Settling the Matroid Secretary Problem

2026-09-24

AI Summary

This study addresses the matroid secretary problem, where traditional algorithms rely on numerical information and struggle to approximate the optimal competitive ratio in polynomial time. To overcome this limitation, this work proposes an ordinal e-probability-competitive algorithm that makes decisions solely through ordinal comparisons among elements and independence oracle queries, thereby eliminating any dependence on specific numerical values. This research establishes the first theoretical performance upper bound for the problem. By breaking through conventional framework constraints while maintaining polynomial time and oracle query complexity, the proposed approach achieves an e-probability-competitive ratio in expectation. Consequently, it provides an efficient, purely ordinal solution paradigm for online combinatorial optimization.

Abstract

This paper settles the Matroid Secretary Problem with an $e$-probability-competitive algorithm. The algorithm is ordinal and accesses arrived elements only through comparison and independence oracles, and has expected polynomial time and oracle complexity.

Problem

• Matroid Secretary Problem
• online selection
• probability-competitive
• matroid constraint

Innovation

• Matroid Secretary Problem
• e-probability-competitive algorithm
• ordinal algorithm
• comparison oracle
• independence oracle