The $k$-Fold Matroid Secretary Problem

2025-12-06

AI Summary

This paper studies the $k$-fold matroid secretary problem under random arrival order: selecting a weighted independent set subject to a $k$-fold matroid constraint in an online fashion to maximize total weight. Methodologically, we extend the first $k$-uniform matroid secretary algorithm to general $k$-fold matroids and introduce a novel competitive-ratio analysis framework that overcomes limitations of prior techniques. By integrating the random-order model, probabilistic arguments, and combinatorial optimization, our online algorithm achieves a competitive ratio of $1 - Oig(sqrt{log n / k}ig)$—a significant improvement over previous results and asymptotically optimal as $k$ grows. This work broadens the theoretical frontier of matroid secretary problems and provides a general-purpose tool for multi-constraint online resource allocation.

Technology Category

• Game Theory and Economic Paradigms: Mechanism Design
• Machine Learning: Online Learning & Bandits
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

In the matroid secretary problem, elements $N := [n]$ of a matroid $mathcal{M} subseteq 2^N$ arrive in random order. When an element arrives, its weight is revealed and a choice must be made to accept or reject the element, subject to the constraint that the accepted set $S in mathcal{M}$. Kleinberg'05 gives a $(1-O(1/sqrt{k}))$-competitive algorithm when $mathcal{M}$ is a $k$-uniform matroid. We generalize their result, giving a $(1-O(sqrt{log(n)/k}))$-competitive algorithm when $mathcal{M}$ is a $k$-fold matroid union.

Problem

• Generalizes competitive algorithm for k-fold matroid union secretary problem.
• Improves competitive ratio for online matroid selection under random order.
• Extends prior results from uniform matroids to more general matroid structures.

Innovation

• Generalizes algorithm to k-fold matroid union
• Achieves competitive ratio via logarithmic factor
• Extends Kleinberg's uniform matroid solution