Hypercontractive Inequality for Pseudo-Boolean Functions of Bounded Fourier Width

A function $f:\ \{-1,1\}^n\rightarrow \mathbb{R}$ is called pseudo-Boolean. It is well-known that each pseudo-Boolean function $f$ can be written as $f(x)=\sum_{I\in {\cal F}}\hat{f}(I)\chi_I(x),$ where ${\cal F}\subseteq \{I:\ I\subseteq [n]\}$,$[n]=\{1,2,...,n\}$, and$\chi_I(x)=\prod_{i\in I}x_i$and$\hat{f}(I)$are non-zero reals. The degree of$f$is$\max \{|I|:\ I\in {\cal F}\}$and the width of$f$is the minimum integer$\rho$such that every$i\in [n]$appears in at most$\rho$sets in$\cal F$. For$i\in [n]$, let$\mathbf{x}_i$be a random variable taking values 1 or -1 uniformly and independently from all other variables$\mathbf{x}_j$,$j\neq i.$Let$\mathbf{x}=(\mathbf{x}_1,...,\mathbf{x}_n)$. The$p$-norm of$f$is$||f||_p=(\mathbb E[|f(\mathbf{x})|^p])^{1/p}$for any$p\ge 1$. It is well-known that$||f||_q\ge ||f||_p$whenever$q> p\ge 1$. However, the higher norm can be bounded by the lower norm times a coefficient not directly depending on$f$: if$f$is of degree$d$and$q> p>1$then$ ||f||_q\le (\frac{q-1}{p-1})^{d/2}||f||_p.$This inequality is called the Hypercontractive Inequality. We show that one can replace$d$by$\rho$in the Hypercontractive Inequality for each$q> p\ge 2$as follows:$ ||f||_q\le ((2r)!\rho^{r-1})^{1/(2r)}||f||_p,$where$r=\lceil q/2\rceil$. For the case$q=4$and$p=2$, which is important in many applications, we prove a stronger inequality:$ ||f||_4\le (2\rho+1)^{1/4}||f||_2.

Computing the partition function of a polynomial on the Boolean cube

For a polynomial f: {-1, 1}^n --> C, we define the partition function as the average of e^{lambda f(x)} over all points x in {-1, 1}^n, where lambda in C is a parameter. We present a quasi-polynomial algorithm, which, given such f, lambda and epsilon >0 approximates the partition function within a relative error of epsilon in N^{O(ln n -ln epsilon)} time provided |lambda| < 1/(2 L sqrt{deg f}), where L=L(f) is a parameter bounding the Lipschitz constant of f from above and N is the number of monomials in f. As a corollary, we obtain a quasi-polynomial algorithm, which, given such an f with coefficients +1 and -1 and such that every variable enters not more than 4 monomials, approximates the maximum of f on {-1, 1}^n within a factor of O(sqrt{deg f}/delta), provided the maximum is N delta for some 0< delta <1. If every variable enters not more than k monomials for some fixed k > 4, we are able to establish a similar result when delta > (k-1)/k.Comment: The final version of this paper is due to be published in the collection of papers "A Journey through Discrete Mathematics. A Tribute to Jiri Matousek" edited by Martin Loebl, Jaroslav Nesetril and Robin Thomas, to be published by Springe

Systems of Linear Equations over F2\mathbb{F}_2 and Problems Parameterized Above Average

In the problem Max Lin, we are given a system $Az=b$ of $m$ linear equations with $n$ variables over $\mathbb{F}_2$ in which each equation is assigned a positive weight and we wish to find an assignment of values to the variables that maximizes the excess, which is the total weight of satisfied equations minus the total weight of falsified equations. Using an algebraic approach, we obtain a lower bound for the maximum excess. Max Lin Above Average (Max Lin AA) is a parameterized version of Max Lin introduced by Mahajan et al. (Proc. IWPEC'06 and J. Comput. Syst. Sci. 75, 2009). In Max Lin AA all weights are integral and we are to decide whether the maximum excess is at least $k$, where $k$ is the parameter. It is not hard to see that we may assume that no two equations in $Az=b$ have the same left-hand side and $n={\rm rank A}$. Using our maximum excess results, we prove that, under these assumptions, Max Lin AA is fixed-parameter tractable for a wide special case: $m\le 2^{p(n)}$ for an arbitrary fixed function $p(n)=o(n)$. Max $r$-Lin AA is a special case of Max Lin AA, where each equation has at most $r$ variables. In Max Exact $r$-SAT AA we are given a multiset of $m$ clauses on $n$ variables such that each clause has $r$ variables and asked whether there is a truth assignment to the $n$ variables that satisfies at least $(1-2^{-r})m + k2^{-r}$ clauses. Using our maximum excess results, we prove that for each fixed $r\ge 2$, Max $r$-Lin AA and Max Exact $r$-SAT AA can be solved in time $2^{O(k \log k)}+m^{O(1)}.$ This improves $2^{O(k^2)}+m^{O(1)}$-time algorithms for the two problems obtained by Gutin et al. (IWPEC 2009) and Alon et al. (SODA 2010), respectively

Beating the random assignment on constraint satisfaction problems of bounded degree

We show that for any odd $k$ and any instance of the Max-kXOR constraint satisfaction problem, there is an efficient algorithm that finds an assignment satisfying at least a $\frac{1}{2} + \Omega(1/\sqrt{D})$ fraction of constraints, where $D$ is a bound on the number of constraints that each variable occurs in. This improves both qualitatively and quantitatively on the recent work of Farhi, Goldstone, and Gutmann (2014), which gave a \emph{quantum} algorithm to find an assignment satisfying a $\frac{1}{2} + \Omega(D^{-3/4})$ fraction of the equations. For arbitrary constraint satisfaction problems, we give a similar result for "triangle-free" instances; i.e., an efficient algorithm that finds an assignment satisfying at least a $\mu + \Omega(1/\sqrt{D})$ fraction of constraints, where $\mu$ is the fraction that would be satisfied by a uniformly random assignment.Comment: 14 pages, 1 figur

Multipartite entanglement in XOR games

We study multipartite entanglement in the context of XOR games. In particular, we study the ratio of the entangled and classical biases, which measure the maximum advantage of a quantum or classical strategy over a uniformly random strategy. For the case of two-player XOR games, Tsirelson proved that this ratio is upper bounded by the celebrated Grothendieck constant. In contrast, Pérez-García et al. proved the existence of entangled states that give quantum players an unbounded advantage over classical players in a three-player XOR game. We show that the multipartite entangled states that are most often seen in today’s literature can only lead to a bias that is a constant factor larger than the classical bias. These states include GHZ states, any state local-unitarily equivalent to combinations of GHZ and maximally entangled states shared between different subsets of the players (e.g., stabilizer states), as well as generalizations of GHZ states of the form ∑iɑi|i〉...|i〉 for arbitrary amplitudes ɑi. Our results have the following surprising consequence: classical three-player XOR games do not follow an XOR parallel repetition theorem, even a very weak one. Besides this, we discuss implications of our results for communication complexity and hardness of approximation. Our proofs are based on novel applications of extensions of Grothendieck’s inequality, due to Blei and Tonge, and Carne, generalizing Tsirelson’s use of Grothendieck’s inequality to bound the bias of two-player XOR games