Related papers: On Circuit Diameter Bounds via Circuit Imbalances

Let $ACC \circ THR$ be the class of constant-depth circuits comprised of AND, OR, and MOD$m$ gates (for some constant $m > 1$), with a bottom layer of gates computing arbitrary linear threshold functions. This class of circuits can be seen…

The problems of computing eccentricity, radius, and diameter are fundamental to graph theory. These parameters are intrinsically defined based on the distance metric of the graph. In this work, we propose quantum algorithms for the diameter…

We derive several upper bounds on the spectral gap of the Laplacian with standard or Dirichlet vertex conditions on compact metric graphs. In particular, we obtain estimates based on the length of a shortest cycle (girth), diameter, total…

We introduce a technically and conceptually simple approach to magnification of circuit and formula lower bounds. Central to the method are so-called distinguishers, sparse matrices that retain some of the key properties of error-correcting…

We prove that the diameter of any unweighted connected graph G is O(k log n/lambda_k), for any k>= 2. Here, lambda_k is the k smallest eigenvalue of the normalized laplacian of G. This solves a problem posed by Gil Kalai.

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…

We consider the problem of finding an orientation with minimum diameter of a connected bridgeless graph. Fomin et. al. discovered a relation between the minimum oriented diameter an the size of a minimal dominating set. We improve their…

Consider an $(n+1)$-dimensional circular cone with opening angle $\alpha \in (0,\pi)$. Using a free-boundary adaptation of the classical calibration method, we prove that, for $n \geq 4$, there exists a threshold $\bar{\alpha}(n) \in…

We obtain a new bound connecting the first non--trivial eigenvalue of the Laplace operator of a graph and the diameter of the graph, which is effective for graphs with small diameter or for graphs, having the number of maximal paths…

Comparator circuit model was originally introduced by Mayr and Subramanian (1992) (and further studied by Cook, Filmus and Le (2012)) to capture problems which are not known to be P-complete but still not known to admit efficient parallel…

We obtain a sharp lower bound on the isoperimetric deficit of a general polygon in terms of the variance of its side lengths, the variance of its radii, and its deviation from being convex. Our technique involves a functional minimization…

We show that there is a language in $\mathsf{S}_2\mathsf{E}/_1$ (symmetric exponential time with one bit of advice) with circuit complexity at least $2^n/n$. In particular, the above also implies the same near-maximum circuit lower bounds…

Min-plus product of two $n\times n$ matrices is a fundamental problem in algorithm research. It is known to be equivalent to APSP, and in general it has no truly subcubic algorithms. In this paper, we focus on the min-plus product on a…

Recent work by Divsalar et al. has shown that properly designed protograph-based low-density parity-check (LDPC) codes typically have minimum (Hamming) distance linearly increasing with block length. This fact rests on ensemble arguments…

We prove a Carbery-Wright style anti-concentration inequality for the unitary Haar measure, by showing that the probability of a polynomial in the entries of a random unitary falling into an $\varepsilon$ range is at most a polynomial in…

We investigate the power of the most important lower bound technique in randomized communication complexity, which is based on an evaluation of the maximal size of approximately monochromatic rectangles, minimized over all distributions on…

We prove exponential lower bounds on the size of homogeneous depth 4 arithmetic circuits computing an explicit polynomial in $VP$. Our results hold for the {\it Iterated Matrix Multiplication} polynomial - in particular we show that any…

We improve Larman's bound on the diameter of a polytope by showing that if $\Delta$ is a normal simplicial complex, all of whose missing faces have size at most $r$, then the diameter of the facet-ridge graph of $\Delta$ is not larger than…

We study the computation of the diameter and radius under the rectilinear link distance within a rectilinear polygonal domain of $n$ vertices and $h$ holes. We introduce a \emph{graph of oriented distances} to encode the distance between…

The computation of the diameter is one of the most central problems in distributed computation. In the standard CONGEST model, in which two adjacent nodes can exchange $O(\log n)$ bits per round (here $n$ denotes the number of nodes of the…