We develop a polynomial time $Omegaleft ( frac 1R log R right)$ approximate algorithm for Max 2CSP-$R$, the problem where we are given a collection of constraints, each involving two variables, where each variable ranges over a set of size $R$, and w
e want to find an assignment to the variables that maximizes the number of satisfied constraints. Assuming the Unique Games Conjecture, this is the best possible approximation up to constant factors. Previously, a $1/R$-approximate algorithm was known, based on linear programming. Our algorithm is based on semidefinite programming (SDP) and on a novel rounding technique. The SDP that we use has an almost-matching integrality gap.
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 Un
ique Games, then she chooses an epsilon-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-epsilon) 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-delta) fraction of all constraints in semi-random instances (we require that the average degree of the graph is Omega(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 epsilon less than some absolute constant. We prove that if epsilon > 1/2, then the problem is hard in one of the models, the result assumes the 2-to-2 conjecture. Finally, we study semi-random instances of Unique Games that are at most (1-epsilon) 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-delta) satisfiable instance if epsilon > c delta.
We give a new algorithm for Unique Games which is based on purely {em spectral} techniques, in contrast to previous work in the area, which relies heavily on semidefinite programming (SDP). Given a highly satisfiable instance of Unique Games, our alg
orithm is able to recover a good assignment. The approximation guarantee depends only on the completeness of the game, and not on the alphabet size, while the running time depends on spectral properties of the {em Label-Extended} graph associated with the instance of Unique Games. We further show that on input the integrality gap instance of Khot and Vishnoi, our algorithm runs in quasi-polynomial time and decides that the instance if highly unsatisfiable. Notably, when run on this instance, the standard SDP relaxation of Unique Games {em fails}. As a special case, we also re-derive a polynomial time algorithm for Unique Games on expander constraint graphs. The main ingredient of our algorithm is a technique to effectively use the full spectrum of the underlying graph instead of just the second eigenvalue, which is of independent interest. The question of how to take advantage of the full spectrum of a graph in the design of algorithms has been often studied, but no significant progress was made prior to this work.
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 gra
ph $W_k$ 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. %$O(frac{n}{k})log n tilde{O}(log log n)$ 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 $frac{n}{k} log n tilde{O}(loglog n)$ where $tilde{O}()$ hides lower order terms. Our bound is within a factor of $tilde{O}(log log n)$ from optimal. This 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.
