Semi-Streaming Matching in a Single Pass II: Greedy is Optimal

2026-07-16

AI Summary

This work resolves a long-standing open problem regarding the upper bound on the approximation ratio for maximum matching in the single-pass semi-streaming model. The authors introduce a novel combinatorial structure termed a “blueprint” and construct the first optimal blueprint, rigorously proving that no single-pass algorithm can achieve an approximation ratio better than 1/2. This result establishes the optimality of the greedy algorithm in this model and simultaneously settles a two-decade-old question concerning the competitive ratio upper bound in preemptive online matching—a central problem in graph streaming. By integrating techniques from semi-streaming and online algorithm analysis, the paper employs combinatorial lower-bound methods to unify and characterize the theoretical limits of both models for the first time.

Abstract

We prove that no single-pass semi-streaming algorithm (deterministic or randomized) can achieve a better-than-half approximation to the maximum matching problem. This implies the optimality of the naive greedy algorithm, answering an outstanding open question in the graph streaming literature since the introduction of the model over two decades ago. Our proof follows the "blueprint framework" introduced previously by the authors, which reduced proving lower bounds for semi-streaming matching to constructing certain combinatorial objects called blueprints. We present an optimal construction of blueprints that when used in this framework implies our semi-streaming matching lower bound. Our results also imply that the optimal competitive ratio of online matching with preemption is half, again matching the naive greedy algorithm, settling this open question as well.

Problem

• semi-streaming
• maximum matching
• approximation ratio
• online matching
• preemption

Innovation

• semi-streaming
• maximum matching
• greedy algorithm
• lower bound
• blueprint framework