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

In “Semi-Streaming Matching in a Single Pass II: Greedy is Optimal,” Sepehr Assadi, Max Jiang, and Mars Xiang of the University of Waterloo settle a two-decade open problem: in the single-pass semi-streaming model, which permits O(n polylog n) memory, no deterministic or randomized algorithm for maximum bipartite matching can achieve a constant approximation strictly above 0.5 with constant probability. Since the standard greedy algorithm already guarantees 0.5, greedy is optimal. The proof advances the authors’ blueprint framework, constructing proper combinatorial blueprints whose value approaches 2/3. The construction encodes vertices as delayed gambler’s-ruin random walks on the line from 0 to 2m, uses reflection symmetry to enforce blueprint ban constraints, and selects the starting-state distribution through a linear program so that the probability of reaching 0 is 2m/(3m+1), converging to 2/3. The resulting lower bound translates through the framework into the 0.5 barrier. The same argument proves that online matching with preemption cannot exceed 0.5 expected competitiveness and, by matching–vertex-cover duality, that single-pass semi-streaming minimum bipartite vertex cover cannot beat approximation factor 2. The paper also reports using Google’s Gemini models, OpenAI’s GPT-5.5 Pro and GPT-5.6 Sol, and Anthropic’s Claude Opus and Claude Fable for brainstorming and technical assistance, while the authors wrote the final mathematical content.