Semi-Robust Communication Complexity of Maximum Matching

TL;DR

The paper studies the one-way two-party communication complexity of Maximum Matching in a semi-robust setting, where OPT edges are randomly split while non-OPT edges are adversarially split. It shows that Alice sending the lexicographically-first maximum matching to Bob yields a 3/4-approximation in expectation, with a tight analysis and a close relation to robust and fully robust regimes. The authors develop an augmenting-path framework and a linear-programming bound to establish the 3/4 guarantee, and also analyze a variant that sends a lexicographically-first maximal matching, achieving 5/8 (tight). The work clarifies the potential and limits of simple LFMM-based protocols, connects to streaming via EDCS techniques, and poses open questions about improving guarantees in fully robust and related models.

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.

Figures (2)

• Figure 1: The hard input graph for $\Pi_{\text{LFMM}}$ in the fully robust setting is the semi-complete graph on $8$ vertices. Importantly, a specific vertex ordering is required that defines the lexicographical ordering of the maximum matchings computed by Alice (see the preliminaries for details). The ordering is indicated by the integers assigned to the vertices. We observe that the positions of $5$ and $6$ are required to be as they are -- the more intuitive ordering with $5$ and $6$ exchanged does not yield an expected approximation factor $< 5/6$.
• Figure 2: LP whose objective value constitutes a lower bound on the approximation factor of $\Pi_{\text{LFMM}}$.

Theorems & Definitions (13)

• Theorem 1
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Theorem 2