We consider the following problem. Given a graph and a rational number μ, 0 < μ ≤ 1, find a connected subgraph of density at least μ with the largest number of vertices. Here, the density of an nvertex graph with m edges is m/ ( n 2 ) . This problem arises in many application contexts such as community detection in social networks. We implement a branch and bound algorithm and tune it for effic...

We discuss optimal constants in a recent result of Rudelson and Vershynin on marginal densities. We show that if f is a probability density on R of the form f(x) = ∏n i=1 fi(xi), where each fi is a density on R, say bounded by one, then the density of any marginal πE(f) is bounded by 2 , where k is the dimension of E. The proof relies on an adaptation of Ball’s approach to cube slicing, carried...

When dealing with modern big data sets, a very common theme is reducing the set through a random process. These generally work by making “many simple estimates” of the full data set, and then judging them as a whole. Perhaps magically, these “many simple estimates” can provide a very accurate and small representation of the large data set. The key tool in showing how many of these simple estima...

As organizations accumulate data over time, the problem of tracking how patterns evolve becomes important. In this paper, we present an algorithm to track the evolution of cluster models in a stream of data. Our algorithm is based on the application of bounds derived using Chernoff’s inequality and makes use of a clustering algorithm that was previously developed by us, namely Practal Clusterin...

A universal upper bound on the MIMO architecture capacity, which is not limited to a particular scenario, is derived in this paper using the correlation matrix approach and the Jensen’s inequality. This bound accounts for both transmit and receive branch correlation in such a way that the impact of these branches can be estimated separately, which simplifies the procedure substantially and also...

We give a simple combinatorial proof of the Chernoff-Hoeffding concentration bound [Che52, Hoe63], which says that the sum of independent {0, 1}-valued random variables is highly concentrated around the expected value. Unlike the standard proofs, our proof does not use the method of higher moments, but rather uses a simple and intuitive counting argument. In addition, our proof is constructive ...

In this work, we study the tradeoffs between the error probabilities of classical-quantum channels and the blocklength n when the transmission rates approach the channel capacity at a rate slower than 1/ √ n, a research topic known as moderate deviation analysis. We show that the optimal error probability vanishes under this rate convergence. Our main technical contributions are a tight quantum...

This paper is concerned with the minimal number of profiles at which a unanimous and anonymous social choice function is manipulable. The lower bound is derived when there are three alternatives to choose from. Examples of social choice functions attaining the lower bound are given. We conjecture that these examples are in fact all minimally manipulable social choice functions. Since some of th...