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arXiv · 2511.04390

Free-Order Online Selection for k-Systems

Kristóf Bérczi·Vasilis Livanos·José A. Soto·Victor Verdugo
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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 bipartite matching, matroid intersection, and matroid secretary problems. In this model, agents and items form a bipartite graph, and the objective is to select a matching that satisfies independence constraints on both sides. We first study the free-order setting under edge-arrivals. For $k$-matroid intersection, we leverage a core lemma by [FSZ, 2022] to design an $Ω(1/k^2)$-competitive algorithm, extending known results for single matroids. Building on this, we introduce $k$-growth systems -- a new class of independence systems that lie properly between $k$-matchoids and $k$-extendible systems and may be of independent combinatorial interest. We establish a generalized core lemma for $k$-growth systems, showing that a suitably defined set of critical elements retains a $Ω(1/k^2)$ fraction of the optimal weight. Using this lemma, we extend our $Ω(1/k^2)$-competitive algorithm to $k$-growth systems. 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 $Ω(β/k^2)$-competitive algorithm for $k$-growth systems, where $β$ denotes the competitiveness of an appropriate type of order-oblivious algorithm for the item-side constraint. Finally, we extend our results to the case of multiple item selection, and obtain constant-competitive algorithms for fundamental cases such as partition matroids and $k$-matching constraints. We also study the closure properties and structural role and of $k$-growth systems within the hierarchy of $k$-systems.

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BibTeXRIS

Kristóf Bérczi, Vasilis Livanos, José A. Soto, Victor Verdugo. 2026-07-15. Free-Order Online Selection for k-Systems. https://arxiv.org/abs/2511.04390

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