Free-order secretary for two-sided independence systems
Kristóf Bérczi, Vasilis Livanos, José A. Soto, Victor Verdugo
TL;DR
The paper introduces a bipartite free-order secretary framework with two-sided independence constraints on agents and items, unifying classic secretary problems, bipartite matching, and matroid-based models. The core technical contributions are the development of hull, primitive hull, and core concepts, culminating in the core lemma for combinations of growth systems and its application to both edge- and agent-arrival models. The results yield $Ω(1/k^2)$-competitive algorithms for $k$-matroid intersections and broader $k$-growth systems, with extensions to combinations of systems and to the agent-arrival setting where the item side uses order-oblivious core-selecting algorithms, achieving $Ω(β/k^2)$ with a parameter $β$. The work also provides constant-competitive guarantees for multiple item selection in simple global constraints (e.g., partition matroids and $k$-matching) and introduces the $k$-growth systems as a unifying class bridging known hierarchies. Overall, the framework offers a versatile toolkit for online combinatorial assignment under two-sided independence constraints, with potential broad impact on online allocation, mechanism design, and related secretary-type problems.
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 $Ω(1/k^2)$-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 $Ω(1/k^2)$ fraction of the optimal weight. Using this lemma, we extend our $Ω(1/k^2)$-competitive algorithm to $k$-growth systems 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 $Ω(β/k^2)$-competitive algorithm for $k$-growth systems, where $β$ 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.