Centrala begrepp
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.
Sammanfattning
The paper presents two main results:
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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.
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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.