arXiv subjects
Publications and source records attributed to Ravi B. Boppana.
We provide a simple proof of a curious inequality for the binary entropy function, an inequality that has been used in two different contexts. In the 1980's, Boppana used this entropy inequality to prove lower bounds on Boolean formulas. More recently, the inequality was used to achieve major progress on Frankl's union-closed sets conjecture. Our proof of the entropy inequality uses basic differential calculus.
We consider a two-player search game on a tree $T$. One vertex (unknown to the players) is randomly selected as the target. The players alternately guess vertices. If a guess $v$ is not the target, then both players are informed in which subtree of $T \smallsetminus v$ the target lies. The winner is the player who guesses the target. When both players play optimally, we show that each of them wins with probability approximately $1/2$. When one player plays optimally and the other plays randomly, we show that the player with the optimal strategy wins with probability between $9/16$ and $2/3$ (asymptotically). When both players play randomly, we show that each wins with probability between $13/30$ and $17/30$ (asymptotically).
Borwein, Bailey, and Girgensohn (2004) asked whether the following infinite series converges: the sum of $(\frac{2}{3} + \frac{1}{3} \sin n)^n / n$ over all positive integers $n$. We prove that their series converges. The proof uses the irrationality measure of $π$.
Let $v_1$, $v_2$, ..., $v_n$ be real numbers whose squares add up to 1. Consider the $2^n$ signed sums of the form $S = \sum \pm v_i$. Boppana and Holzman (2017) proved that at least 13/32 of these sums satisfy $|S| \le 1$. Here we improve their bound to $0.427685$.
We bound the performance guarantees that follow from Turán-like bounds for unweighted and weighted independent sets in bounded-degree graphs. In particular, a randomized approach of Boppana forms a simple 1-round distributed algorithm, as well as a streaming and preemptive online algorithm. We show it gives a tight $(Δ+1)/2$-approximation in unweighted graphs of maximum degree $Δ$, which is best possible for 1-round distributed algorithms. For weighted graphs, it gives only a $Δ$-approximation, but a simple modification results in an asymptotic expected $0.529 Δ$-approximation. This compares with a recent, more complex $Δ$-approximation~\cite{BCGS17}, which holds deterministically.
Let $v_1$, $v_2$, ..., $v_n$ be real numbers whose squares add up to 1. Consider the $2^n$ signed sums of the form $S = \sum \pm v_i$. Holzman and Kleitman (1992) proved that at least 3/8 of these sums satisfy $|S| \le 1$. This 3/8 bound seems to be the best their method can achieve. Using a different method, we improve the bound to 13/32, thus breaking the 3/8 barrier.