Showing papers by "Alexander A. Razborov published in 2008"

On the minimal density of triangles in graphs

Abstract: For a fixed ρ ∈ [0, 1], what is (asymptotically) the minimal possible density g3(ρ) of triangles in a graph with edge density ρ? We completely solve this problem by proving that $$ g_3(\rho) =\frac{(t-1)\ofb{t-2\sqrt{t(t-\rho(t+1))}}\ofb{t+\sqrt{t(t-\rho(t+1))}}^2}{t^2(t+1)^2},$$ where $t\df \lfloor 1/(1-\rho)\rfloor$ is the integer such that $\rho\in\bigl[ 1-\frac 1t,1-\frac 1{t+1}\bigr]$ .

Resolution Is Not Automatizable Unless W[P] Is Tractable

Abstract: We show that neither resolution nor tree-like resolution is automatizable unless the class W[P] from the hierarchy of parameterized problems is fixed-parameter tractable by randomized algorithms with one-sided error.

The Sign-Rank of AC^O

Abstract: The sign-rank of a matrix A = [Aij] with plusmn1 entries is the least rank of a real matrix B = [Bij] with AijBij > 0 for all i, j. We obtain the first exponential lower bound on the sign-rank of a function in AC0. Namely, let f(x, y) = Lambdai=1 m Lambdaj=1 m 2(xij Lambda yij). We show that the matrix [f(x, y)]x, y has sign-rank 2Omega(m). This in particular implies that Sigma2 ccnsubeUPPcc, which solves a long-standing open problem posed by Babai, Frankl, and Simon (1986). Our result additionally implies a lower bound in learning theory. Specifically, let Phi1,..., Phir : {0, 1}n rarrRopf be functions such that every DNF formula f : {0, 1}n rarr {-1, +1} of polynomial size has the representation f equiv sign(a1Phi1 + hellip + arPhir) for some reals a1,..., ar. We prove that then r ges 2Omega(n 1/3 ), which essentially matches an upper bound of 2Otilde(n 1/3 ) due to Klivans and Servedio (2001). Finally, our work yields the first exponential lower bound on the size of threshold-of-majority circuits computing a function in AC0. This substantially generalizes and strengthens the results of Krause and Pudlak (1997).

Almost Euclidean subspaces of e N 1 via expander codes

Abstract: We give an explicit (in particular, deterministic polynomial time) construction of subspaces X ⊆ ℝN of dimension (1 - o(1))N such that for every x ∈ X, {display equation}. If we are allowed to use N1/log log N ≤ N°(1) random bits and dim(X) ≥ (1 - η)N for any fixed constant η, the lower bound can be further improved to (log N)-°(1) √N||x||2. Our construction makes use of unbalanced bipartite graphs to impose local linear constraints on vectors in the subspace, and our analysis relies on expansion properties of the graph. This is inspired by similar constructions of error-correcting codes.

Almost Euclidean subspaces of eN1 via expander codes

Abstract: We give an explicit (in particular, deterministic polynomial time) construction of subspaces X ⊆ ℝN of dimension (1 - o(1))N such that for every x ∈ X, {display equation}. If we are allowed to use N1/log log N ≤ N°(1) random bits and dim(X) ≥ (1 - η)N for any fixed constant η, the lower bound can be further improved to (log N)-°(1) √N||x||2. Our construction makes use of unbalanced bipartite graphs to impose local linear constraints on vectors in the subspace, and our analysis relies on expansion properties of the graph. This is inspired by similar constructions of error-correcting codes.

A simple proof of Bazzi's theorem.

Abstract: Linial and Nisan [1990] asked if any polylog-wise independent distribution fools any function in AC0. In a recent remarkable development, Bazzi solved this problem for the case of DNF formulas. The aim of this note is to present a simplified version of his proof.

The Sign-Rank of AC^0.

Abstract: The sign-rank of a matrix $A=[A_{ij}]$ with $\pm1$ entries is the least rank of a real matrix $B=[B_{ij}]$ with $A_{ij}B_{ij}>0$ for all $i,j$. We obtain the first exponential lower bound on the sign-rank of a function in $\mathsf{AC}^0$. Namely, let $f(x,y)=\bigwedge_{i=1,\dots,m}\bigvee_{j=1,\dots,m^2}(x_{ij}\wedge y_{ij})$. We show that the matrix $[f(x,y)]_{x,y}$ has sign-rank $\exp(\Omega(m))$. This in particular implies that $\Sigma_2^{cc} ot\subseteq\mathsf{UPP}^{cc}$, which solves a longstanding open problem in communication complexity posed by Babai, Frankl, and Simon [Proceedings of the 27th Symposium on Foundations of Computer Science (FOCS), 1986, pp. 337-347]. Our result additionally implies a lower bound in learning theory. Specifically, let $\phi_1,\dots,\phi_r:\{0,1\}^n\to\mathbb{R}$ be functions such that every DNF formula $f:\{0,1\}^n\to\{-1,+1\}$ of polynomial size has the representation $f\equiv\mathrm{sgn}(a_1\phi_1+\dots+a_r\phi_r)$ for some reals $a_1,\dots,a_r$. We prove that then $r\geqslant\exp(\Omega(n^{1/3}))$, which essentially matches an upper bound of $\exp(\tilde{O}(n^{1/3}))$, due to Klivans and Servedio [J. Comput. System Sci., 68 (2004), pp. 303-318]. Finally, our work yields the first exponential lower bound on the size of threshold-of-majority circuits computing a function in $\mathsf{AC}^0$. This substantially generalizes and strengthens the results of Krause and Pudlak [Theoret. Comput. Sci., 174 (1997), pp. 137-156].