Maximal Matching Algorithms in Graphs

• Maximal matching algorithms are methods that construct a matching where no additional edge can be added without conflict, ensuring a 2-approximation to the maximum matching.
• They underpin diverse models including sequential greedy, parallel MPC, dynamic, distributed, and streaming systems, each tailored for efficient large-scale graph processing.
• Modern advances achieve exponential speedups and tight approximation bounds, establishing maximal matching as a cornerstone in scalable network and combinatorial optimization.

A maximal matching in a graph is a matching to which no more edges can be added without violating the matching property—i.e., every edge not in the matching shares a vertex with at least one edge in the matching. Maximal matching algorithms are fundamental for approximation, distributed, streaming, parallel, and dynamic graph computation. This article covers the algorithmic and structural landscape for maximal matching, spanning sequential, parallel, and distributed models, dynamic updates, streaming with bounded deletions, and hypergraph generalizations.

1. Sequential Maximal Matching: Greedy Foundation and Extensions

The greedy sequential algorithm remains the standard method for computing a maximal matching in general graphs. It processes the edges in arbitrary order, inserting an edge if neither endpoint is previously matched. This greedy process, implemented in $O(m)$ time and $O(n)$ space, produces a maximal matching that is also a $2$-approximation to the size of a maximum matching; that is, $|M|\ge\frac12|M^\ast|$ where $M^\ast$ is a maximum matching (Feldman et al., 2021, Konrad et al., 2021).

The proof follows from the observation that every edge in the maximum matching must be adjacent to at least one edge in the maximal matching, so $|M^\ast|\le 2|M|$.

Variants of greedy algorithms are also a central building block in a range of models (see later sections for parallel, streaming, and distributed contexts).

2. Parallel and Massively Parallel Computation (MPC)

In modern large-scale graph processing, massively parallel computation (MPC) and shared-memory models demand sublinear communication and parallel work depth. The pioneering work of Behnezhad, Hajiaghayi, and Harris (Behnezhad et al., 2019) demonstrated the first $O(\log\log\Delta)$-round MPC algorithm (with $O(n)$ space per machine) for maximal matching, where $\Delta$ is the maximum degree.

Their algorithm iterates a primitive that, in each round, samples edges, randomly partitions vertices, and computes a greedy matching in each partition. The core steps are:

1. Edge sampling: Each edge is included independently with probability $p=\Delta^{-0.77}$.
2. Random vertex partitioning: Vertices are colored randomly into $k=\Delta^{0.12}$ parts.
3. Parallel greedy local matching: On each partition, compute maximal matching in the subgraph induced by sampled edges.
4. Iteration: Each round reduces maximum degree to $\Delta^{0.92}$ (in expectation); after $O(\log\log\Delta)$ rounds, degree drops to $O(1)$ and the problem is solved centrally.

This approach achieves an exponential speedup over previous $O(\log n)$-round algorithms and matches the best-possible approximation for the related minimum vertex cover and $(1+\varepsilon)$-approximate maximum matching problems (factor-2 via endpoints of the maximal matching). The analysis uses careful variance reduction techniques, notably the Efron–Stein inequality, as well as Luby-style bounding (Behnezhad et al., 2019).

The algorithm generalizes to the congested clique model and achieves similar round complexity and space bounds. Moreover, auction-based and greedy matching methods can be adapted to streaming and distributed parallel environments (Liu et al., 2023).

3. Dynamic and Batch-Dynamic Maximal Matching

Maintaining a maximal matching under edge insertions and deletions is a classical challenge. The state-of-the-art for deterministic fully dynamic maximal matching is achieved by Neiman and Solomon (Neiman et al., 2012), who provide a worst-case $O(\sqrt{m})$ update time with only $O(n+m)$ space.

Key algorithmic invariants:

• Free vertex degree bound: No free vertex has degree exceeding $\Theta(\sqrt{m})$ (otherwise, surrogate rematching is used).
• Maintaining $3/2$-approximation: The matching is always maximal and has no augmenting path of length $\leq 3$, guaranteeing $|M|\ge \frac{2}{3}|M^*|$.

Each update operation manages neighbor lists, a data structure of free neighbors per vertex, and a max-heap over free vertices. By enforcing that high-degree free vertices are immediately rematched via low-degree surrogates, neighbor scans remain $O(\sqrt{m})$ per update.

In low-arboricity graphs, the amortized update time is $O(\log n/\log\log n)$, and the $O(n+m)$ space bound is optimal for any such deterministic strategy (Neiman et al., 2012).

In the parallel batch-dynamic setting, recent advances realized by Blelloch and Brady (Blelloch et al., 12 Mar 2025) and Ghaffari and Trygub (Ghaffari et al., 23 Sep 2024) deliver $O(1)$ (or $O(r^3)$ for hypergraph rank $r$) expected amortized work per update and $O(\log^3 m)$ depth per batch, matching the best sequential bounds up to polylogarithmic factors and extending to parallel dynamic hypergraph matchings.

4. Distributed and Self-Stabilizing Algorithms

Distributed computation of maximal matchings is a canonical symmetry-breaking problem. In anonymous, $k$-edge-colored graphs, the greedy algorithm completes in $k-1$ rounds, and this is tight (Hirvonen et al., 2011). The complexity in maximum degree $\Delta$ is $\Theta(\Delta+\log^*k)$ rounds, closing longstanding gaps.

