Counting independent sets in hypergraphs

Let $G$ be a triangle-free graph with $n$ vertices and average degree $t$. We show that $G$ contains at least $$e^{(1-n^{-1/12})\frac{1}{2}\frac{n}{t}\ln t (\frac{1}{2}\ln t-1)}$$ independent sets. This improves a recent result of the first and third authors \cite{countingind}. In particular, it implies that as $n \to \infty$, every triangle-free graph on $n$ vertices has at least $e^{(c_1-o(1)) \sqrt{n} \ln n}$ independent sets, where $c_1 = \sqrt{\ln 2}/4 = 0.208138..$. Further, we show that for all $n$, there exists a triangle-free graph with $n$ vertices which has at most $e^{(c_2+o(1))\sqrt{n}\ln n}$ independent sets, where $c_2 = 1+\ln 2 = 1.693147..$. This disproves a conjecture from \cite{countingind}. Let $H$ be a $(k+1)$-uniform linear hypergraph with $n$ vertices and average degree $t$. We also show that there exists a constant $c_k$ such that the number of independent sets in $H$ is at least $$e^{c_{k} \frac{n}{t^{1/k}}\ln^{1+1/k}{t}}.$$ This is tight apart from the constant $c_k$ and generalizes a result of Duke, Lefmann, and R\"odl \cite{uncrowdedrodl}, which guarantees the existence of an independent set of size $\Omega(\frac{n}{t^{1/k}} \ln^{1/k}t)$. Both of our lower bounds follow from a more general statement, which applies to hereditary properties of hypergraphs

The complexity of approximating bounded-degree Boolean #CSP

AbstractThe degree of a CSP instance is the maximum number of times that any variable appears in the scopes of constraints. We consider the approximate counting problem for Boolean CSP with bounded-degree instances, for constraint languages containing the two unary constant relations {0} and {1}. When the maximum allowed degree is large enough (at least 6) we obtain a complete classification of the complexity of this problem. It is exactly solvable in polynomial time if every relation in the constraint language is affine. It is equivalent to the problem of approximately counting independent sets in bipartite graphs if every relation can be expressed as conjunctions of {0}, {1} and binary implication. Otherwise, there is no FPRAS unless NP=RP. For lower degree bounds, additional cases arise, where the complexity is related to the complexity of approximately counting independent sets in hypergraphs

Distributed Lower Bounds for Ruling Sets

Given a graph $G = (V,E)$, an $(\alpha, \beta)$-ruling set is a subset $S \subseteq V$ such that the distance between any two vertices in $S$ is at least $\alpha$, and the distance between any vertex in $V$ and the closest vertex in $S$ is at most $\beta$. We present lower bounds for distributedly computing ruling sets. More precisely, for the problem of computing a $(2, \beta)$-ruling set in the LOCAL model, we show the following, where $n$ denotes the number of vertices, $\Delta$ the maximum degree, and $c$ is some universal constant independent of $n$ and $\Delta$. $\bullet$ Any deterministic algorithm requires $\Omega\left(\min \left\{ \frac{\log \Delta}{\beta \log \log \Delta} , \log_\Delta n \right\} \right)$ rounds, for all $\beta \le c \cdot \min\left\{ \sqrt{\frac{\log \Delta}{\log \log \Delta}} , \log_\Delta n \right\}$. By optimizing $\Delta$, this implies a deterministic lower bound of $\Omega\left(\sqrt{\frac{\log n}{\beta \log \log n}}\right)$ for all $\beta \le c \sqrt[3]{\frac{\log n}{\log \log n}}$. $\bullet$ Any randomized algorithm requires $\Omega\left(\min \left\{ \frac{\log \Delta}{\beta \log \log \Delta} , \log_\Delta \log n \right\} \right)$ rounds, for all $\beta \le c \cdot \min\left\{ \sqrt{\frac{\log \Delta}{\log \log \Delta}} , \log_\Delta \log n \right\}$. By optimizing $\Delta$, this implies a randomized lower bound of $\Omega\left(\sqrt{\frac{\log \log n}{\beta \log \log \log n}}\right)$ for all $\beta \le c \sqrt[3]{\frac{\log \log n}{\log \log \log n}}$. For $\beta > 1$, this improves on the previously best lower bound of $\Omega(\log^* n)$ rounds that follows from the 30-year-old bounds of Linial [FOCS'87] and Naor [J.Disc.Math.'91]. For $\beta = 1$, i.e., for the problem of computing a maximal independent set, our results improve on the previously best lower bound of $\Omega(\log^* n)$ on trees, as our bounds already hold on trees

