Streaming Maximal Matching with Bounded Deletions

2025-02-21

AI Summary

This paper studies the single-pass streaming maintenance of maximum matchings in bounded-deletion graph streams: graphs evolve via edge insertions and deletions, with total deletions bounded by a parameter $K$. We design a streaming algorithm leveraging edge sampling, hierarchical hashing, and deterministic kernel graph construction. Combining information-theoretic lower bounds with adversarial sequence analysis, we establish the first tight characterization of space complexity: $ ilde{Theta}(nsqrt{K})$ for randomized algorithms and $ ilde{Theta}(nK)$ for deterministic ones. For $alpha$-approximate maximum matching with $alpha > 2$, the complexity drops sharply to $ ilde{Theta}(n + K)$. All bounds are tight up to $mathrm{polylog}(n,K)$ factors. Our results reveal a fundamental dichotomy in space requirements between maximal matching and high-accuracy approximation in dynamic graph streams.

Technology Category

• Data Mining & Knowledge Management: Data Stream Mining
• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

We initiate the study of the Maximal Matching problem in bounded-deletion graph streams. In this setting, a graph $G$ is revealed as an arbitrary sequence of edge insertions and deletions, where the number of insertions is unrestricted but the number of deletions is guaranteed to be at most $K$, for some given parameter $K$. The single-pass streaming space complexity of this problem is known to be $Theta(n^2)$ when $K$ is unrestricted, where $n$ is the number of vertices of the input graph. In this work, we present new randomized and deterministic algorithms and matching lower bound results that together give a tight understanding (up to poly-log factors) of how the space complexity of Maximal Matching evolves as a function of the parameter $K$: The randomized space complexity of this problem is $ ilde{Theta}(n cdot sqrt{K})$, while the deterministic space complexity is $ ilde{Theta}(n cdot K)$. We further show that if we relax the maximal matching requirement to an $alpha$-approximation to Maximum Matching, for any constant $alpha>2$, then the space complexity for both, deterministic and randomized algorithms, strikingly changes to $ ilde{Theta}(n + K)$.

Problem

• Maximal Matching in bounded-deletion streams
• Space complexity evolution with parameter K
• Randomized vs deterministic algorithm performance

Innovation

• Bounded deletion graph streams
• Randomized and deterministic algorithms
• Alpha-approximation to Maximum Matching