Fractional Fully Online Matching

Abstract

This paper studies fractional matching on general graphs in the fully online model of Huang et al. (JACM 2020), in which all vertices arrive online and remain available for only a limited time. The algorithm must make irrevocable fractional matching decisions while the relevant vertices are simultaneously available. We extend the classic Water-Filling algorithm, also known as Balance and originally introduced by Kalyanasundaram and Pruhs (TCS 2000), to the fully online setting. Using an online primal-dual framework, we prove that the generalized Water-Filling algorithm achieves a competitive ratio of $2-\sqrt{2}\approx 0.586$ in the fully online model, and that this analysis is tight. To surpass the $2-\sqrt{2}$ barrier, we incorporate the ideas of eager matching and history-based pricing into Water-Filling. We show that the resulting algorithm achieves an improved competitive ratio of $0.599$, thereby establishing that Water-Filling is not optimal in the fully online setting. On the hardness side, we further improve the known upper bound for fractional fully online matching, reducing the previous best bound of $0.6297$ due to Eckl et al. (ORL 2021) to $0.6132$.