## C&O Reading Group - Rian Neogi

Speaker: |
Rian Neogi |

Affiliation: |
University of Waterloo |

Location: |
MC 6029 |

**Abstract:** In this talk, we will cover the paper of Svensson and Tarnawski that shows that perfect matching in general (non-bipartite) graphs in is quasi-NC. Similar to the work of Fenner, Gurjar and Thierauf (covered earlier in the reading group), the approach is to derandomize the isolation lemma for the perfect matching polytope by applying weight functions to iteratively restrict to subfaces of the polytope. However, the perfect matching polytope in general graphs is not as well-structured as it is in bipartite graphs. The faces of the polytope no longer correspond to subgraphs and now involve additional tight odd set constraints that need to be dealt with. This makes it so that a cycle with non-zero circulation may still exist in the support of the new face. Additionally, the existence of odd cycles in the graph breaks the cycle counting argument used in the paper of Fenner, Gurjar, Thierauf. We will see how Svensson and Tarnawski deal with these issues in the talk.