Almost Tight Bounds for Online Hypergraph Matching

TL;DR

This work studies online hypergraph matching where hyperedges of size $k$ arrive adversarially and must be irrevocably either accepted or rejected. It establishes near-tight bounds for the integral version, showing a fundamental limit at $(2+o(1))/k$ for randomized online algorithms, and a robust baseline of $1/k$ for deterministic schemes; for the fractional version, it presents a $$(1-o(1))/\ln(k)$$-competitive online algorithm and proves a matching $(1+o(1))/\ln(k)$ hardness, effectively closing the asymptotic gap. The authors introduce a Water-Filling style algorithm for fractional matching, supported by primal-dual analysis, and extend the approach to weighted fractional matching under free disposal. These results distinguish the fractional and integral settings, offer near-optimal strategies in the online, adversarial setting, and identify key open questions, including whether the $1/k$ barrier in the integral case can be surpassed for larger $k$ or under additional arrival models. Overall, the paper advances our understanding of competitive ratios in online packing problems on hypergraphs and informs efficient resource allocation under sequential demands.

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 naïve 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.

Figures (2)

• Figure 1: Shown is gadget $G_{10}$ proving that a competitive ratio of $\frac{4}{k} + \epsilon$ is imposisble for $k = 10$. The numbers indicate in which phase each edge was added. The lightly shaded areas represent the vertex sets $A_1, \ldots, A_5$ which are useful for the construction of $H_k$.
• Figure 2: Shown is the upper-bounding construction with $k = 10$, $l = 3$, $\delta = 0.5$. In each step we replace the blue edges with as many red edges of $\frac{1}{1 + \delta}$ times the size as possible. Then we pick the $l$ red edges that the algorithm puts the most weight on, make those the new blue edges and repeat until only singleton edges are left.

Theorems & Definitions (15)

• Theorem : Informal
• Theorem : Informal
• theorem thmcountertheorem: Folklore
• proof
• lemma thmcounterlemma: Yao's Principle
• theorem thmcountertheorem
• proof
• theorem thmcountertheorem
• proof : Proof Sketch
• theorem thmcountertheorem