Related papers: Some Closure Results for Polynomial Factorization …

It has been known for almost three decades that many $\mathrm{NP}$-hard optimization problems can be solved in polynomial time when restricted to structures of constant treewidth. In this work we provide the first extension of such results…

Let $\mathrm{R}$ be a real closed field, and $\mathrm{D} \subset \mathrm{R}$ an ordered domain. We describe an algorithm that given as input a polynomial $P \in \mathrm{D} [ X_{1},\ldots,X_{k} ]$, and a finite set, $\mathcal{A}= \{ p_{1},…

We study how the complexity of modular circuits computing AND depends on the depth of the circuits and the prime factorization of the modulus they use. In particular our construction of subexponential circuits of depth 2 for AND helps us to…

Let $V$ be a vector space over a finite field $k$. We give a condition on a subset $A \subset V$ that allows for a local criterion for checking when a function $f:A \to k$ is a restriction of a polynomial function of degree $<m$ on $V$. In…

Let V $\subset$ C n be an equidimensional algebraic set and g be an n-variate polynomial with rational coefficients. Computing the critical points of the map that evaluates g at the points of V is a cornerstone of several algorithms in real…

General factors are a generalization of matchings. Given a graph $G$ with a set $\pi(v)$ of feasible degrees, called a degree constraint, for each vertex $v$ of $G$, the general factor problem is to find a (spanning) subgraph $F$ of $G$…

Polynomial identity testing and arithmetic circuit lower bounds are two central questions in algebraic complexity theory. It is an intriguing fact that these questions are actually related. One of the authors of the present paper has…

For the General Factor problem we are given an undirected graph $G$ and for each vertex $v\in V(G)$ a finite set $B_v$ of non-negative integers. The task is to decide if there is a subset $S\subseteq E(G)$ such that $deg_S(v)\in B_v$ for…

We study the size blow-up that is necessary to convert an algebraic circuit of product-depth $\Delta+1$ to one of product-depth $\Delta$ in the multilinear setting. We show that for every positive $\Delta = \Delta(n) = o(\log n/\log \log…

We study deterministic polynomial identity testing (PIT) and reconstruction algorithms for depth-$4$ arithmetic circuits of the form \[ \Sigma^{[r]}\!\wedge^{[d]}\!\Sigma^{[s]}\!\Pi^{[\delta]}. \] This model generalizes Waring…

In 1979 Valiant showed that the complexity class VP_e of families with polynomially bounded formula size is contained in the class VP_s of families that have algebraic branching programs (ABPs) of polynomially bounded size. Motivated by the…

Let C be a depth-3 circuit with n variables, degree d and top fanin k (called sps(k,d,n) circuits) over base field F. It is a major open problem to design a deterministic polynomial time blackbox algorithm that tests if C is identically…

It is often useful to have polynomial upper or lower bounds on a one-dimensional function that are valid over a finite interval, called a trust region. A classical way to produce polynomial bounds of degree $k$ involves bounding the range…

The central open question of algebraic complexity is whether VP is unequal to VNP, which is saying that the permanent cannot be represented by families of polynomial-size algebraic circuits. For symmetric algebraic circuits, this has been…

In this paper we study arithmetic computations in the nonassociative, and noncommutative free polynomial ring $\mathbb{F}\{x_1,x_2,\ldots,x_n\}$. Prior to this work, nonassociative arithmetic computation was considered by Hrubes, Wigderson,…

Polynomial Identity Testing (PIT) is a fundamental computational problem. The famous depth-$4$ reduction result by Agrawal and Vinay (FOCS 2008) has made PIT for depth-$4$ circuits an enticing pursuit. A restricted depth-4 circuit computing…

We present a new probabilistic algorithm to find a finite set of points intersecting the closure of each connected component of the realization of every sign condition over a family of real polynomials defining regular hypersurfaces that…

We exhibit a monotone function computable by a monotone circuit of quasipolynomial size such that any monotone circuit of polynomial depth requires exponential size. This is the first size-depth tradeoff result for monotone circuits in the…

The problem of finding a nontrivial factor of a polynomial f(x) over a finite field F_q has many known efficient, but randomized, algorithms. The deterministic complexity of this problem is a famous open question even assuming the…

Proving super-polynomial size lower bounds for $\textsf{TC}^0$, the class of constant-depth, polynomial-size circuits of Majority gates, is a notorious open problem in complexity theory. A major frontier is to prove that $\textsf{NEXP}$…