Our result contains technical ingredients that may be of independent interest.

We initiate a study of when the value of mathematical relaxations such as linear and semi-definite programs for constraint satisfaction problems (CSPs) is approximately preserved when restricting the instance to a sub-instance induced by a small random subsample of the variables. Let C be a family of CSPs such as 3SAT, Max-Cut, etc.., and let Π be a mathematical program that is a relaxation for C, in the sense that for every instance P ∈ C, Π(P) is a number in [0, 1] upper bounding the maximum fraction of satisfiable constraints of P. Loosely speaking, we say that subsampling holds for C and Π if for every sufficiently dense instance P ∈ C and every ε> 0, if we let P ′ be the instance obtained by restricting P to a sufficiently large constant number of variables, then Π(P ′ ) ∈ (1 ± ε)Π(P). We say that weak subsampling holds if the above guarantee is replaced with Π(P ′ ) = 1 − Θ(γ) whenever Π(P) = 1 − γ, where Θ hides only absolute constants. We obtain both positive and negative results, showing that: 1. Subsampling holds for the BasicLP and BasicSDP programs. BasicSDP is a variant of the semidefinite program considered by Raghavendra (2008), who showed it gives an optimal approximation factor for every constraint-satisfaction problem under the unique games conjecture. BasicLP is the linear programming analog of BasicSDP.

Graph-partitioning problems are a central topic of research in the study of approximation algorithms. They are of interest to both theoreticians, for their far-reaching connections to different areas of mathematics, and to practitioners, as algorithms for graph partitioning can be used as fundamental building blocks in many applications, such as image segmentation and clustering. While many theoretical approximation algorithms exist for graph partitioning, they often rely on multicommodity-flow computations that run in quadratic time in the worst case and are too time-consuming for the massive graphs that are prevalent in today’s applications. In this dissertation, we study the design of approximation algorithms that yield strong approximation guarantees, while running in subquadratic time and relying on computational procedures that are often fast in practice. The results that we describe encompass two different approaches to the construction of such fast algorithms. Our first result exploits the Cut-Matching game of Khandekar, Rao and Vazirani [41], an elegant framework for designing graph-partitioning algorithms that rely on single-commodity, rather than multicommodity, maximum flow. Within this framework, we give two novel algorithms that achieve an O(log n)-approximation for the problem of finding the cut of minimum

The spectral profile of a graph is a natural generalization of the classical notion of its Rayleigh quotient. Roughly speaking, given a graph G, for each 0 < δ < 1, the spectral profile ΛG(δ) minimizes the Rayleigh quotient (from the variational characterization) of the spectral gap of the Laplacian matrix of G over vectors with support at most δ over a suitable probability measure. Formally, the spectral profile ΛG of a graph G is a function ΛG: [0, 1/2] → R defined as: ΛG(δ) def = min x∈R V d(supp(x))�δ P gij(xi − xj) where