Weighted Matching in the Random-Order Streaming and Robust Communication Models

TL;DR

A $(2/3-\epsilon)-approximation algorithm for maximum weight matching in random-order streams, using space $O(n \log n \log R)$, where $R$ is the ratio between the heaviest and the lightest edge in the graph.

Abstract

We study the maximum weight matching problem in the random-order semi-streaming model and in the robust communication model. Unlike many other sublinear models, in these two frameworks, there is a large gap between the guarantees of the best known algorithms for the unweighted and weighted versions of the problem. In the random-order semi-streaming setting, the edges of an $n$-vertex graph arrive in a stream in a random order. The goal is to compute an approximate maximum weight matching with a single pass over the stream using $O(n\text{ polylog } n)$ space. Our main result is a $(2/3-ε)$-approximation algorithm for maximum weight matching in random-order streams, using space $O(n \log n \log R)$, where $R$ is the ratio between the heaviest and the lightest edge in the graph. Our result nearly matches the best known unweighted $(2/3+ε_0)$-approximation (where $ε_0 \sim 10^{-14}$ is a small constant) achieved by Assadi and Behnezhad [ICALP 2021], and significantly improves upon previous weighted results. Our techniques also extend to the related robust communication model, in which the edges of a graph are partitioned randomly between Alice and Bob. Alice sends a single message of size $O(n\text{ polylog }n)$ to Bob, who must compute an approximate maximum weight matching. We achieve a $(5/6-ε)$-approximation using $O(n \log n \log R)$ words of communication, matching the results of Azarmehr and Behnezhad [ICALP 2023] for unweighted graphs.

Figures (4)

• Figure 1: An example of a weighted graph $G$ and its unfolding $\phi(G)$.
• Figure 2: An example of a subgraph $H \subseteq \phi(G)$ and its refolding $\mathcal{R}(H) \subseteq G$. In this example, $H = \{(u^1,v^2)\}$. Then $\mathcal{R}(H) = \{(u,v)\}$.
• Figure 3: Refolding does not in general preserve matching size in non-bipartite graphs. Consider for example the blue subgraph $H = \{(x^1,z^2), (z^1,y^2), (y^1,x^2) \} \subseteq \phi(G)$ shown in the diagram. Then $\mu(H) = 3$, but $\mu_w(\mathcal{R}(H)) =2.$
• Figure 4: Illustration of the reduction to the bipartite case. We show that $\widetilde{H} \cup \widetilde{U}$ contains a matching of size at least $\left(\frac{2}{3}-\epsilon \right) \mu_w(G)$. Since $\widetilde{G}$ is bipartite, we can refold $\widetilde{H} \cup \widetilde{U}$ without reducing the matching size.

Theorems & Definitions (62)

• Theorem 1
• Theorem 2
• Theorem 3
• Definition 2.1: Graph Unfolding KLST
• Definition 2.2: Refolding BDL
• Lemma 2.3: BDL
• Definition 2.3: $b$-batch random-order stream model
• Definition 2.4
• Lemma 2.5: Informal version of Lemma 5.7 in BDL
• Definition 2.6: EDCS BS_EDCSbip