Deterministic $(1+\varepsilon)$-Approximate Maximum Matching with $\mathsf{poly}(1/\varepsilon)$ Passes in the Semi-Streaming Model and Beyond

TL;DR

This work presents a deterministic $(1+\varepsilon)$-approximation algorithm for maximum matching in the semi-streaming model with poly$(1/\varepsilon)$ passes and memory $n\cdot\text{poly}(1/\varepsilon)$. It introduces a phase-based Hopcroft-Karp-inspired approach augmented with structures for free nodes, on-hold mechanisms, and jumping to extend active paths, while handling odd cycles without blossom contraction. The paper also develops a general framework that simulates the streaming approach in CONGEST and MPC, achieving polynomial dependence on $1/\varepsilon$ and improving over prior exponential pass complexities. The results advance the theoretical understanding of graph streaming and distributed computation, providing a scalable path to near-optimal matching with deterministic guarantees.

Abstract

We present a deterministic $(1+\varepsilon)$-approximate maximum matching algorithm in $\mathsf{poly} 1/\varepsilon$ passes in the semi-streaming model, solving the long-standing open problem of breaking the exponential barrier in the dependence on $1/\varepsilon$. Our algorithm exponentially improves on the well-known randomized $(1/\varepsilon)^{O(1/\varepsilon)}$-pass algorithm from the seminal work by McGregor~[APPROX05], the recent deterministic algorithm by Tirodkar with the same pass complexity~[FSTTCS18]. Up to polynomial factors in $1/\varepsilon$, our work matches the state-of-the-art deterministic $(\log n / \log \log n) \cdot (1/\varepsilon)$-pass algorithm by Ahn and Guha~[TOPC18], that is allowed a dependence on the number of nodes $n$. Our result also makes progress on the Open Problem 60 at sublinear.info. Moreover, we design a general framework that simulates our approach for the streaming setting in other models of computation. This framework requires access to an algorithm computing an $O(1)$-approximate maximum matching and an algorithm for processing disjoint $(\mathsf{poly} 1 / \varepsilon)$-size connected components. Instantiating our framework in $\mathsf{CONGEST}$ yields a $\mathsf{poly}(\log{n}, 1/\varepsilon)$ round algorithm for computing $(1+\varepsilon$)-approximate maximum matching. In terms of the dependence on $1/\varepsilon$, this result improves exponentially state-of-the-art result by Lotker, Patt-Shamir, and Pettie~[LPSP15]. Our framework leads to the same quality of improvement in the context of the Massively Parallel Computation model as well.

Figures (4)

• Figure 1: Nodes $\alpha$, $\beta$, and $\gamma$ are free. The black single-segments are unmatched and black (full) double-segments are matched edges. The path $P'$ corresponding to a DFS branch of $\gamma$ is shown by the red solid spline. Since the edge $a_5$ is part of the path, the current DFS branch of $\gamma$ cannot be extended up to the free node $\beta$ along the dashed blue line. Furthermore, the path from $\gamma$ to the edge $a_3$ can potentially "block" a longer DFS search path of $\alpha$ illustrated with a solid blue line. However, the edges along the DFS searches of $\alpha$ and $\gamma$ can be combined to find an augmenting path between $\alpha$ and $\gamma$.
• Figure 2: Assume that we first perform a DFS style search over the red (dashed) path from $\alpha$ to $a_3$. Moreover, assume that the algorithm maintains the (shortest) path length of the DFS as labels on edges, but not of arcs. In that case, this red path sets labels $\ell(a_1) = 1$, $\ell(a_2) = 2$ and $\ell(a_3) = 3$. Eventually, the search backtracks to $a_1$ and continues over $a_3$ setting the label to $\ell(a_3) = 2$. However, this DFS branch cannot continue to $a_4$, even though the path exists, due to the distance label $\ell(a_2) < \ell(a_3) + 1$. By considering labels on arcs, we allow $\alpha$ to extend its DFS search over $(\alpha, a_1, a_3, a_2, a_4)$.
• Figure 3: In this example, $\alpha$ is a free node, black (full) single-segments are unmatched and black (full) double-segments are matched edges. Assume that the algorithm first explores the red (dashed) path from $\alpha$ to $a_4$. This red path sets labels $\ell(\overleftarrow{a_5}) = 1$ and $\ell(a_4) = 2$. After this exploration, in the next two passes $\alpha$ backtracks along the red path and sets the active path to be $(\alpha)$. Then, $\alpha$ continues extending its active path (over three $\textsc{Pass-Bundle}\xspace$s) along $a_1$, $a_2$ and $a_3$, setting labels $\ell(a_1) = 1$, $\ell(a_2) = 2$ and $\ell(a_3) = 3$, illustrated with the blue solid line. However, notice that $\ell(a_4) < 4$ and hence $\alpha$ does not extend the active path to become $(\alpha, a_1, a_2, a_3, a_4)$. On the other hand, $\ell(a_5) = \infty$ and our algorithm should enable $\alpha$ to reach $a_5$. This is achieved by enabling $\alpha$ to "jump" over $a_4$ and let the active path become $(\alpha, a_1, a_2, a_3, a_4, a_5)$, illustrated with the blue dashed line. We also set $\ell(a_5) \coloneqq 5$. (In this example, the arc $a_4$ plays role of $b_1$ in the description of Extend-Active-Paths.) Subsequently, in the next $\textsc{Pass-Bundle}\xspace$$\alpha$ will extend its active path to $(\alpha, a_1, a_2, a_3, a_4, a_5, a_6)$.
• Figure 4: Illustrations for the inductive step of \ref{['lemma: non-active cycle']}. The upper picture illustrates the case where the alternating path $P(a_{i + 1}) = (b_1, \ldots, b_h)$ intersects the suffix $(c_1, \ldots, c_{h'})$ alternating path $P(\overleftarrow{a_{i + 1}})$ at some $\overleftarrow{b_j} = c_{j'}$. In this case, we can complete the prefix $(b_1, \ldots, b_{j})$ into a path to $\overleftarrow{a_1}$ using the prefix of $P(\overleftarrow{a_{i + 1}})$. In the lower, we consider the case that $b_j = c_{j'}$. In this case, we can complete the prefix $(b_1, \ldots, b_{j})$ into an alternating path to $\overleftarrow{a_1}$ using the suffix of $P(\overleftarrow{a_{i + 1}})$ and the path $\overleftarrow{a_1, \ldots, a_{i + 1}}$. In the proof, we show that $(b_1, \ldots, b_{j - 1}) \circ (c_{j'}, \ldots c_{h'})$ cannot intersect the path $(a_1, \ldots, a_{i + 1})$.

Theorems & Definitions (61)

• Theorem 1.1
• Theorem 1.2: Restatement of \ref{['theorem:framework']}
• Definition 2.1: An Unmatched Edge and a Free Node
• Definition 2.2: Alternating Path, Alternating Length, Alternating Distance
• Definition 2.3: Concatenation of Alternating Paths
• Definition 2.4: Path Reverse
• Definition 2.5: The Label of an Arc
• Definition 2.6: Structure of a Free Node
• Definition 2.8: Active and Inactive Arcs and Vertices
• Definition 2.9: Removing-directions operator