# Dynamic Algorithms for Maximum Matching Size

@article{Behnezhad2022DynamicAF, title={Dynamic Algorithms for Maximum Matching Size}, author={Soheil Behnezhad}, journal={ArXiv}, year={2022}, volume={abs/2207.07607} }

We study fully dynamic algorithms for maximum matching. This is a well-studied problem, known to admit several update-time/approximation trade-oﬀs. For instance, it is known how to maintain a 1/2-approximate matching in (poly log n ) time or a 2 / 3-approximate matching in O ( √ n ) time, where n is the number of vertices. Improving either of these bounds has been a long-standing open problem. In this paper, we show that when the goal is to maintain just the size of the matching instead of its…

## References

SHOWING 1-10 OF 44 REFERENCES

### Faster Fully Dynamic Matchings with Small Approximation Ratios

- Computer Science, MathematicsSODA
- 2016

A fully dynamic deterministic algorithm that maintains a (3/2 + e)-approximation in amortized update time O(m1/4e--2.5) and manages to be polynomially faster than all existing deterministic algorithms, while still maintaining a better-than-2 approximation.

### Beating Greedy Matching in Sublinear Time

- Computer Science, MathematicsArXiv
- 2022

This work designs a less “adaptive” augmentation algorithm for maximum matching that might be of independent interest and can be implemented in O ( n 1+ ε ) time, where n is the number of vertices and the constant ε > 0 can be made arbitrarily small.

### Fully Dynamic (1+ e)-Approximate Matchings

- Computer Science2013 IEEE 54th Annual Symposium on Foundations of Computer Science
- 2013

The main result is a data structure that maintains a (1+ϵ) approximation of maximum matching under edge insertions/deletions in worst case Õ(√mϵ-2) time per update, which improves the 3/2 approximation given by Neiman and Solomon [20] which runs in similar time.

### New deterministic approximation algorithms for fully dynamic matching

- Computer Science, MathematicsSTOC
- 2016

This work presents two deterministic dynamic algorithms for the maximum matching problem and is the first deterministic algorithm that can maintain an o(logn)-approximate maximum matching with polylogarithmic update time.

### Simple deterministic algorithms for fully dynamic maximal matching

- Computer Science, MathematicsSTOC '13
- 2013

This paper shows the first deterministic fully dynamic algorithm that outperforms the trivial one and provides a deterministic worst-case update time of O(√m), which maintains a matching which is in fact a 3/2-approximate maximum cardinality matching (MCM).

### Constant-Time Approximation Algorithms via Local Improvements

- Computer Science, Mathematics2008 49th Annual IEEE Symposium on Foundations of Computer Science
- 2008

This work gives the first constant-time algorithm that for the class of graphs of degree bounded by d, computes the maximum matching size to within epsIVn, for any epsivn > 0, where n is the number of nodes in the graph.

### A near-optimal sublinear-time algorithm for approximating the minimum vertex cover size

- Computer Science, MathematicsSODA
- 2012

An algorithm is given that outputs a (2, e)-estimate of the size of a minimum vertex cover whose query complexity and running time are O(n) · poly(1/e) and the result is nearly optimal.

### Fully Dynamic Maximal Matching in O (log n) Update Time

- Computer Science2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
- 2011

This is the first polylog update time for maximal matching that implies an exponential improvement from the previous results and can maintain a factor two approximate maximum matching in $O(\log n )$ expected amortized time per update as a direct corollary of the maximal matching scheme.

### New Trade-Offs for Fully Dynamic Matching via Hierarchical EDCS

- Computer ScienceSODA
- 2022

A new approach to designing fully dynamic approximate matching algorithms that in a uniﬁed manner not only recovers all previously known trade-o ﬀ s that were achieved via very diﬀ erent techniques, but reveals some new ones as well.

### Deterministic Fully Dynamic Data Structures for Vertex Cover and Matching

- Computer ScienceSODA
- 2015

We present the first deterministic data structures for maintaining approximate minimum vertex cover and maximum matching in a fully dynamic graph in o([EQUATION]m) time per update. In particular, for…