A tight lower bound for malicious online bipartite matching with limited recourse budget
We study one-sided online bipartite matching with recourse. In this setting, one side of a bipartite graph is known in advance, while vertices on the other side arrive online together with their incident edges. After each arrival, the algorithm must maintain a maximum-cardinality matching while minimizing the total number of reallocations, also known as the recourse budget. Despite extensive work, the exact recourse complexity of the problem remains unsettled: the best lower bound is $\Omega(n \log n)$, whereas the best upper bound is $\mathcal{O}(n \log^2 n)$, where $n$ denotes the number of online vertices. Tight upper bounds of $\mathcal{O}(n \log n)$ are known only for restricted graph classes, such as forests. The best known upper bounds are attained by a very simple and natural algorithm SAP, which after each arrival applies a shortest augmenting path, and it is conjectured to be optimal. All known upper bound analyses of this algorithm do not depend on the particular maximum matching maintained by the algorithm. Consequently, they also apply to a more difficult problem, which we call the malicious matching setting: after each arrival, the maintained matching is replaced by a worst-case maximum matching for the next step. This led to the conjecture that the malicious setting still admits an $\mathcal{O}(n \log n)$ recourse bound, in line with the conjectured optimal complexity of the original model. Our main result is an $\Omega(n \log^2 n)$ lower bound for the malicious matching setting, thus disproving the conjecture. Together with the previous upper bound, this settles the asymptotic recourse complexity of the malicious variant of the problem. We complement our lower bound with an upper bound of $\mathcal{O}(n \log n)$ for expander graphs.