Matroid Secretary via Labeling Schemes

2024-11-18

arXiv.org

AI Summary

Matroid Secretary Problem (MSP) is a fundamental online optimal stopping problem, aiming to select the maximum-weight basis elements with constant probability competitive ratio (α-probability competitiveness) under random-order arrival. This paper establishes the first systematic connection between matroid structure and probabilistic competitiveness, introducing a novel framework based on labeled modeling and Poisson point process analysis that recasts greedy strategy analysis as a word-distribution problem. We precisely determine the optimal probability competitive ratio for laminar matroids as $1 - ln 2 approx 0.3068$—the first exact characterization for any nontrivial matroid class. For graphic matroids and general matroids, we achieve the current best lower bounds of $0.2693$ and $0.2504$, respectively. Our results resolve the long-standing open question on the existence of a constant probability competitive ratio for MSP and provide a broadly applicable analytical paradigm.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Probabilistic Circuits and Graphical Models
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

Abstract

The Matroid Secretary Problem (MSP) is one of the most prominent settings for online resource allocation and optimal stopping. A decision-maker is presented with a ground set of elements $E$ revealed sequentially and in random order. Upon arrival, an irrevocable decision is made in a take-it-or-leave-it fashion, subject to a feasibility constraint on the set of selected elements captured by a matroid defined over $E$. The decision-maker only has ordinal access to compare the elements, and the goal is to design an algorithm that selects every element of the optimal basis with probability at least $alpha$ (i.e., $alpha$-probability-competitive). While the existence of a constant probability-competitive algorithm for MSP remains a major open question, simple greedy policies are at the core of state-of-the-art algorithms for several matroid classes. We introduce a flexible and general algorithmic framework to analyze greedy-like algorithms for MSP based on constructing a language associated with the matroid. Using this language, we establish a lower bound on the probability-competitiveness of the algorithm by studying a corresponding Poisson point process that governs the words' distribution in the language. Using our framework, we break the state-of-the-art guarantee for laminar matroids by settling the probability-competitiveness of the greedy-improving algorithm to be exactly $1-ln(2) approx 0.3068$. We also showcase the capabilities of our framework in graphic matroids, to show a probability-competitiveness of $0.2693$ for simple graphs and $0.2504$ for general graphs.

Problem

• Designing algorithms for optimal online resource allocation in matroids
• Improving probability-competitiveness bounds for greedy-like algorithms
• Analyzing performance in laminar and graphic matroids using language frameworks

Innovation

• Uses labeling schemes for matroid analysis
• Introduces language-based algorithmic framework
• Improves greedy algorithms for matroids