Related papers: Monotone Complexity of Spanning Tree Polynomial Re…
We show that sharp thresholds for Boolean functions directly imply average-case circuit lower bounds. More formally we show that any Boolean function exhibiting a sharp enough threshold at \emph{arbitrary} critical density cannot be…
We show that any one-round algorithm that computes a minimum spanning tree (MST) in the unicast congested clique must use a link bandwidth of $\Omega(\log^3 n)$ bits in the worst case. Consequently, computing an MST under the standard…
We consider the number of spanning trees in circulant graphs of $\beta n$ vertices with generators depending linearly on $n$. The matrix tree theorem gives a closed formula of $\beta n$ factors, while we derive a formula of $\beta-1$…
Let $P \subseteq \mathbb{R}^2$ be a set of points and $T$ be a spanning tree of $P$. The \emph{stabbing number} of $T$ is the maximum number of intersections any line in the plane determines with the edges of $T$. The \emph{tree stabbing…
We characterize the monotone bounded depth formula complexity for graph homomorphism and colored isomorphism polynomials using a graph parameter called the cost of bounded product depth baggy elimination tree. Using this characterization,…
Motivated by applications to sensor networks, as well as to many other areas, this paper studies the construction of minimum-degree spanning trees. We consider the classical node-register state model, with a weakly fair scheduler, and we…
The study of Markov processes and broadcasting on trees has deep connections to a variety of areas including statistical physics, graphical models, phylogenetic reconstruction, Markov Chain Monte Carlo, and community detection in random…
We study the minimum spanning tree (MST) problem in the massively parallel computation (MPC) model. Our focus is particularly on the *strictly sublinear* regime of MPC where the space per machine is $O(n^\delta)$. Here $n$ is the number of…
We prove that for any decision tree calculating a boolean function $f:\{-1,1\}^n\to\{-1,1\}$, \[ \Var[f] \le \sum_{i=1}^n \delta_i \Inf_i(f), \] where $\delta_i$ is the probability that the $i$th input variable is read and $\Inf_i(f)$ is…
Cayley's formula states that there are $n^{n-2}$ spanning trees in the complete graph on $n$ vertices; it has been proved in more than a dozen different ways over its 150 year history. The complete graphs are a special case of threshold…
We consider a problem of approximating the size of the largest clique in a graph, with a monotone circuit. Concretely, we focus on distinguishing a random Erd\H{o}s-Renyi graph $\mathcal{G}_{n,p}$, with $p=n^{-\frac{2}{\alpha-1}}$ chosen…
The \emph{sum-of-squares (SoS) complexity} of a $d$-multiquadratic polynomial $f$ (quadratic in each of $d$ blocks of $n$ variables) is the minimum $s$ such that $f = \sum_{i=1}^s g_i^2$ with each $g_i$ $d$-multilinear. In the case $d=2$,…
Given a directed graph $G$ on $n$ vertices with a special vertex $s$, the directed minimum degree spanning tree problem requires computing a incoming spanning tree rooted at $s$ whose maximum tree in-degree is the smallest among all such…
This paper is our third step towards developing a theory of testing monomials in multivariate polynomials and concentrates on two problems: (1) How to compute the coefficients of multilinear monomials; and (2) how to find a maximum…
Graph polynomials encode fundamental combinatorial invariants of graphs. Their computation is investigated using tree and path decomposition frameworks, with formal definitions of treewidth, k-trees, and pathwidth establishing the…
Robust sunflowers are a generalization of combinatorial sunflowers that have applications in monotone circuit complexity, DNF sparsification, randomness extractors, and recent advances on the Erd\H{o}s-Rado sunflower conjecture. The recent…
We show that several new classes of groups are measure strongly treeable. In particular, finitely generated groups admitting planar Cayley graphs, elementarily free groups, and the group of isometries of the hyperbolic plane and all its…
We prove that any extended formulation that approximates the matching polytope on $n$-vertex graphs up to a factor of $(1+\varepsilon)$ for any $\frac2n \le \varepsilon \le 1$ must have at least $\binom{n}{{\alpha}/{\varepsilon}}$ defining…
Inspired by the combinatorial constructions in earlier work of the authors that generalized the classical Alexander polynomial to a large class of spatial graphs with a balanced weight on edges, we show that the value of the Alexander…
Given a graph $G=(V,E)$ and a set $C$ of unordered pairs of edges regarded as being in conflict, a stable spanning tree in $G$ is a set of edges $T$ inducing a spanning tree in $G$, such that for each $\left\lbrace e_i, e_j \right\rbrace…