# NC Algorithms for Computing a Perfect Matching and a Maximum Flow in One-Crossing-Minor-Free Graphs

@article{Eppstein2018NCAF, title={NC Algorithms for Computing a Perfect Matching and a Maximum Flow in One-Crossing-Minor-Free Graphs}, author={David Eppstein and Vijay V. Vazirani}, journal={SIAM J. Comput.}, year={2018}, volume={50}, pages={1014-1033} }

In 1988, Vazirani gave an NC algorithm for computing the number of perfect matchings in $K_{3,3}$-minor-free graphs by building on Kasteleyn's scheme for planar graphs, and stated that this "opens up the possibility of obtaining an NC algorithm for finding a perfect matching in $K_{3,3}$-free graphs." In this paper, we finally settle this 30-year-old open problem. Building on recent NC algorithms for planar and bounded-genus perfect matching by Anari and Vazirani and later by Sankowski, we…

## 2 Citations

### Finding Maximal Sets of Laminar 3-Separators in Planar Graphs in Linear Time

- MathematicsSODA
- 2019

This work considers decomposing a 3-connected planar graph G using laminar separators of size three and shows how to find a maximal set of 3-separators disjoint from v which are laminAR and satisfy: every vertex in a separator $X$ has two neighbours not in the unique component of G-X containing $v$.

### Reachability and Matching in Single Crossing Minor Free Graphs

- MathematicsFSTTCS
- 2021

The maximum matching result improves upon the recent result of Eppstein and Vazirani, where they show an NC bound for constructing perfect matching in general single crossing minor free graphs.

## References

SHOWING 1-10 OF 66 REFERENCES

### NC Algorithms for Computing the Number of Perfect Matchings in K_3,3-Free Graphs and Related Problems

- Mathematics, Computer ScienceInf. Comput.
- 1989

### Some perfect matchings and perfect half-integral matchings in NC

- Mathematics, Computer ScienceChic. J. Theor. Comput. Sci.
- 2008

We show that for any class of bipartite graphs which is closed under edge deletion and where the number of perfect matchings can be counted in NC, there is a deterministic NC algorithm for finding a…

### All-Pairs Minimum Cuts in Near-Linear Time for Surface-Embedded Graphs

- Computer Science, MathematicsSoCG
- 2016

The algorithm constructs a Gomory-Hu tree for the given graph, providing a data structure with space $O(n)$ that can answer minimum-cut queries in constant time.

### Exponential Speedup of Fixed-Parameter Algorithms for Classes of Graphs Excluding Single-Crossing Graphs as Minors

- MathematicsAlgorithmica
- 2004

A fixed-parameter algorithm that constructively solves the k-dominating set problem on any class of graphs excluding a single-crossing graph as a minor in O(4^{9.55\sqrt{k}}n^{O(1)})$ time is presented.

### Counting the Number of Perfect Matchings in K5-Free Graphs

- Mathematics, Computer Science2014 IEEE 29th Conference on Computational Complexity (CCC)
- 2014

It is shown that the number of perfect matchings in K5-free graphs can be computed in polynomial time and this work parallelizes the sequential algorithm and shows that the problem is in TC2.

### NC Algorithms for Weighted Planar Perfect Matching and Related Problems

- MathematicsICALP
- 2018

A new relatively simple but versatile framework is developed that handles the combinatorial structure of matchings directly and needs to only know weights of appropriately defined matchings from algebraic subroutines.

### The Matching Problem in General Graphs Is in Quasi-NC

- Mathematics, Computer Science2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)
- 2017

It is shown that the perfect matching problem in general graphs is in Quasi-NC, and the result is obtained by a derandomization of the Isolation Lemma for perfect matchings to obtain a Randomized NC algorithm.

### Finding Maximal Sets of Laminar 3-Separators in Planar Graphs in Linear Time

- MathematicsSODA
- 2019

This work considers decomposing a 3-connected planar graph G using laminar separators of size three and shows how to find a maximal set of 3-separators disjoint from v which are laminAR and satisfy: every vertex in a separator $X$ has two neighbours not in the unique component of G-X containing $v$.

### Optimization and Recognition for K 5-minor Free Graphs in Linear Time

- Mathematics, Computer ScienceLATIN
- 2008

A linear time algorithm which determines whether an inputgraph contains K5 as aminor and outputs a K5-model if the input graph contains one and constructs a tree decomposition such that each node of the tree corresponds to a planar graph or a graph with eight vertices.

### Planar Graph Perfect Matching Is in NC

- Computer Science2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)
- 2018

This paper gives an NC algorithm for finding a perfect matching in a planar graph at which many new conditions, involving constraints of the polytope, are simultaneously satisfied.