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Robust Communication Complexity of Matching: EDCS Achieves 5/6 Approximation

Authors Amir Azarmehr, Soheil Behnezhad

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  • 15 pages

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Amir Azarmehr
  • Northeastern University, Boston, MA, USA
Soheil Behnezhad
  • Northeastern University, Boston, MA, USA


The second author thanks David Wajc for enlightening discussions about going beyond 2/3-approximations via EDCS.

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Amir Azarmehr and Soheil Behnezhad. Robust Communication Complexity of Matching: EDCS Achieves 5/6 Approximation. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 14:1-14:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


We study the robust communication complexity of maximum matching. Edges of an arbitrary n-vertex graph G are randomly partitioned between Alice and Bob independently and uniformly. Alice has to send a single message to Bob such that Bob can find an (approximate) maximum matching of the whole graph G. We specifically study the best approximation ratio achievable via protocols where Alice communicates only Õ(n) bits to Bob. There has been a growing interest on the robust communication model due to its connections to the random-order streaming model. An algorithm of Assadi and Behnezhad [ICALP'21] implies a (2/3+ε₀ ∼ .667)-approximation for a small constant 0 < ε₀ < 10^{-18}, which remains the best-known approximation for general graphs. For bipartite graphs, Assadi and Behnezhad [Random'21] improved the approximation to .716 albeit with a computationally inefficient (i.e., exponential time) protocol. In this paper, we study a natural and efficient protocol implied by a random-order streaming algorithm of Bernstein [ICALP'20] which is based on edge-degree constrained subgraphs (EDCS) [Bernstein and Stein; ICALP'15]. The result of Bernstein immediately implies that this protocol achieves an (almost) (2/3 ∼ .666)-approximation in the robust communication model. We present a new analysis, proving that it achieves a much better (almost) (5/6 ∼ .833)-approximation. This significantly improves previous approximations both for general and bipartite graphs. We also prove that our analysis of Bernstein’s protocol is tight.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Random order and robust communication complexity
  • Maximum Matching
  • Robust Communication Complexity
  • Edge Degree Constrained Subgraph


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  1. Sepehr Assadi and Soheil Behnezhad. Beating Two-Thirds For Random-Order Streaming Matching. In 48th International Colloquium on Automata, Languages, and Programming, ICALP 2021, July 12-16, 2021, Glasgow, Scotland (Virtual Conference), pages 19:1-19:13, 2021. URL:
  2. Sepehr Assadi and Soheil Behnezhad. On the Robust Communication Complexity of Bipartite Matching. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2021, August 16-18, 2021, University of Washington, Seattle, Washington, USA (Virtual Conference), pages 48:1-48:17, 2021. URL:
  3. Sepehr Assadi and Aaron Bernstein. Towards a Unified Theory of Sparsification for Matching Problems. In 2nd Symposium on Simplicity in Algorithms, SOSA 2019, January 8-9, 2019, San Diego, CA, USA, pages 11:1-11:20, 2019. URL:
  4. Amir Azarmehr and Soheil Behnezhad. Robust Communication Complexity of Matching: EDCS Achieves 5/6 Approximation, 2023. URL:
  5. Soheil Behnezhad. Improved Analysis of EDCS via Gallai-Edmonds Decomposition. CoRR, abs/2110.05746, 2021. URL:
  6. Aaron Bernstein. Improved Bounds for Matching in Random-Order Streams. In 47th International Colloquium on Automata, Languages, and Programming, ICALP 2020, July 8-11, 2020, Saarbrücken, Germany (Virtual Conference), volume 168 of LIPIcs, pages 12:1-12:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL:
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  13. Michael Kapralov. Space Lower Bounds for Approximating Maximum Matching in the Edge Arrival Model. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, Virtual Conference, January 10 - 13, 2021, pages 1874-1893. SIAM, 2021. URL:
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