A Simple and Efficient Semi-Streaming Algorithm for Approximating Maximum (Weighted) Matching

Core Concepts

This paper presents a simple and efficient semi-streaming algorithm that can (1-ε)-approximate the maximum (weighted) matching in a graph using O(log(n)/ε) passes and O(n log(n)/ε) bits of space.

Abstract

The paper presents two main results:

1. For maximum cardinality bipartite matching (MBM):

   • The algorithm uses a "sample-and-solve" approach, where it samples a small set of edges and computes a maximum matching on the sample.
   • If the sampled matching is not large enough, the algorithm increases the importance of the uncovered edges and repeats the sampling.
   • The algorithm uses a primal-dual analysis based on the duality between matchings and vertex covers in bipartite graphs to show that it converges in O(log(n)/ε) passes.
   • The semi-streaming implementation uses O(n log(n)/ε) bits of space and O(log(n)/ε) passes.

2. For maximum weight general matching:

   • The algorithm extends the ideas from the bipartite case to general graphs, using the duality between matchings and odd-set covers.
   • It samples edges with probabilities proportional to their weights and importance, and updates the importance of uncovered edges.
   • The analysis uses the Cunningham-Marsh theorem to bound the number of potential odd-set covers that need to be considered.
   • The semi-streaming implementation uses O(n log^2(n)/ε) bits of space and O(log(n)/ε) passes.

The key contribution of the paper is providing simple and efficient semi-streaming algorithms for these fundamental problems, improving upon the prior more complex algorithms.

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Key Insights Distilled From

A Simple $(1-ε)$-Approximation Semi-Streaming Algorithm for Maximum (Weighted) Matching

by Sepehr Assad... at arxiv.org 04-23-2024

https://arxiv.org/pdf/2307.02968.pdf

Deeper Inquiries

Can the dependence on ε in the pass complexity of these algorithms be further improved, perhaps by using more sophisticated techniques beyond the multiplicative weight update method

The dependence on ε in the pass complexity of these algorithms can potentially be improved by exploring more sophisticated techniques beyond the multiplicative weight update method. One possible approach could involve incorporating ideas from online learning algorithms, such as online convex optimization or online gradient descent. These techniques have been successfully applied in various optimization problems to achieve better convergence rates and performance guarantees. By adapting these methods to the semi-streaming model for graph optimization, it may be possible to reduce the pass complexity further and improve the approximation guarantees for ε.

Are there any lower bounds that rule out algorithms with better dependence on ε in the semi-streaming model for these problems

There are lower bounds that restrict the possibility of achieving algorithms with significantly better dependence on ε in the semi-streaming model for these problems. These lower bounds stem from the inherent limitations of the streaming model itself, such as the restricted access to the input data and the limited memory available for processing. For example, recent research has shown that achieving sublinear dependence on ε for certain graph optimization problems in the semi-streaming model may be inherently challenging due to the nature of the model and the complexity of the problems. These lower bounds serve as important theoretical barriers that guide the development of algorithms in the streaming setting.

How can these algorithmic ideas be applied to other related graph optimization problems in the streaming setting

The algorithmic ideas presented in the context of maximum matching problems in the semi-streaming model can be extended and applied to other related graph optimization problems in the streaming setting. For example, these techniques can be adapted for problems like minimum vertex cover, maximum independent set, or maximum cut in graphs. By formulating these problems in the context of semi-streaming algorithms and leveraging similar sampling and approximation strategies, it is possible to develop efficient algorithms with provable approximation guarantees. Additionally, the concepts of fractional matchings and vertex covers can be generalized to more complex optimization problems in graphs, opening up avenues for exploring a wide range of graph optimization tasks in the streaming model.