An Asymptotically Optimal Algorithm for Maximum Matching in Dynamic Streams
We present an algorithm for the maximum matching problem in dynamic (insertion-deletions) streams with asymptotically optimal space: for any n-vertex graph, our algorithm with high probability outputs an α-approximate matching in a single pass using O(n²/α³) bits of space.
A long line of work on the dynamic streaming matching problem has reduced the gap between space upper and lower bounds first to n^{o(1)} factors [Assadi-Khanna-Li-Yaroslavtsev; SODA 2016] and subsequently to polylog factors [Dark-Konrad; CCC 2020]. Our upper bound now matches the Dark-Konrad lower bound up to O(1) factors, thus completing this research direction.
Our approach consists of two main steps: we first (provably) identify a family of graphs, similar to the instances used in prior work to establish the lower bounds for this problem, as the only "hard" instances to focus on. These graphs include an induced subgraph which is both sparse and contains a large matching. We then design a dynamic streaming algorithm for this family of graphs which is more efficient than prior work. The key to this efficiency is a novel sketching method, which bypasses the typical loss of polylog(n)-factors in space compared to standard L₀-sampling primitives, and can be of independent interest in designing optimal algorithms for other streaming problems.
Graph streaming algorithms
Sketching
Maximum matching
Theory of computation~Graph algorithms analysis
Theory of computation~Approximation algorithms analysis
Theory of computation~Sketching and sampling
9:1-9:23
Regular Paper
We thank the anonymous reviewers of ITCS 2022 for their many insightful suggestions that helped with improving the presentation of this paper.
Sepehr
Assadi
Sepehr Assadi
Department of Computer Science, Rutgers University, Piscataway, NJ, USA
https://www.cs.rutgers.edu/~sa1497/
Research supported in part by a NSF CAREER Grant CCF-2047061, and a gift from Google Research.
Vihan
Shah
Vihan Shah
Department of Computer Science, Rutgers University, Piscataway, NJ, USA
Research supported in part by a NSF CAREER Grant CCF-2047061.
10.4230/LIPIcs.ITCS.2022.9
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Sepehr Assadi and Vihan Shah
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