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.

翻译：在在线超图匹配问题中，大小为$k$的超边在公共基集上以对抗顺序在线到达。目标是获得最大匹配（不相交的超边集）。针对该问题的朴素贪心算法实现了$\frac{1}{k}$的竞争比。我们证明，没有（随机化）在线算法能够获得优于$\frac{2+o(1)}{k}$的竞争比。如果允许边分数分配，我们给出一个确定性在线算法，其竞争比为$\frac{1-o(1)}{\ln(k)}$，并证明没有任何在线算法的竞争比能严格优于$\frac{1+o(1)}{\ln(k)}$。最后，在自由处置假设下，我们针对分数边加权版本的问题给出了一个$\frac{1-o(1)}{\ln(k)}$竞争算法。