The Bruss-Robertson Inequality: Elaborations, Extensions, and Applications
J. Michael Steele

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.
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