Tight Bounds for Online Matching in Bounded-Degree Graphs with Vertex Capacities

TL;DR

This paper establishes tight bounds for online $b$-matching in bipartite graphs with bounded degrees, showing nearly optimal solutions are achievable online as server capacities grow, improving previous competitive ratios.

Contribution

The authors develop a new primal-dual online algorithm for $b$-matching in bounded-degree bipartite graphs, achieving asymptotically optimal competitive ratios close to 1, surpassing prior bounds.

Findings

Optimal competitive ratio approaches 1 as server capacities increase

New online algorithm with primal-dual analysis improves bounds

Results extend to weighted matching and auction models

Abstract

We study the $b$-matching problem in bipartite graphs $G=(S,R,E)$. Each vertex $s\in S$ is a server with individual capacity $b_s$. The vertices $r\in R$ are requests that arrive online and must be assigned instantly to an eligible server. The goal is to maximize the size of the constructed matching. We assume that $G$ is a $(k,d)$-graph~\cite{NW}, where $k$ specifies a lower bound on the degree of each server and $d$ is an upper bound on the degree of each request. This setting models matching problems in timely applications. We present tight upper and lower bounds on the performance of deterministic online algorithms. In particular, we develop a new online algorithm via a primal-dual analysis. The optimal competitive ratio tends to~1, for arbitrary $k\geq d$, as the server capacities increase. Hence, nearly optimal solutions can be computed online. Our results also hold for the vertex-weighted problem extension, and thus for AdWords and auction problems in which each bidder issues individual, equally valued bids. Our bounds improve the previous best competitive ratios. The asymptotic competitiveness of~1 is a significant improvement over the previous factor of $1-1/e^{k/d}$, for the interesting range where $k/d\geq 1$ is small. Recall that $1-1/e\approx 0.63$. Matching problems that admit a competitive ratio arbitrarily close to~1 are rare. Prior results rely on randomization or probabilistic input models.