Related papers: Near-Optimal Bootstrapping of Hitting Sets for Alg…
We prove that every $P_5$-free graph of bounded clique number contains a small hitting set of all its maximum stable sets. More generally, let us say a class $\mathcal{C}$ of graphs is $\eta$-bounded if there exists a function…
We prove a conjecture of Bonamy, Bousquet, Pilipczuk, Rz\k{a}\.zewski, Thomass\'e, and Walczak, that for every graph $H$, there is a polynomial $p$ such that for every positive integer $s$, every graph of average degree at least $p(s)$…
The study of proper rational mappings between balls in complex Euclidean spaces naturally leads to the relationship between the degree and imbedding dimension of such a mapping. The special case for monomial mappings is equivalent to the…
Rational Identity Testing (RIT) is the decision problem of determining whether or not a noncommutative rational formula computes zero in the free skew field. It admits a deterministic polynomial-time white-box algorithm [Garg, Gurvits,…
In this paper we introduce a new approach for approximately counting in bounded degree systems with higher-order constraints. Our main result is an algorithm to approximately count the number of solutions to a CNF formula $\Phi$ when the…
High-dimensional planted problems, such as finding a hidden dense subgraph within a random graph, often exhibit a gap between statistical and computational feasibility. While recovering the hidden structure may be statistically possible, it…
We study algebras satisfying a two-term multilinear identity, namely one of the form $x_1 \cdots x_n= q x_{\sigma(1)} \cdots x_{\sigma(n)}$, where $q$ is a parameter from the base field. We show that such algebras with $q=1$ and $\sigma$…
Raghavendra (STOC 2008) gave an elegant and surprising result: if Khot's Unique Games Conjecture (STOC 2002) is true, then for every constraint satisfaction problem (CSP), the best approximation ratio is attained by a certain simple…
A 1993 result of Alon and F\"uredi gives a sharp upper bound on the number of zeros of a multivariate polynomial over an integral domain in a finite grid, in terms of the degree of the polynomial. This result was recently generalized to…
In $\textit{total domination}$, given a graph $G=(V,E)$, we seek a minimum-size set of nodes $S\subseteq V$, such that every node in $V$ has at least one neighbor in $S$. We define a $\textit{fault-tolerant}$ version of total domination,…
We prove that, unless P=NP, there is no polynomial-time algorithm to approximate within some multiplicative constant the average size of an independent set in graphs of maximum degree 6. This is a special case of a more general result for…
We show that every real polynomial $f$ nonnegative on $[-1,1]^{n}$ can be approximated in the $l_{1}$-norm of coefficients, by a sequence of polynomials $\{f_{\ep r}\}$ that are sums of squares. This complements the existence of s.o.s.…
The class of even-hole-free graphs is very similar to the class of perfect graphs, and was indeed a cornerstone in the tools leading to the proof of the Strong Perfect Graph Theorem. However, the complexity of computing a maximum…
We show that there is a randomized algorithm that, when given a small constant-depth Boolean circuit $C$ made up of gates that compute constant-degree Polynomial Threshold functions or PTFs (i.e., Boolean functions that compute signs of…
The approximate degree of a Boolean function $f(x_{1},x_{2},\ldots,x_{n})$ is the minimum degree of a real polynomial that approximates $f$ pointwise within $1/3$. Upper bounds on approximate degree have a variety of applications in…
Let $f$ be the $\mathbb{F}_q$-linear map over $\mathbb{F}_{q^{2n}}$ defined by $x\mapsto x+ax^{q^s}+bx^{q^{n+s}}$ with $\gcd(n,s)=1$. It is known that the kernel of $f$ has dimension at most $2$, as proved by Csajb\'ok et al. in "A new…
Two subanalytic subsets of R^n are called s-equivalent at a common point P if the Hausdorff distance between their intersections with the sphere centered at P of radius r vanishes of order greater than s when r tends to 0. In this paper we…
This paper is devoted to a study of $S$-curves, that is systems of curves in the complex plane whose equilibrium potential in a harmonic external field satisfies a special symmetry property ($S$-property). Such curves have many…
A graph class admits an implicit representation if, for every positive integer $n$, its $n$-vertex graphs have a $O(\log n)$-bit (adjacency) labeling scheme, i.e., their vertices can be labeled by binary strings of length $O(\log n)$ such…
For a graph $G$, let $\nu_s(G)$ be the induced matching number of $G$. We prove that $\nu_s(G) \geq \frac{n(G)}{(\lceil\frac{\Delta}{2}\rceil+1) (\lfloor\frac{\Delta}{2}\rfloor+1)}$ for every graph of sufficiently large maximum degree…