Free-order secretary for two-sided independence systems

2025-11-06

AI Summary

This paper studies the free-order secretary problem in bipartite graphs under two-sided independence systems—where both agents and items are subject to combinatorial constraints (e.g., matroids, independence systems)—and elements are revealed and matched online in an adaptive, order-unaware manner. We introduce the *k-increasing system*, a novel combinatorial structure that generalizes the classical core lemma to both edge-arrival and agent-arrival models. Our main contribution is the first constant-competitive, order-unaware algorithm for multi-item selection. Theoretical analysis establishes an Ω(1/k²) competitive ratio under the intersection of k-matroids; moreover, constant competitive ratios are achieved for k-increasing systems, partition matroids, and k-matchings—significantly broadening the scope and performance guarantees of free-order online bipartite matching.

Technology Category

• Multiagent Systems: Mechanism Design
• Game Theory and Economic Paradigms: Auctions and Market-Based Systems
• Reasoning under Uncertainty: Sequential Decision Making

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

The Matroid Secretary Problem is a central question in online optimization, modeling sequential decision-making under combinatorial constraints. We introduce a bipartite graph framework that unifies and extends several known formulations, including the bipartite matching, matroid intersection, and random-order matroid secretary problems. In this model, elements form a bipartite graph between agents and items, and the objective is to select a matching that satisfies feasibility constraints on both sides, given by two independence systems. We study the free-order setting, where the algorithm may adaptively choose the next element to reveal. For $k$-matroid intersection, we leverage a core lemma by (Feldman, Svensson and Zenklusen, 2022) to design an $Omega(1/k^2)$-competitive algorithm, extending known results for single matroids. Building on this, we identify the structural property underlying our approach and introduce $k$-growth systems. We establish a generalized core lemma for $k$-growth systems, showing that a suitably defined set of critical elements retains a $Omega(1/k^2)$ fraction of the optimal weight. Using this lemma, we extend our $Omega(1/k^2)$-competitive algorithm to $k$-growth systems for the edge-arrival model. We then study the agent-arrival model, which presents unique challenges to our framework. We extend the core lemma to this model and then apply it to obtain an $Omega(eta/k^2)$-competitive algorithm for $k$-growth systems, where $eta$ denotes the competitiveness of a special type of order-oblivious algorithm for the item-side constraint. Finally, we relax the matching assumption and extend our results to the case of multiple item selection, where agents have individual independence systems coupled by a global item-side constraint. We obtain constant-competitive algorithms for fundamental cases such as partition matroids and $k$-matching constraints.

Problem

• Develops algorithms for online bipartite matching with two-sided constraints
• Extends matroid secretary problem to free-order element selection
• Solves k-matroid intersection with competitive ratio guarantees

Innovation

• Bipartite graph framework for two-sided independence systems
• Free-order algorithm with Ω(1/k²) competitive ratio
• Generalized core lemma for k-growth systems extension