Secretary problem: graphs, matroids and greedoids

TL;DR

This paper generalizes the secretary problem using greedoids, matroids, and antimatroids to model the selection of the best k independent elements, with applications to various combinatorial structures.

Contribution

It introduces a unified framework for the secretary problem based on greedoids, extending previous models to include matroids and antimatroids.

Findings

Models the secretary problem using greedoids, matroids, and antimatroids.

Provides examples with uniform, graphical matroids, and binary trees.

Extends classical secretary problem to more complex combinatorial structures.

Abstract

In the paper the generalisation of the well known "secretary problem" is considered. The aim of the paper is to give a generalised model in such a way that the chosen set of the possible best $k$ elements have to be independent of all rejected elements. This condition is formulated using the theory of greedoids and in their special cases -- matroids and antimatroids. Examples of some special cases of greedoids (uniform, graphical matroids and binary trees) are considered.