Improved Bounds for Online Matching with Random Edge Arrivals

We study the unweighted online matching problem with random edge arrivals, where each edge arrives at an independent time drawn uniformly from $[0,1]$, and the algorithm must immediately and irrevocably accept or reject it while maintaining a matching. Guruganesh and Singla (IPCO, 2017) first broke the $1/2$ barrier of the naive Greedy algorithm by giving a (1/2+ε)-competitive algorithm for a small constant ε>0, a guarantee that has remained the best known for nearly a decade. We give a simple three-phase algorithm that achieves a competitive ratio of $0.5445$ by deliberately rejecting selected early edges to reserve their endpoints for later matching opportunities. With a different choice of parameters, the same algorithm is $0.5535$-competitive on triangle-free graphs, including bipartite graphs. We also establish competitive ratio upper bounds below $0.6964$ for general graphs and $0.7043$ for bipartite graphs via exact rational evaluation of an optimal-policy recurrence on fixed graphs, even when the input graph is known up to isomorphism.