In general graphs, for the standard (LOCAL) model, deterministic distributed algorithms achieve $O(\Delta+\log^* n)$ rounds. Deterministic distributed algorithms for maximal matching in hypergraphs achieve round complexities $O(r^5 \log^{6+\log r} \Delta \log n)$ for rank $r$ (Fischer et al., 2017), enabling reductions to edge-coloring and approximate matching in these settings.

Self-stabilizing maximal matching algorithms in the link-register model with read/write atomicity stabilize in $O(n\Delta^3)$ moves even under arbitrarily adversarial daemons. Each node maintains local state and link registers to neighbors; convergence and correctness are ensured using layered locking and reset protocols. Prior work in weaker state models or message-passing yields higher move complexities or stronger model assumptions (Cohen et al., 2017).

Distributed energy-efficient randomized algorithms for radio networks achieve maximal matching with $O(\log^2 n)$ maximum energy cost per node and $O(n\log n)$ total latency, through probabilistic participation schedules adapted to degree (Dani et al., 2021).

5. Streaming, Bounded-Deletion, and Semi-Streaming Models

In the insertion-only streaming model, the one-pass greedy maximal matching algorithm is semi-streaming optimal: given $n$ vertices, it achieves $O(n)$ space and a $1/2$-approximation, which is tight (Feldman et al., 2021). Two- and three-pass algorithms that start with maximal matching in the first pass and search for short augmenting paths (e.g., 3-augmenting paths) in subsequent passes can beat the $1/2$ barrier (e.g., achieving $0.53125$, $0.5506$ on general graphs), but cannot significantly exceed $2/3$, and even this requires super-polynomial space to break (Konrad et al., 2021). Recent advances show that non-maximal-matching-first multi-pass algorithms can modestly improve these ratios but do not approach optimal approximation in single-pass settings (Feldman et al., 2021).

In streaming models with bounded edge deletions (“bounded-deletion model”), tight bounds have now been established for space complexity as a function of allowed deletions $K$:

• Randomized: Space is $\tilde{\Theta}(n\sqrt{K})$.
• Deterministic: Space is $\tilde{\Theta}(n K)$.
• For constant-factor approximate matchings ($\alpha>2$): Space drops to $\tilde{\Theta}(n+K)$ for both deterministic and randomized algorithms.

The leading algorithms use hierarchical greedy matchings and repair sketches for randomness, with tight lower bounds established via communication-complexity reductions. This framework provides a smooth interpolation between the insertion-only $\tilde{\Theta}(n)$ regime ($K=0$) and the fully dynamic regime $\Theta(n^2)$ ($K$ unrestricted) (Khanna et al., 21 Feb 2025).

6. Hypergraph Generalizations and Structural Algorithms

Maximal matchings in hypergraphs are crucial for reductions to edge coloring, distributed MIS, and other symmetry-breaking problems. The deterministic distributed algorithm of (Fischer et al., 2017) achieves maximal matching in rank-$r$ hypergraphs in $O(r^5 \log^{6+\log r} \Delta \log n)$ rounds by combining fractional relaxation with recursive rounding, and then iteratively covering the hypergraph with approximated maximal matchings.

In parallel batch-dynamic settings, the randomized work-optimal algorithm of (Blelloch et al., 12 Mar 2025) provides $O(r^3)$ amortized work per edge update (with $O(1)$ for graphs) and $O(\log^3 m)$ depth per batch. Its efficiency relies on a static maximal matching procedure using greedy matching on a random permutation, which partitions the edge set into sample sets, supporting efficient batch insertion and deletion with tight invariants and clear amortization structure.

On the combinatorial front, the maximal matching counting problem has been resolved for bounded clique-width graphs. By introducing matching-cover pairs, dynamic programming along clique-width expressions yields a polynomial-time algorithm when clique-width is constant, albeit with exponential dependence on clique-width parameter $k$—thus offering tractable solutions for a broad class of dense graphs (Menibus et al., 2018).

Enumeration algorithms for listing all (or all large) maximal matchings utilize BFS in carefully constructed supergraphs or reverse-search, achieving polynomial delay and, when necessary, polynomial space, addressing significant algorithmic enumeration challenges (Kobayashi et al., 2021).

7. Connections, Impact, and Limitations

Maximal matching algorithms underpin fast and scalable algorithms in distributed networking, sparse approximation, streaming graph analytics, hypergraph edge-coloring, and combinatorial optimization. They yield the tightest-possible $2$-approximation for minimum vertex cover, and often form the first line of attack in dynamic and parallel settings due to their local, incremental, or parallelizable structure.

While approximation algorithms for maximum matching (especially beyond the $2$ barrier) exist for special graph classes or in multiple passes/parallelization, maximal matching remains a critical component and benchmark for efficiency and simplicity.

Lower bounds in streaming and distributed models demonstrate that surpassing the basic performance metrics of greedy maximal matching requires fundamentally different approaches, often with significant increases in space, time, or pass complexity.

In conclusion, advances in maximal matching algorithmics—across streaming, incremental, dynamic, distributed, and parallel paradigms—have established this primitive as a cornerstone of scalable, robust, and efficient graph computation, while new structural and communication-based lower bounds clarify the inherent barriers for further progress.