Online Matroid Embeddings

2024-07-14

arXiv.org

AI Summary

This paper studies the online matroid embedding problem—how to embed an unknown matroid, whose structure is revealed incrementally over time, into a known larger matroid, thereby reducing online secretary problems under unknown constraints to those under known matroid constraints. Method: We formally introduce the notion of *online matroid embedding*, analyze exact embeddability for binary and laminar matroids, and establish impossibility for general matroids. We propose an *approximate embedding* framework and tightly characterize the optimal approximation ratio for embedding binary matroids into partition matroids (with matching upper and lower bounds). Results: We demonstrate constant-factor equivalence among diverse secretary problems over binary matroids—including settings with unknown/known structures and pairwise-independent weight distributions. Our work establishes sharp existence boundaries for online embeddings and introduces a novel reduction paradigm for online selection under unknown constraints.

Technology Category

• Machine Learning: Online Learning & Bandits
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Economics, Online Markets and Human Computation: Sustainability of Web economics
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We introduce the notion of an online matroid embedding, which is an algorithm for mapping an unknown matroid that is revealed in an online fashion to a larger-but-known matroid. The existence of such embedding enables a reduction from the version of the matroid secretary problem where the matroid is unknown to the version where the matroid is known in advance. We show that online matroid embeddings exist for binary (and hence graphic) and laminar matroids. We also show a negative result showing that no online matroid embedding exists for the class of all matroids. Finally, we define the notion of an approximate matroid embedding, generalizing the notion of {alpha}-partition property, and provide an upper bound on the approximability of binary matroids by a partition matroid, matching the lower bound of Dughmi et al.

Problem

• Develops online algorithms for embedding unknown matroids into known ones
• Establishes equivalences between different binary matroid secretary problem variants
• Connects adversarial weight settings with pairwise-independent distributions

Innovation

• Online embedding maps unknown matroids to known ones
• Binary matroid embedding enables secretary problem equivalences
• Connects adversarial and pairwise-independent weight distributions