arXiv · 1801.03365
We discuss five ways of proving Chernoff's bound and show how they lead to different extensions of the basic bound.
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Wolfgang Mulzer. 2019-05-02. Five Proofs of Chernoff's Bound with Applications. https://arxiv.org/abs/1801.03365
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We consider the two-machine open-shop scheduling problem in which one machine is unavailable during a fixed interval. We study the resumable setting: an operation interrupted by the unavailability interval may resume, without penalty, when the machine becomes available. The objective is to minimize the makespan. Although the problem is NP-hard and several approximation algorithms are known, whether it admits a fully polynomial-time approximation scheme (FPTAS) has remained open for two decades. We resolve this question affirmatively by giving the first FPTAS, thereby strengthening the previously known polynomial-time approximation scheme (PTAS). As an intermediate result, we develop a new pseudo-polynomial dynamic program with seven state dimensions, improving on the ten-dimensional formulation in the literature.
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Let $X$ be a finite set with $|X|=n$ and let $\mathrm{Jac}(a,b)=|a\,\triangle\, b|/|a\cup b|$ be the Jaccard distance on the power set $2^X$. Lladser and Paradise recently proved that the metric dimension of $(2^X,\mathrm{Jac})$ is $\Theta(n/\ln n)$, with the constant left open; their bounds are $(\ln 2)\,n/\ln n\lesssim \beta(2^X,\mathrm{Jac})\lesssim 2\ln(2e)\,n/\ln n$. We determine the constant: \[ \beta(2^X,\mathrm{Jac})=\frac{2n}{\log_2 n}\,(1+o(1))=(2\ln 2)\,\frac{n}{\ln n}\,(1+o(1)). \] The proof identifies the problem, on each ``slice'' of subsets of fixed cardinality, with the Erd\H{o}s--R\'enyi coin-weighing problem for a spring scale (the problem of \emph{detecting matrices}). The lower bound is the Erd\H{o}s--R\'enyi entropy argument applied to the middle slice; the upper bound follows from the explicit detecting families of Lindstr\"om and of Cantor and Mills, augmented by a single extra landmark that reveals cardinality.