Secretary Matching with General Arrivals

TL;DR

This paper develops online algorithms for secretary matching in general weighted graphs under vertex and edge arrival models, achieving improved competitive ratios and demonstrating that vertex arrival can outperform bipartite models.

Contribution

It introduces the first $5/12$-competitive algorithm for vertex arrival secretary matching in general graphs and a $1/4$-competitive algorithm for edge arrival, surpassing previous bounds.

Findings

Vertex arrival algorithm is tight at 5/12 competitiveness.

Edge arrival algorithm achieves 1/4 competitiveness.

Vertex arrival in general graphs outperforms bipartite models with 1/e guarantee.

Abstract

We provide online algorithms for secretary matching in general weighted graphs, under the well-studied models of vertex and edge arrivals. In both models, edges are associated with arbitrary weights that are unknown from the outset, and are revealed online. Under vertex arrival, vertices arrive online in a uniformly random order; upon the arrival of a vertex $v$, the weights of edges from $v$ to all previously arriving vertices are revealed, and the algorithm decides which of these edges, if any, to include in the matching. Under edge arrival, edges arrive online in a uniformly random order; upon the arrival of an edge $e$, its weight is revealed, and the algorithm decides whether to include it in the matching or not. We provide a $5/12$-competitive algorithm for vertex arrival, and show it is tight. For edge arrival, we provide a $1/4$-competitive algorithm. Both results improve upon state of the art bounds for the corresponding settings. Interestingly, for vertex arrival, secretary matching in general graphs outperforms secretary matching in bipartite graphs with 1-sided arrival, where $1/e$ is the best possible guarantee.