The Bruss-Robertson Inequality: Elaborations, Extensions, and Applications

TL;DR

This paper explores the Bruss-Robertson inequality, extending its applicability to dependent variables and diverse distributions, and reviews its use in combinatorial optimization problems like the sequential knapsack and subsequence selection.

Contribution

It introduces extensions of the Bruss-Robertson inequality that do not require independence or identical distributions of summands, broadening its theoretical and practical scope.

Findings

Extended the inequality to dependent variables

Applied the inequality to combinatorial optimization problems

Reviewed multiple applications in optimization contexts

Abstract

The Bruss-Robertson inequality gives a bound on the maximal number of elements of a random sample whose sum is less than a specified value, and the extension of that inequality which is given here neither requires the independence of the summands nor requires the equality of their marginal distributions. A review is also given of the applications of the Bruss-Robertson inequality, especially the applications to problems of combinatorial optimization such as the sequential knapsack problem and the sequential monotone subsequence selection problem.