Greedy Online Bipartite Matching on Random Graphs

We study the average performance of online greedy matching algorithms on $G(n,n,p)$, the random bipartite graph with $n$ vertices on each side and edges occurring independently with probability $p=p(n)$. In the online model, vertices on one side of the graph are given up front while vertices on the other side arrive sequentially; when a vertex arrives its edges are revealed and it must be immediately matched or dropped. We begin by analyzing the \textsc{oblivious} algorithm, which tries to match each arriving vertex to a random neighbor, even if the neighbor has already been matched. The algorithm is shown to have a performance ratio of at least $1-1/e$ for all monotonic functions $p(n)$, where the performance ratio is defined asymptotically as the ratio of the expected matching size given by the algorithm to the expected maximum matching size. Next we show that the conventional \textsc{greedy} algorithm, which assigns each vertex to a random unmatched neighbor, has a performance ratio of at least 0.837 for all monotonic functions $p(n)$. Under the $G(n,n,p)$ model, the performance of \textsc{greedy} is equivalent to the performance of the well known \textsc{ranking} algorithm, so our results show that \textsc{ranking} has a performance ratio of at least 0.837. We finally consider vertex-weighted bipartite matching. Our proofs are based on simple differential equations that describe the evolution of the matching process.