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Register a new userBoolean constant degree functions on the slice are juntas
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Added by
Yuval Filmus
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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We show that a Boolean degree $d$ function on the slice $binom{[n]}{k} = { (x_1,ldots,x_n) in {0,1} : sum_{i=1}^n x_i = k }$ is a junta, assuming that $k,n-k$ are large enough. This generalizes a classical result of Nisan and Szegedy on the hypercube. Moreover, we show that the maximum number of coordinates that a Boolean degree $d$ function can depend on is the same on the slice and the hypercube.
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We show that if $fcolon S_n to {0,1}$ is $epsilon$-close to linear in $L_2$ and $mathbb{E}[f] leq 1/2$ then $f$ is $O(epsilon)$-close to a union of mostly disjoint cosets, and moreover this is sharp: any such union is close to linear. This constitutes a sharp Friedgut-Kalai-Naor theorem for the symmetric group. Using similar techniques, we show that if $fcolon S_n to mathbb{R}$ is linear, $Pr[f otin {0,1}] leq epsilon$, and $Pr[f = 1] leq 1/2$, then $f$ is $O(epsilon)$-close to a union of mostly disjoint cosets, and this is also sharp; and that if $fcolon S_n to mathbb{R}$ is linear and $epsilon$-close to ${0,1}$ in $L_infty$ then $f$ is $O(epsilon)$-close in $L_infty$ to a union of disjoint cosets.
Classification of Non-linear Boolean functions is a long-standing problem in the area of theoretical computer science. In this paper, effort has been made to achieve a systematic classification of all n-variable Boolean functions, where only one affine Boolean function belongs to each class. Two different methods are proposed to achieve this classification. The first method is a recursive procedure that uses the Cartesian product of sets starting from the set of 1-variable Boolean function and in the second method classification is achieved through a set of invariant bit positions with respect to an affine function belonging to that class. The invariant bit positions also provide information concerning the size and symmetry properties of the classes/sub-classes, such that the members of classes/sub-classes satisfy certain similar properties.
Let $G$ be a finite, undirected $d$-regular graph and $A(G)$ its normalized adjacency matrix, with eigenvalues $1 = lambda_1(A)geq dots ge lambda_n ge -1$. It is a classical fact that $lambda_n = -1$ if and only if $G$ is bipartite. Our main result provides a quantitative separation of $lambda_n$ from $-1$ in the case of Cayley graphs, in terms of their expansion. Denoting $h_{out}$ by the (outer boundary) vertex expansion of $G$, we show that if $G$ is a non-bipartite Cayley graph (constructed using a group and a symmetric generating set of size $d$) then $lambda_n ge -1 + ch_{out}^2/d^2,,$ for $c$ an absolute constant. We exhibit graphs for which this result is tight up to a factor depending on $d$. This improves upon a recent result by Biswas and Saha who showed $lambda_n ge -1 + h_{out}^4/(2^9d^8),.$ We also note that such a result could not be true for general non-bipartite graphs.
Dimension is a standard and well-studied measure of complexity of posets. Recent research has provided many new upper bounds on the dimension for various structurally restricted classes of posets. Bounded dimension gives a succinct representation of the poset, admitting constant response time for queries of the form is $x<y$?. This application motivates looking for stronger notions of dimension, possibly leading to succinct representations for more general classes of posets. We focus on two: boolean dimension, introduced in the 1980s and revisited in recent research, and local dimension, a very new one. We determine precisely which values of dimension/boolean dimension/local dimension imply that the two other parameters are bounded.
For a graph $G=(V,E)$, $kin mathbb{N}$, and a complex number $w$ the partition function of the univariate Potts model is defined as [ {bf Z}(G;k,w):=sum_{phi:Vto [k]}prod_{substack{uvin E phi(u)=phi(v)}}w, ] where $[k]:={1,ldots,k}$. In this paper we give zero-free regions for the partition function of the anti-ferromagnetic Potts model on bounded degree graphs. In particular we show that for any $Deltain mathbb{N}$ and any $kgeq eDelta+1$, there exists an open set $U$ in the complex plane that contains the interval $[0,1)$ such that ${bf Z}(G;k,w) eq 0$ for any $win U$ and any graph $G$ of maximum degree at most $Delta$. (Here $e$ denotes the base of the natural logarithm.) For small values of $Delta$ we are able to give better results. As an application of our results we obtain improved bounds on $k$ for the existence of deterministic approximation algorithms for counting the number of proper $k$-colourings of graphs of small maximum degree.
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