Degree-bounded Online Bipartite Matching: OCS vs. Ranking

TL;DR

The paper investigates online bipartite matching on degree-bounded graphs, focusing on Ranking and Online Correlated Selection (OCS) in the (d,d)-bounded setting. It proves a consistent performance advantage for OCS over Ranking across all fixed d≥2, providing tight upper bounds for Ranking (e.g., 0.8264 at d=2 and 0.8161 as d→∞) and strong per-vertex lower bounds for OCS (0.8352 for small d and 0.8976 as d→∞). The main technical contributions include a Markov-chain-based analysis of Ranking, a general and a refined small-d hard-instance construction, and a candidate-function framework that enables computing an optimal f* in O(d^2), yielding a competitive ratio of 1 − 1/f*(d) for OCS. The results extend to (k,d)-bounded graphs and highlight a principled separation between two canonical online matching paradigms, with implications for designing robust online algorithms under degree constraints.

Abstract

We revisit the online bipartite matching problem on $d$-regular graphs, for which Cohen and Wajc (SODA 2018) proposed an algorithm with a competitive ratio of $1-2\sqrt{H_d/d} = 1-O(\sqrt{(\log d)/d})$ and showed that it is asymptotically near-optimal for $d=ω(1)$. However, their ratio is meaningful only for sufficiently large $d$, e.g., the ratio is less than $1-1/e$ when $d\leq 168$. In this work, we study the problem on $(d,d)$-bounded graphs (a slightly more general class of graphs than $d$-regular) and consider two classic algorithms for online matching problems: \Ranking and Online Correlated Selection (OCS). We show that for every fixed $d\geq 2$, the competitive ratio of OCS is at least $0.835$ and always higher than that of \Ranking. When $d\to \infty$, we show that OCS is at least $0.897$-competitive while \Ranking is at most $0.816$-competitive. We also show some extensions of our results to $(k,d)$-bounded graphs.

Figures (9)

• Figure 1: The comparison of competitive ratios of OCS, Ranking, High-Degreejournals/teco/NaorW18 and Markingconf/soda/CohenW18 when $d \leq 200$ and $d\leq 20$, respectively, where L.B. and U.B. stand for lower bound and upper bound. The general upper bound is from conf/soda/CohenW18, who analyzed a hard instance when $d\to \infty$. In Appendix \ref{['sec:problem-hardness']} we refine their analysis to give an upper bound for every $d\geq 2$.
• Figure 2: Comparing OCS and Ranking on a simple example graph, where the number next to an edge represents the probability that the edge is selected by the algorithm, conditioned on both offline neighbors being available when the corresponding online request arrives.
• Figure 3: Hard instance for Ranking: an illustrating example with $d=5$.
• Figure 4: An Markov chain based illustration of $X(i,j)$.
• Figure 5: Hard instance for Ranking: an illustrating example when $d=3$.

Theorems & Definitions (38)

• Definition 2.1: $(k,d)$-bounded graphs
• Theorem 3.1
• Corollary 3.2
• Definition 3.3
• Lemma 3.4
• Lemma 3.5
• proof
• Definition 3.6
• Lemma 3.7
• Definition 4.1: Candidate Function