Distributed Matching Algorithms

• Distributed matching algorithms are techniques designed for pairing resources and solving combinatorial optimization problems in distributed systems, covering maximal, weighted, and stable matching scenarios.
• They leverage various paradigms such as greedy methods, primal-dual and LP-based approaches, augmenting-path searches, and composable coresets to achieve efficient approximations and exact solutions.
• These algorithms are implemented across different computational models like LOCAL, CONGEST, MapReduce/MPC, and asynchronous message-passing, balancing trade-offs between speed, accuracy, and communication constraints.

Distributed matching algorithms are a central area of study in theoretical computer science and practical large-scale systems, encompassing resource matching in distributed computing, peer-to-peer pairing, and combinatorial optimization under distributed control. These algorithms address both classical combinatorial matching (maximum cardinality, maximum weight, perfect, stable) as well as generalized settings (hypergraphs, b-matching, multi-agent and two-sided preferences), with a diverse array of models (synchronous LOCAL and CONGEST, asynchronous message-passing, MapReduce/MPC, and communication-constrained networks).

1. Distributed Matching Models and Problem Formulations

Distributed matching problems generally fall into several categories:

• Maximum Cardinality and Weighted Matching: Given a graph $G=(V,E)$, a matching is a subset of edges $M\subseteq E$ such that no two edges share a vertex. The objective is to maximize $|M|$ (cardinality) or, with weights $w:E\to\mathbb{R}_+$, to maximize $\sum_{e\in M}w(e)$ (Even et al., 2014, Assadi et al., 2019).
• Perfect and Minimum-Weight Perfect Matching: A perfect matching covers all vertices. The distributed Edmonds' Blossom algorithm computes minimum-weight perfect matchings exactly in asynchronous, message-passing systems (Peterson et al., 2022).
• Stable Matching: Each agent has (possibly private) preference lists. The classical Gale-Shapley deferred acceptance (DA) algorithm constructs stable matchings; recent work provides efficient, truthful, and fair distributed mechanisms for the regime of common preferences on one side (Hirvonen et al., 2024).
• Generalizations: Hypergraph matching (edges of arbitrary size), $b$-matching (vertex capacities), multi-view and graph-matching (feature or permutation assignment), and resource allocation with group activities or in directed functional graphs are all addressed in scalable distributed and parallel regimes (Hanguir et al., 2020, Fischer, 2017, Morge et al., 2017, Tran et al., 2020).

Distributed computation models include:

• LOCAL: Unbounded messages, arbitrary local computation, with synchronous rounds (Even et al., 2014, Fischer, 2017, Fischer et al., 2017).
• CONGEST: $O(\log n)$-bit messages on graph edges per round, modeling bandwidth constraints (Izumi et al., 2023, Kitamura et al., 2021).
• MapReduce/MPC: Parallel computation with locality and space restrictions, typically sublinear memory per machine, $O(1)$ or $O(\log n)$ rounds (Assadi et al., 2019, Hanguir et al., 2020).
• Asynchronous Message-Passing: Event-driven, with local state, arbitrary latency, and potentially non-FIFO channels (Peterson et al., 2022).

2. Algorithmic Paradigms and Complexity Regimes

Distributed matching algorithms leverage several key paradigms:

• Greedy and Greedy-like: Simple maximal matching can be computed optimally by a greedy approach; in the edge-colored case, the trivial greedy is round-optimal ($k-1$ for $M\subseteq E$0 colors) (Hirvonen et al., 2011). For weighted matching, distributed greedy yields a 2-approximation (Assadi et al., 2019, Koufogiannakis et al., 2020).
• Primal-Dual and LP-based: Fractional packing-covering LPs for matching are solved and rounded via local distributed subroutines (Fischer, 2017, Koufogiannakis et al., 2020). Deterministic distributed rounding (power-of-two halving, decomposition, local coloring) enables deterministic $M\subseteq E$1 approximations in polylogarithmic rounds (Fischer, 2017), while primal-dual methods support 2-approximations with $M\subseteq E$2 randomized rounds (Koufogiannakis et al., 2020).
• Augmenting-Path Search: Recent algorithms use distributed constructions of alternating base trees and sparse certificates to efficiently find short augmenting paths, enabling nearly linear-in-$M\subseteq E$3 round exact matching in the CONGEST model (Izumi et al., 2023), and subquadratic $M\subseteq E$4-round algorithms for exact maximum matching (Kitamura et al., 2021).
• Composable Coresets and Batch Sampling: In MapReduce/MPC, composable randomized coresets allow for parallel, two-round $M\subseteq E$5-approximate maximum weight matching, scaling to graphs with trillions of edges (Assadi et al., 2019).
• Primal-Dual Stack Algorithms: For $M\subseteq E$6-matching, StackMR leverages primal-dual dual raising and stack layering, supporting $M\subseteq E$7-approximation in $M\subseteq E$8 MapReduce steps, with bounded violation of capacity constraints (Morales et al., 2011).
• High-rank and Hypergraph Matching: Deterministic maximal matching in $M\subseteq E$9-rank hypergraphs (and, by reduction, $|M|$0 edge-coloring) is achieved in $|M|$1 rounds, using recursively-designed deterministic rounding for fractional solutions (Fischer et al., 2017).
• Asynchronous Exact Algorithms: The distributed Blossom algorithm (Peterson et al., 2022) implements Edmonds’ primal-dual framework with dual weights, alternating trees, and cycle contractions via actor processes and supervisor protocols, achieving correctness and polynomial-time in an asynchronous environment.

3. Approximation, Optimality, and Lower Bounds

The state-of-the-art round complexity and approximation guarantees, along with tight lower bounds, are summarized in the following table.

Problem/Class	Round Complexity (Model)	Approximation	Determinism	Reference
Maximal Matching	$|M|$2 (LOCAL)	maximal	deterministic	(Fischer, 2017)
$|M|$3-approx MC Matching	$|M|$4 (LOCAL)	$|M|$5	deterministic	(Even et al., 2014)
$|M|$6-approx MW Matching	$|M|$7 (LOCAL)	$|M|$8	deterministic	(Fischer, 2017)
Max Weighted Matching	$|M|$9 (message-passing)	$w:E\to\mathbb{R}_+$0	randomized	(Koufogiannakis et al., 2020)
MW Matching in MapReduce	2 rounds (MPC/MapReduce)	$w:E\to\mathbb{R}_+$1	randomized	(Assadi et al., 2019)
Max Matching in Bounded $w:E\to\mathbb{R}_+$2	$w:E\to\mathbb{R}_+$3 (LOCAL)	$w:E\to\mathbb{R}_+$4	deterministic	(Even et al., 2014)
Exact Maximum Matching	$w:E\to\mathbb{R}_+$5 (CONGEST)	exact	randomized	(Izumi et al., 2023)
Min-Weight Perf. Matching	$w:E\to\mathbb{R}_+$6 (asynchronous)	exact	deterministic	(Peterson et al., 2022)
Max Matching in $w:E\to\mathbb{R}_+$7-UH Hypergraph	$w:E\to\mathbb{R}_+$8 (MPC, $w:E\to\mathbb{R}_+$9 space/mach)	$\sum_{e\in M}w(e)$0	randomized	(Hanguir et al., 2020)
Maximal Matching in $\sum_{e\in M}w(e)$1-rank Hypergraph	$\sum_{e\in M}w(e)$2 (LOCAL)	maximal	deterministic	(Fischer et al., 2017)
Stable Matching (common prefs)	$\sum_{e\in M}w(e)$3 (CONGEST)	stable	deterministic (truthful)	(Hirvonen et al., 2024)
Maximal Matching (edge-colored,