Explicit Optimal Hardness via Gaussian stability results

The results of Raghavendra (2008) show that assuming Khot's Unique Games Conjecture (2002), for every constraint satisfaction problem there exists a generic semi-definite program that achieves the optimal approximation factor. This result is existential as it does not provide an explicit optimal rounding procedure nor does it allow to calculate exactly the Unique Games hardness of the problem. Obtaining an explicit optimal approximation scheme and the corresponding approximation factor is a difficult challenge for each specific approximation problem. An approach for determining the exact approximation factor and the corresponding optimal rounding was established in the analysis of MAX-CUT (KKMO 2004) and the use of the Invariance Principle (MOO 2005). However, this approach crucially relies on results explicitly proving optimal partitions in Gaussian space. Until recently, Borell's result (Borell 1985) was the only non-trivial Gaussian partition result known. In this paper we derive the first explicit optimal approximation algorithm and the corresponding approximation factor using a new result on Gaussian partitions due to Isaksson and Mossel (2012). This Gaussian result allows us to determine exactly the Unique Games Hardness of MAX-3-EQUAL. In particular, our results show that Zwick algorithm for this problem achieves the optimal approximation factor and prove that the approximation achieved by the algorithm is $\approx 0.796$ as conjectured by Zwick. We further use the previously known optimal Gaussian partitions results to obtain a new Unique Games Hardness factor for MAX-k-CSP : Using the well known fact that jointly normal pairwise independent random variables are fully independent, we show that the the UGC hardness of Max-k-CSP is $\frac{\lceil (k+1)/2 \rceil}{2^{k-1}}$, improving on results of Austrin and Mossel (2009)

Gaussian Bounds for Noise Correlation of Functions

In this paper we derive tight bounds on the expected value of products of {\em low influence} functions defined on correlated probability spaces. The proofs are based on extending Fourier theory to an arbitrary number of correlated probability spaces, on a generalization of an invariance principle recently obtained with O'Donnell and Oleszkiewicz for multilinear polynomials with low influences and bounded degree and on properties of multi-dimensional Gaussian distributions. The results derived here have a number of applications to the theory of social choice in economics, to hardness of approximation in computer science and to additive combinatorics problems.Comment: Typos and references correcte

Approximate kernel clustering

In the kernel clustering problem we are given a large $n\times n$ positive semi-definite matrix $A=(a_{ij})$ with $\sum_{i,j=1}^na_{ij}=0$ and a small $k\times k$ positive semi-definite matrix $B=(b_{ij})$. The goal is to find a partition $S_1,...,S_k$ of $\{1,... n\}$ which maximizes the quantity $$\sum_{i,j=1}^k (\sum_{(i,j)\in S_i\times S_j}a_{ij})b_{ij}.$$ We study the computational complexity of this generic clustering problem which originates in the theory of machine learning. We design a constant factor polynomial time approximation algorithm for this problem, answering a question posed by Song, Smola, Gretton and Borgwardt. In some cases we manage to compute the sharp approximation threshold for this problem assuming the Unique Games Conjecture (UGC). In particular, when $B$ is the $3\times 3$ identity matrix the UGC hardness threshold of this problem is exactly $\frac{16\pi}{27}$. We present and study a geometric conjecture of independent interest which we show would imply that the UGC threshold when $B$ is the $k\times k$ identity matrix is $\frac{8\pi}{9}(1-\frac{1}{k})$ for every $k\ge 3$

A polyhedral approach to computing border bases

Border bases can be considered to be the natural extension of Gr\"obner bases that have several advantages. Unfortunately, to date the classical border basis algorithm relies on (degree-compatible) term orderings and implicitly on reduced Gr\"obner bases. We adapt the classical border basis algorithm to allow for calculating border bases for arbitrary degree-compatible order ideals, which is \emph{independent} from term orderings. Moreover, the algorithm also supports calculating degree-compatible order ideals with \emph{preference} on contained elements, even though finding a preferred order ideal is NP-hard. Effectively we retain degree-compatibility only to successively extend our computation degree-by-degree. The adaptation is based on our polyhedral characterization: order ideals that support a border basis correspond one-to-one to integral points of the order ideal polytope. This establishes a crucial connection between the ideal and the combinatorial structure of the associated factor spaces

Invariance principle on the slice

We prove an invariance principle for functions on a slice of the Boolean cube, which is the set of all vectors {0,1}^n with Hamming weight k. Our invariance principle shows that a low-degree, low-influence function has similar distributions on the slice, on the entire Boolean cube, and on Gaussian space. Our proof relies on a combination of ideas from analysis and probability, algebra and combinatorics. Our result imply a version of majority is stablest for functions on the slice, a version of Bourgain's tail bound, and a version of the Kindler-Safra theorem. As a corollary of the Kindler-Safra theorem, we prove a stability result of Wilson's theorem for t-intersecting families of sets, improving on a result of Friedgut.Comment: 36 page