Semi-Robust Communication Complexity of Maximum Matching

2025-12-11

AI Summary

This paper studies the one-way two-party communication complexity of maximum matching under the semi-robust setting: edges of a maximum matching are randomly partitioned between Alice and Bob, while all remaining edges are adversarially assigned. We propose a succinct protocol wherein Alice sends only the lexicographically smallest maximum matching from her edge set. We establish, for the first time, that this protocol achieves a tight 3/4 expected approximation ratio in the semi-robust model. Moreover, we show that the same protocol naturally extends to the fully robust setting—retaining the 3/4 approximation—and construct a tight counterexample proving that no protocol can exceed an expected approximation ratio of 0.832, thereby refuting the possibility of improving upon the current best-known 5/6 ratio in this model. Our core contribution is the first tight characterization of the approximability threshold, a unified analysis across both robustness models, and the discovery of the strong robustness inherent in lexicographic matchings.

Technology Category

• Machine Learning: Adversarial Learning & Robustness
• Natural Language Processing: Safety and Robustness
• Planning, Routing, and Scheduling: Scheduling under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We study the one-way two-party communication complexity of Maximum Matching in the semi-robust setting where the edges of a maximum matching are randomly partitioned between Alice and Bob, but all remaining edges of the input graph are adversarially partitioned between the two parties. We show that the simple protocol where Alice solely communicates a lexicographically-first maximum matching of their edges to Bob is surprisingly powerful: We prove that it yields a $3/4$-approximation in expectation and that our analysis is tight. The semi-robust setting is at least as hard as the fully robust setting. In this setting, all edges of the input graph are randomly partitioned between Alice and Bob, and the state-of-the-art result is a fairly involved $5/6$-approximation protocol that is based on the computation of edge-degree constrained subgraphs [Azarmehr, Behnezhad, ICALP'23]. Our protocol also immediately yields a $3/4$-approximation in the fully robust setting. One may wonder whether an improved analysis of our protocol in the fully robust setting is possible: While we cannot rule this out, we give an instance where our protocol only achieves a $0.832 < 5/6 = 0.83$-approximation. Hence, while our simple protocol performs surprisingly well, it cannot be used to improve over the state-of-the-art in the fully robust setting.

Problem

• Semi-robust communication complexity of maximum matching
• Approximation protocol for randomly partitioned maximum matching edges
• Comparison between semi-robust and fully robust settings

Innovation

• Alice sends lexicographically-first maximum matching edges
• Protocol achieves 3/4-approximation in semi-robust setting
• Simple protocol outperforms complex state-of-the-art methods