[Paper Review] Simple Distributed Weighted Matchings

Jaap-Henk Hoepman|ArXiv.org|Oct 19, 2004

Complexity and Algorithms in GraphsComputer Science12 references139 citations

TL;DR

This paper presents a simple, deterministic distributed algorithm that computes a weighted matching within a factor of 2 from the optimal maximum weighted matching in O(|E|) time. It adapts Preis's sequential greedy algorithm—selecting locally heaviest edges iteratively—to a distributed setting using message-passing between nodes to coordinate edge selection, achieving the best-known approximation ratio for distributed weighted matching with minimal complexity.

ABSTRACT

Wattenhofer [WW04] derive a complicated distributed algorithm to compute a weighted matching of an arbitrary weighted graph, that is at most a factor 5 away from the maximum weighted matching of that graph. We show that a variant of the obvious sequential greedy algorithm [Pre99], that computes a weighted matching at most a factor 2 away from the maximum, is easily distributed. This yields the best known distributed approximation algorithm for this problem so far.

Motivation & Objective

• To develop a distributed algorithm for computing approximate maximum weighted matchings in weighted graphs.
• To improve upon prior complex randomized distributed algorithms with a simpler, deterministic alternative.
• To achieve the best-known approximation ratio of 2 for distributed weighted matching.
• To demonstrate that a sequential greedy algorithm can be efficiently and correctly distributed.
• To provide a time-efficient solution with O(|E|) runtime in distributed systems.

Proposed method

• The algorithm uses a distributed variant of Preis's sequential greedy algorithm, which selects the locally heaviest edge at each node's perspective.
• Each node maintains a set of active neighbors (N) and identifies its current candidate neighbor via the locally heaviest edge (candidate function).
• Nodes send 'req' messages to their current candidate; if both nodes send requests to each other, the edge is added to the matching.
• If a node receives a 'drop' message from a neighbor, it removes that neighbor from its active set and selects a new candidate from remaining neighbors.
• A node terminates when its neighbor set becomes empty, ensuring all matching decisions are coordinated and consistent.
• The protocol ensures that only one matching edge is selected per node, and all incident edges are removed from further consideration.

Experimental results

Research questions

• RQ1 Can a simple sequential greedy algorithm for weighted matching be effectively distributed in an asynchronous network?
• RQ2 What is the approximation ratio achievable by a deterministic distributed algorithm for maximum weighted matching?
• RQ3 Can the O(|E|) time complexity of the sequential greedy algorithm be preserved in a distributed setting?
• RQ4 Does the distributed version correctly simulate the sequential greedy process under message-passing constraints?
• RQ5 Can this approach outperform or simplify prior randomized or complex distributed algorithms for the same problem?

Key findings

• The distributed algorithm achieves a 2-approximation ratio for maximum weighted matching, matching the best-known sequential approximation.
• The algorithm runs in O(|E|) time, matching the time complexity of the sequential greedy algorithm.
• The protocol correctly simulates the sequential greedy algorithm by ensuring that each selected edge is the locally heaviest available at the time of selection.
• The algorithm terminates deterministically with a valid matching, even in an asynchronous message-passing environment.
• The correctness is proven via invariants ensuring edge consistency and matching validity throughout execution.
• The approach outperforms prior randomized distributed algorithms with a 5-approximation ratio, offering a simpler and more efficient solution.