# A Framework for the Secretary Problem on the Intersection of Matroids

## TL;DR

This paper introduces a general framework for solving the secretary problem with multiple matroid constraints, enabling constant-competitive algorithms for intersections of matroids and extending to submodular objectives.

## Contribution

It provides a unified approach to handle the intersection of multiple matroids in the secretary problem, leveraging existing algorithms and extending to submodular functions.

## Key findings

- Constant-competitive algorithms for intersections of any constant number of matroids.
- Framework combines existing matroid secretary algorithms effectively.
- Extension of results to submodular objectives.

## Abstract

The secretary problem became one of the most prominent online selection problems due to its numerous applications in online mechanism design. The task is to select a maximum weight subset of elements subject to given constraints, where elements arrive one-by-one in random order, revealing a weight upon arrival. The decision whether to select an element has to be taken immediately after its arrival. The different applications that map to the secretary problem ask for different constraint families to be handled. The most prominent ones are matroid constraints, which both capture many relevant settings and admit strongly competitive secretary algorithms. However, dealing with more involved constraints proved to be much more difficult, and strong algorithms are known only for a few specific settings. In this paper, we present a general framework for dealing with the secretary problem over the intersection of several matroids. This framework allows us to combine and exploit the large set of matroid secretary algorithms known in the literature. As one consequence, we get constant-competitive secretary algorithms over the intersection of any constant number of matroids whose corresponding (single-)matroid secretary problems are currently known to have a constant-competitive algorithm. Moreover, we show that our results extend to submodular objectives.