Formal Barriers to Simple Algorithms for the Matroid Secretary Problem

TL;DR

This paper investigates the fundamental limitations of certain algorithmic frameworks in solving the matroid secretary problem, showing that some natural classes of algorithms cannot achieve the conjectured constant-competitiveness.

Contribution

It introduces the first impossibility results for broad classes of algorithms, highlighting inherent barriers in resolving the matroid secretary conjecture.

Findings

Impossibility results for greedy algorithms.

Impossibility results for randomized partition algorithms.

Limits of existing algorithmic frameworks for the problem.

Abstract

Babaioff et al. [BIK2007] introduced the matroid secretary problem in 2007, a natural extension of the classic single-choice secretary problem to matroids, and conjectured that a constant-competitive online algorithm exists. The conjecture still remains open despite substantial partial progress, including constant-competitive algorithms for numerous special cases of matroids, and an $O(\log \log \text{rank})$-competitive algorithm in the general case. Many of these algorithms follow principled frameworks. The limits of these frameworks are previously unstudied, and prior work establishes only that a handful of particular algorithms cannot resolve the matroid secretary conjecture. We initiate the study of impossibility results for frameworks to resolve this conjecture. We establish impossibility results for a natural class of greedy algorithms and for randomized partition algorithms, both of which contain known algorithms that resolve special cases.