Almost Tight Bounds for Online Hypergraph Matching

Abstract: In the online hypergraph matching problem, hyperedges of size $k$ over a common ground set arrive online in adversarial order. The goal is to obtain a maximum matching (disjoint set of hyperedges). A na\"ive greedy algorithm for this problem achieves a competitive ratio of $\frac{1}{k}$. We show that no (randomized) online algorithm has competitive ratio better than $\frac{2+o(1)}{k}$. If edges are allowed to be assigned fractionally, we give a deterministic online algorithm with competitive ratio $\frac{1-o(1)}{\ln(k)}$ and show that no online algorithm can have competitive ratio strictly better than $\frac{1+o(1)}{\ln(k)}$. Lastly, we give a $\frac{1-o(1)}{\ln(k)}$ competitive algorithm for the fractional edge-weighted version of the problem under a free disposal assumption.