Improved algorithms and analysis for the laminar matroid secretary problem

TL;DR

This paper introduces a new algorithm for the laminar matroid secretary problem that achieves a provably constant competitive ratio, advancing the understanding of online selection under matroid constraints.

Contribution

The paper presents the first constant-competitive algorithm specifically designed for the laminar matroid secretary problem.

Findings

Achieves a 0.053 competitive ratio for the laminar matroid secretary problem.

Provides analysis and proof of the algorithm's competitive performance.

Extends the class of matroid secretary problems with known constant-competitive algorithms.

Abstract

In a matroid secretary problem, one is presented with a sequence of objects of various weights in a random order, and must choose irrevocably to accept or reject each item. There is a further constraint that the set of items selected must form an independent set of an associated matroid. Constant-competitive algorithms (algorithms whose expected solution weight is within a constant factor of the optimal) are known for many types of matroid secretary problems. We examine the laminar matroid and show an algorithm achieving provably 0.053 competitive ratio.