A Note on the Pseudorandomness of Low-Degree Polynomials over the Integers

We initiate the study of a problem called the Polynomial Independence Distinguishing Problem (PIDP). The problem is parameterized by a set of polynomials $\mathcal{Q}=(q_1,\ldots, q_m)$ where each $q_i:\mathbb{R}^n\rightarrow \mathbb{R}$ and an input distribution $\mathcal{D}$ over the reals. The goal of the problem is to distinguish a tuple of the form $\{ q_i,q_i(\mathbf{x})\}_{i\in [m]}$ from $\{ q_i,q_i(\mathbf{x}_i)\}_{i\in [m]}$ where $\mathbf{x}, \mathbf{x}_1,\ldots , \mathbf{x}_m$ are each sampled independently from the distribution $\mathcal{D}^n$. Refutation and search versions of this problem are conjectured to be hard in general for polynomial time algorithms (Feige, STOC 02) and are also subject to known theoretical lower bounds for various hierarchies (such as Sum-of-Squares and Sherali-Adams). Nevertheless, we show polynomial time distinguishers for the problem in several scenarios, including settings where such lower bounds apply to the search or refutation versions of the problem. Our results apply to the setting when each polynomial is a constant degree multilinear polynomial. We show that this natural problem admits polynomial time distinguishing algorithms for the following scenarios: Non-trivial Distinguishers: We define a non-trivial distinguisher to be an algorithm that runs in time $n^{O(1)}$ and distinguishes between the two distributions with probability at least $n^{-O(1)}$. We show that such non-trivial distinguishers exist for large classes of worst-case families of polynomials, and essentially any non-trivial input distribution that is symmetric around zero, and isn\u27t equivalent to a distribution over Boolean values. In particular, we show that when $m\geq n$ and the sets of indices corresponding to the variables present in each monomial exhibit a weak expansion property with expansion factor greater than $1/2$ for unions of at most $4$ sets, then a non-trivial distinguisher exists. Overwhelming Distinguishers: Next we consider the problem of amplifying the success probability of the distinguisher, to guarantee that it succeeds with probability $1-n^{-\omega(1)}$. We obtain such an overwhelming distinguisher for natural random classes of homogeneous multilinear constant degree $d$ polynomials, denoted by $\mathcal{Q}_{n,d,p}$, and natural input distributions $\mathcal{D}$ such as discrete Gaussians or uniform distributions over bounded intervals. The polynomials are chosen by independently sampling each coefficient to be $0$ with probability $p$ and uniformly from \cD otherwise. For these polynomials, we show a surprisingly simple distinguisher that requires $p> n\log n/\binom{n}{d}$ and $m\geq \tilde{O}(n^{2})$ samples, independent of the degree $d$. This is in contrast with the setting for refutation, where we have sum-of-squares lower bounds against constant degree sum-of-squares algorithms (Grigoriev, TCS 01; Schoenebeck, FOCS 08) for this parameter regime for degree $d>6$

Circuit Lower Bounds, Help Functions, and the Remote Point Problem

We investigate the power of Algebraic Branching Programs (ABPs) augmented with help polynomials, and constant-depth Boolean circuits augmented with help functions. We relate the problem of proving explicit lower bounds in both these models to the Remote Point Problem (introduced by Alon, Panigrahy, and Yekhanin (RANDOM '09)). More precisely, proving lower bounds for ABPs with help polynomials is related to the Remote Point Problem w.r.t. the rank metric, and for constant-depth circuits with help functions it is related to the Remote Point Problem w.r.t. the Hamming metric. For algebraic branching programs with help polynomials with some degree restrictions we show exponential size lower bounds for explicit polynomials

Jacobian hits circuits: Hitting-sets, lower bounds for depth-D occur-k formulas & depth-3 transcendence degree-k circuits

We present a single, common tool to strictly subsume all known cases of polynomial time blackbox polynomial identity testing (PIT) that have been hitherto solved using diverse tools and techniques. In particular, we show that polynomial time hitting-set generators for identity testing of the two seemingly different and well studied models - depth-3 circuits with bounded top fanin, and constant-depth constant-read multilinear formulas - can be constructed using one common algebraic-geometry theme: Jacobian captures algebraic independence. By exploiting the Jacobian, we design the first efficient hitting-set generators for broad generalizations of the above-mentioned models, namely: (1) depth-3 (Sigma-Pi-Sigma) circuits with constant transcendence degree of the polynomials computed by the product gates (no bounded top fanin restriction), and (2) constant-depth constant-occur formulas (no multilinear restriction). Constant-occur of a variable, as we define it, is a much more general concept than constant-read. Also, earlier work on the latter model assumed that the formula is multilinear. Thus, our work goes further beyond the results obtained by Saxena & Seshadhri (STOC 2011), Saraf & Volkovich (STOC 2011), Anderson et al. (CCC 2011), Beecken et al. (ICALP 2011) and Grenet et al. (FSTTCS 2011), and brings them under one unifying technique. In addition, using the same Jacobian based approach, we prove exponential lower bounds for the immanant (which includes permanent and determinant) on the same depth-3 and depth-4 models for which we give efficient PIT algorithms. Our results reinforce the intimate connection between identity testing and lower bounds by exhibiting a concrete mathematical tool - the Jacobian - that is equally effective in solving both the problems on certain interesting and previously well-investigated (but not well understood) models of computation