Semi-Streaming Algorithms for Hypergraph Matching

TL;DR

This paper introduces two novel one-pass streaming algorithms for hypergraph matching, offering improved approximation guarantees and reduced memory usage compared to naive methods, with practical implementation and testing on diverse hypergraphs.

Contribution

The paper presents two new streaming algorithms for hypergraph matching with provable approximation guarantees and efficient memory usage, advancing the state-of-the-art in streaming hypergraph algorithms.

Findings

Streaming algorithms outperform naive methods in solution quality.

Memory consumption is significantly reduced, by a factor of 13 on average.

Algorithms outperform offline greedy in running time.

Abstract

We propose two one-pass streaming algorithms for the $\mathcal{NP}$-hard hypergraph matching problem. The first algorithm stores a small subset of potential matching edges in a stack using dual variables to select edges. It has an approximation guarantee of $\frac{1}{d(1+\varepsilon)}$ and requires $\mathcal{O}((\frac{n}{\varepsilon}) \log^2{n})$ bits of memory, where $n$ is the number of vertices in the hypergraph, $d$ is the maximum number of vertices in a hyperedge, and $\epsilon > 0$ is a parameter to be chosen. The second algorithm computes, stores, and updates a single matching as the edges stream, with an approximation ratio dependent on a parameter $\alpha$. Its best approximation guarantee is $\frac{1}{(2d-1) + 2 \sqrt{d(d-1)}}$, and it requires only $\mathcal{O}(n)$ memory. We have implemented both algorithms and compared them with respect to solution quality, memory consumption, and running times on two diverse sets of hypergraphs with a non-streaming greedy and a naive streaming algorithm. Our results show that the streaming algorithms achieve much better solution quality than naive algorithms when facing adverse orderings. Furthermore, these algorithms reduce the memory required by a factor of 13 in the geometric mean on our test problems, and also outperform the offline Greedy algorithm in running time.