arXiv subjects

Kevin P. Costello

Publications and source records attributed to Kevin P. Costello.

6 recordsLinked to original sources

arXiv2022

Avoiding Monochromatic Solutions to 3-term Equations

Given an equation, the integers $[n] = \{1, 2, \dots, n\}$ as inputs, and the colors red and blue, how can we color $[n]$ in order to minimize the number of monochromatic solutions to the equation, and what is the minimum? The answer is only known for a handful of equations, but much progress has been made on improving upper and lower bounds on minima for various equations. A well-studied characteristic an equation, which has its roots in graph Ramsey theory, is to determine if the minimum number of monochromatic solutions can be achieved (asymptotically) by uniformly random colorings. Such equations are called common. We prove that no 3-term equations are common and provide a lower bound for a specific class of 3-term equations.

math.CO↗

arXiv2015

Faster Information Gathering in Ad-Hoc Radio Tree Networks

We study information gathering in ad-hoc radio networks. Initially, each node of the network has a piece of information called a rumor, and the overall objective is to gather all these rumors in the designated target node. The ad-hoc property refers to the fact that the topology of the network is unknown when the computation starts. Aggregation of rumors is not allowed, which means that each node may transmit at most one rumor in one step. We focus on networks with tree topologies, that is we assume that the network is a tree with all edges directed towards the root, but, being ad-hoc, its actual topology is not known. We provide two deterministic algorithms for this problem. For the model that does not assume any collision detection nor acknowledgement mechanisms, we give an $O(n\log\log n)$-time algorithm, improving the previous upper bound of $O(n\log n)$. We also show that this running time can be further reduced to $O(n)$ if the model allows for acknowledgements of successful transmissions.

cs.DS↗

arXiv2009

Concentration of random determinants and permanent estimators

We show that the absolute value of the determinant of a matrix with random independent (but not necessarily iid) entries is strongly concentrated around its mean. As an application, we show that the Godsil-Gutman and Barvinok estimators for the permanent of a strictly positive matrix give sub-exponential approximation ratios with high probability.

math.PR↗

arXiv2009

Bilinear and Quadratic Variants on the Littlewood-Offord Problem

If f(x_1, x_2, ..., x_n) is a polynomial dependent on a large number of independent Bernoulli random variables, what can be said about the maximum concentration of f on any single value? For linear polynomials, this reduces to one version of the classical Littlewood-Offord problem: Given nonzero constants a_1 through a_n, what is the maximum number of sums of the form +/- a_1 +/- a_2 +/-... +/- a_n which take on any single value? Here we consider the case where f is either a bilinear form or a quadratic form. For the bilinear case, we show that the only forms having concentration significantly larger than n^{-1} are those which are in a certain sense very close to being degenerate. For the quadratic case, we show that no form having many nonzero coefficients has concentration significantly larger than n^{-1/2}. In both cases the results are nearly tight.

math.PR↗

arXiv2007

On the Rank of Random Sparse Matrices

We investigate the rank of random (symmetric) sparse matrices. Our main finding is that with high probability, any dependency that occurs in such a matrix is formed by a set of few rows that contains an overwhelming number of zeros. This allows us to obtain an exact estimate for the co-rank.

math.PR↗

arXiv2006

The Rank of Random Graphs

We show that almost surely the rank of the adjacency matrix of the Erdös-Rényi random graph $G(n,p)$ equals the number of non-isolated vertices for any $c\ln n/n<p<1/2$, where $c$ is an arbitrary positive constant larger than 1/2. In particular, the giant component (a.s.) has full rank in this range.

math.PR↗