Beating the Folklore Algorithm for Dynamic Matching
The maximum matching problem in dynamic graphs subject to edge updates (insertions and deletions) has received much attention over the last few years; a multitude of approximation/time tradeoffs were obtained, improving upon the folklore algorithm, which maintains a maximal (and hence 2-approximate) matching in O(n) worst-case update time in n-node graphs.
We present the first deterministic algorithm which outperforms the folklore algorithm in terms of both approximation ratio and worst-case update time. Specifically, we give a (2-Ω(1))-approximate algorithm with O(m^{3/8}) = O(n^{3/4}) worst-case update time in n-node, m-edge graphs. For sufficiently small constant ε > 0, no deterministic (2+ε)-approximate algorithm with worst-case update time O(n^{0.99}) was known. Our second result is the first deterministic (2+ε)-approximate weighted matching algorithm with O_ε(1)⋅ O(∜{m}) = O_ε(1)⋅ O(√n) worst-case update time. Neither of our results were previously known to be achievable by a randomized algorithm against an adaptive adversary.
Our main technical contributions are threefold: first, we characterize the tight cases for kernels, which are the well-studied matching sparsifiers underlying much of the (2+ε)-approximate dynamic matching literature. This characterization, together with multiple ideas - old and new - underlies our result for breaking the approximation barrier of 2. Our second technical contribution is the first example of a dynamic matching algorithm whose running time is improved due to improving the recourse of other dynamic matching algorithms. Finally, we show how to use dynamic bipartite matching algorithms as black-box subroutines for dynamic matching in general graphs without incurring the natural 3/2 factor in the approximation ratio which such approaches naturally incur (reminiscent of the integrality gap of the fractional matching polytope in general graphs).
dynamic matching
dynamic graph algorithms
sublinear algorithms
Theory of computation~Dynamic graph algorithms
111:1-111:23
Regular Paper
This research is partially supported by NSF award 1812919, ONR award N000141912550, and a gift from Cisco Research.
https://arxiv.org/abs/2106.10321
The authors thank Sayan Bhattacharya and Aaron Bernstein for helpful discussions, and Aaron Bernstein for sharing a preprint of [Bernstein et al., 2021].
Mohammad
Roghani
Mohammad Roghani
Stanford University, CA, USA
roghani@stanford.edu
Amin
Saberi
Amin Saberi
Stanford University, CA, USA
saberi@stanford.edu
David
Wajc
David Wajc
Stanford University, CA, USA
wajc@stanford.edu
10.4230/LIPIcs.ITCS.2022.111
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Mohammad Roghani, Amin Saberi, and David Wajc
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