We study the satisfiability of ordering constraint satisfaction problems (CSPs) above average. We prove the conjecture of Gutin, van Iersel, Mnich, and Yeo that the satisfiability above average of ordering CSPs of arity $k$ is fixed-parameter tractab
le for every $k$. Previously, this was only known for $k=2$ and $k=3$. We also generalize this result to more general classes of CSPs, including CSPs with predicates defined by linear inequalities. To obtain our results, we prove a new Bonami-type inequality for the Efron-Stein decomposition. The inequality applies to functions defined on arbitrary product probability spaces. In contrast to other variants of the Bonami Inequality, it does not depend on the mass of the smallest atom in the probability space. We believe that this inequality is of independent interest.
We study vertex cut and flow sparsifiers that were recently introduced by Moitra, and Leighton and Moitra. We improve and generalize their results. We give a new polynomial-time algorithm for constructing O(log k / log log k) cut and flow sparsifiers
, matching the best existential upper bound on the quality of a sparsifier, and improving the previous algorithmic upper bound of O(log^2 k / log log k). We show that flow sparsifiers can be obtained from linear operators approximating minimum metric extensions. We introduce the notion of (linear) metric extension operators, prove that they exist, and give an exact polynomial-time algorithm for finding optimal operators. We then establish a direct connection between flow and cut sparsifiers and Lipschitz extendability of maps in Banach spaces, a notion studied in functional analysis since 1930s. Using this connection, we prove a lower bound of Omega(sqrt{log k/log log k}) for flow sparsifiers and a lower bound of Omega(sqrt{log k}/log log k) for cut sparsifiers. We show that if a certain open question posed by Ball in 1992 has a positive answer, then there exist tilde O(sqrt{log k}) cut sparsifiers. On the other hand, any lower bound on cut sparsifiers better than tilde Omega(sqrt{log k}) would imply a negative answer to this question.
In this note we improve a recent result by Arora, Khot, Kolla, Steurer, Tulsiani, and Vishnoi on solving the Unique Games problem on expanders. Given a $(1-varepsilon)$-satisfiable instance of Unique Games with the constraint graph $G$, our algorit
hm finds an assignment satisfying at least a $1- C varepsilon/h_G$ fraction of all constraints if $varepsilon < c lambda_G$ where $h_G$ is the edge expansion of $G$, $lambda_G$ is the second smallest eigenvalue of the Laplacian of $G$, and $C$ and $c$ are some absolute constants.
