We examine a hierarchy of equivalence classes of quasi-random properties of Boolean Functions. In particular, we prove an equivalence between a number of properties including balanced influences, spectral discrepancy, local strong regularity, homomorphism enumerations of colored or weighted graphs and hypergraphs associated with Boolean functions as well as the $k$th-order strict avalanche criterion amongst others. We further construct families of quasi-random boolean functions which exhibit the properties of our equivalence theorem and separate the levels of our hierarchy.
Towards a Proof of the Fourier--Entropy Conjecture?
The total influence of a function is a central notion in analysis of Boolean functions, and characterizing functions that have small total influence is one of the most fundamental questions associated with it. The KKL theorem and the Friedgut junta theorem give a strong characterization of such functions whenever the bound on the total influence is $o(\log n)$ . However, both results become useless when the total influence of the function is $ω(\log n)$ . The only case in which this logarithmic barrier has been broken for an interesting class of functions was proved by Bourgain and Kalai, who …
On the Fourier Entropy Influence Conjecture for Extremal Classes
The Fourier Entropy-Influence (FEI) Conjecture of Friedgut and Kalai states that ${\bf H}[f] \leq C \cdot {\bf I}[f]$ holds for every Boolean function $f$ , where ${\bf H}[f]$ denotes the spectral entropy of $f$ , ${\bf I}[f]$ is its total influence, and $C > 0$ is a universal constant. Despite significant interest in the conjecture it has only been shown to hold for some classes of Boolean functions such as symmetric functions and read-once formulas. In this work, we prove the conjecture for extremal cases, functions with small influence and functions with high entropy. Specifically, we show …
Computing the partition function of a polynomial on the Boolean cube
对于多项式f：{-1，1}^n --> C，我们将配分函数定义为e^{lambda f（x）}在{-1，1}^n中所有点x上的平均值，其中C中的lambda是一个参数。我们提出了一个拟多项式算法，在给定f，lambda和λ>0的情况下，只要|lambda| < 1/（2 L sqrt{deg f}），该算法在N^{O（ln n -ln n）}时间内以相对误差λ逼近配分函数，其中L=L（f）是从上面限定f的Lipschitz常数的参数，N是f中单项式的个数.作为推论，我们获得了一个准多项式算法，给定系数为+1和-1的f，并且每个变量输入不超过4个单式，该算法在O（sqrt{deg f}/delta）的因子内逼近{-1，1}^n上f的最大值，如果对于某些0< delta 4，最大值为N delta<1. If every variable enters not more than k monomials for some fixed k >，则当delta >（k-1）/k时，我们能够建立类似的结果。
We introduce the notion of a scheduling problem which is a boolean function $S$ over atomic formulas of the form $x_i \leq x_j$ . Considering the $x_i$ as jobs to be performed, an integer assignment satisfying $S$ schedules the jobs subject to the constraints of the atomic formulas. The scheduling counting function counts the number of solutions to $S$ . We prove that this counting function is a polynomial in the number of time slots allowed. Scheduling polynomials include the chromatic polynomial of a graph, the zeta polynomial of a lattice, the Billera-Jia-Reiner polynomial of a matroid. To …
A Lower Bound for the Fourier Entropy of Boolean Functions on the Biased Hypercube
School of Statistics and Data Science, Nankai University（南开大学统计与数据科学学院）；Extremal Combinatorics and Probability Group (ECOPRO), Institute for Basic Science (IBS)（基础科学研究所极组合与概率组）
