The Multiple-Choice Matroid Secretary Problem

2026-07-16

AI Summary

This work addresses the multi-choice matroid secretary problem ((J,κ)-MSP), where elements arrive online in random order and the goal is to select a maximum-weight independent set under both multiple-selection and global cardinality constraints. The paper innovatively decouples online storage from the final solution, introducing a multi-track algorithmic framework and modeling the selection process via a union of matroids, combined with a single-threshold routing strategy. It establishes the first systematic theoretical foundation for this problem: when capacity constraints are absent, it proves the existence of an exact optimal algorithm for transversal matroids, whose competitive ratio matches the success probability of the classical J-choice secretary problem; it identifies rank-one instances as worst-case scenarios; and it derives performance guarantees—exponentially converging to 1 for transversal matroids, and approximation ratios of \(1 - O(e^{-J/(k e)})\) for k-column-sparse matroids and \(1 - O(e^{-J/e})\) for laminar matroids—while quantifying the rank loss induced by global capacity and its coupling with parameters \(b\), \(J\), and \(\kappa\).

Abstract

We introduce and study the multiple-choice matroid secretary problem, denoted $(J,κ)$-MSP. For rank-one matroids and $κ=\infty$, it reduces to the classical secretary problem with $J$ choices. Elements arrive in uniformly random order. Algorithms may keep a candidate pool $\mathrm{AUX}$ feasible in the $J$-fold union matroid $\mathcal{M}^{(J)}$ satisfying $|\mathrm{AUX}|\le κ\cdot\mathrm{rank}(\mathcal{M})$. Finally, one extracts the maximum-weight independent subset of $\mathrm{AUX}$ in $\mathcal{M}$. This model separates online storage from the final feasible solution. We study two multiple-choice implementations: multi-track algorithms (maintaining $J$ independent sets of $\mathcal{M}$) and union-based algorithms (maintaining the pool directly in $\mathcal{M}^{(J)}$). Our main result is an exact optimal algorithm for transversal matroids in the uncapacitated $(J,\infty)$ setting. For fixed $J$, its probability-competitive ratio equals the optimal success probability of the classical $J$-choice secretary problem. Thus, rank-one instances are the worst case for the whole transversal class, and the optimal guarantee converges exponentially fast to $1$ as $J$ grows. We also analyze a simple single-threshold routing algorithm for capacitated transversal matroids with local capacities $b$ and global capacity $κ\cdot\mathrm{rank}(\mathcal{M})$. Its analysis provides explicit finite-parameter bounds and asymptotic formulas, showing how finite-rank loss caused by global capacity decays, and how $b$, $J$, and $κ$ interact. Finally, we instantiate the multi-track approach for $k$-column-sparse matroids (guarantee $1-O(e^{-J/(ke)})$) and the union-based approach for laminar matroids (guarantee $1-O(e^{-J/e})$).

Problem

• Multiple-Choice
• Matroid Secretary Problem
• Online Selection
• Transversal Matroids
• Competitive Ratio

Innovation

• matroid secretary problem
• multiple-choice selection
• transversal matroids
• competitive ratio
• online algorithms