Almost Tight Bounds for Online Hypergraph Matching

TL;DR

This paper establishes near-tight bounds for online hypergraph matching, showing the limitations of algorithms and proposing new competitive algorithms for fractional and weighted variants.

Contribution

It provides new upper bounds on the competitive ratio for online hypergraph matching and introduces algorithms with near-optimal ratios for fractional and weighted cases.

Findings

No online algorithm can beat a 2/k competitive ratio.

A deterministic algorithm achieves (1-o(1))/ln(k) ratio for fractional matching.

A similar ratio applies to the weighted fractional case under free disposal.

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.