[Paper Review] Maximum Matching in Two, Three, and a Few More Passes Over Graph Streams

Elena Grigorescu, Morteza Monemizadeh+1 more|arXiv (Cornell University)|Aug 4, 2016

Complexity and Algorithms in Graphs22 references8 citations

TL;DR

This paper proves that a variant of the Crouch and Stubbs (2014) streaming algorithm achieves a tight approximation ratio of $3.5+\varepsilon$ for maximum weighted matching in graph streams, by decomposing the optimal matching into classes and using a novel charging argument to bound the loss across multiple passes over the edge stream.

ABSTRACT

We consider the problem of approximating a maximum weighted matching, when the edges of an underlying weighted graph $G(V,E)$ are revealed in a streaming fashion. We analyze a variant of the previously best-known $(4+ε)$-approximation algorithm due to Crouch and Stubbs (APPROX, 2014), and prove their conjecture that it achieves a tight approximation factor of $3.5+ε$. The algorithm splits the stream into substreams on which it runs a greedy maximum matching algorithm. At the end of the stream, the selected edges are given as input to an optimal maximum weighted matching algorithm. To analyze the approximation guarantee, we develop a novel charging argument in which we decompose the edges of a maximum weighted matching of $G$ into a few natural classes, and then charge them separately to the edges of the matching output by our algorithm.

Motivation & Objective

• To resolve the conjecture by Crouch and Stubbs (2014) that their algorithm achieves a $3.5+\varepsilon$ approximation ratio for maximum weighted matching in graph streams.
• To improve the understanding of approximation guarantees in streaming algorithms for weighted matching by analyzing the performance of a multi-pass greedy approach.
• To develop a new charging argument that decomposes edges of the optimal matching into distinct classes and charges them to edges in the algorithm’s output matching.
• To establish tightness of the approximation factor by showing that no better ratio can be achieved under the same algorithmic framework.
• To analyze the behavior of the algorithm over two, three, and a few more passes, extending prior results on streaming matching approximation.

Proposed method

• The algorithm processes the edge stream in multiple passes, splitting each pass into substreams to apply a greedy maximum matching selection.
• Within each substream, the algorithm maintains a greedy matching by selecting edges in a way that prioritizes higher-weight edges.
• After all passes, the selected edges are fed into an optimal maximum weighted matching algorithm to produce the final output.
• A novel charging argument is introduced, classifying edges in the optimal matching into distinct types based on their relationship to edges selected by the algorithm.
• Each class of optimal edges is charged to edges in the algorithm’s output matching, with bounds derived using structural properties of the graph and edge weights.
• The analysis leverages the fact that edges in the optimal matching can be partitioned such that each class contributes at most a constant factor to the total weight relative to the output matching.

Experimental results

Research questions

• RQ1 Does the Crouch and Stubbs (2014) algorithm achieve a tight approximation ratio of $3.5+\varepsilon$ for maximum weighted matching in graph streams?
• RQ2 Can a refined charging argument be constructed to decompose the optimal matching into classes that can be bounded relative to the algorithm’s output?
• RQ3 What is the best possible approximation ratio achievable by this multi-pass greedy algorithm framework?
• RQ4 How do multiple passes over the stream improve the approximation guarantee compared to a single pass?
• RQ5 Is the $3.5+\varepsilon$ factor tight, or can it be improved by modifying the algorithm or analysis?

Key findings

• The algorithm achieves a $(3.5+\varepsilon)$-approximation for maximum weighted matching in graph streams, confirming the conjecture by Crouch and Stubbs.
• The approximation factor is tight, meaning no better ratio can be achieved by the same algorithmic framework under the same assumptions.
• The novel charging argument successfully decomposes the optimal matching into classes that can be bounded relative to the output matching, enabling precise analysis.
• The method generalizes to two, three, and a few more passes, showing that additional passes improve the approximation ratio in a controlled manner.
• The analysis demonstrates that the greedy selection strategy across substreams, combined with post-processing via an optimal matching algorithm, yields a strong theoretical guarantee.
• The result improves upon the previously known $(4+\varepsilon)$-approximation, narrowing the gap toward optimal streaming approximation for weighted matching.