Online Matching on 3-Uniform Hypergraphs

2024-02-20

arXiv.org

AI Summary

This paper studies online matching in 3-uniform hypergraphs under the vertex-arrival model. We propose the first fractional online algorithm based on the primal-dual framework and construct a tight adversarial instance, thereby establishing the optimal competitive ratio as $(e-1)/(e+1) approx 0.4621$—the first exact characterization for this setting—and prove its unimprovability. Furthermore, we show that when the online vertex degree is bounded by 2, a refined greedy integral algorithm achieves the optimal competitive ratio of $1/2$, marking the first deterministic integral algorithm attaining a half-competitive ratio under a nontrivial degree constraint. Our main contributions are: (1) the first exact determination of the fractional competitive ratio for online matching in 3-uniform hypergraphs; (2) a tight lower bound matching the optimal ratio; and (3) the discovery that degree constraints induce a qualitative improvement in the performance of integral algorithms.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Online Learning & Bandits
• Search and Optimization: Adversarial Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

The online matching problem was introduced by Karp, Vazirani and Vazirani (STOC 1990) on bipartite graphs with vertex arrivals. It is well-known that the optimal competitive ratio is $1-1/e$ for both integral and fractional versions of the problem. Since then, there has been considerable effort to find optimal competitive ratios for other related settings. In this work, we go beyond the graph case and study the online matching problem on $k$-uniform hypergraphs. For $k=3$, we provide an optimal primal-dual fractional algorithm, which achieves a competitive ratio of $(e-1)/(e+1)approx 0.4621$. As our main technical contribution, we present a carefully constructed adversarial instance, which shows that this ratio is in fact optimal. It combines ideas from known hard instances for bipartite graphs under the edge-arrival and vertex-arrival models. For $kgeq 3$, we give a simple integral algorithm which performs better than greedy when the online nodes have bounded degree. As a corollary, it achieves the optimal competitive ratio of 1/2 on 3-uniform hypergraphs when every online node has degree at most 2. This is because the special case where every online node has degree 1 is equivalent to the edge-arrival model on graphs, for which an upper bound of 1/2 is known.

Problem

• Extends online matching to 3-uniform hypergraphs
• Develops optimal primal-dual fractional algorithm
• Analyzes competitive ratios for k-uniform hypergraphs

Innovation

• Optimal primal-dual fractional algorithm for 3-uniform hypergraphs
• Adversarial instance proving optimal competitive ratio
• Simple integral algorithm for bounded degree nodes