Dynamic Complexity of Expansion
Abstract
Dynamic Complexity was introduced by Immerman and Patnaik \cite{PatnaikImmerman97} (see also \cite{DongST95}). It has seen a resurgence of interest in the recent past, see \cite{DattaHK14,ZeumeS15,MunozVZ16,BouyerJ17,Zeume17,DKMSZ18,DMVZ18,BarceloRZ18,DMSVZ19,SchmidtSVZK20,DKMTVZ20} for some representative examples. Use of linear algebra has been a notable feature of some of these papers. We extend this theme to show that the gap version of spectral expansion in bounded degree graphs can be maintained in the class $\DynACz$ (also known as $\dynfo$, for domain independent queries) under batch changes (insertions and deletions) of $O(\frac{\log{n}}{\log{\log{n}}})$ many edges. The spectral graph theoretic material of this work is based on the paper by KaleSeshadri \cite{KaleS11}. Our primary technical contribution is to maintain up to logarithmic powers of the transition matrix of a bounded degree undirected graph in $\DynACz$.
 Publication:

arXiv eprints
 Pub Date:
 August 2020
 DOI:
 10.48550/arXiv.2008.05728
 arXiv:
 arXiv:2008.05728
 Bibcode:
 2020arXiv200805728D
 Keywords:

 Computer Science  Computational Complexity
 EPrint:
 29 pages