The approximate uniform sampling of graphs with a given degree sequence is awell-known, extensively studied problem in theoretical computer science . In this work we study an extension of the problem, where degree intervals are specified rather than a single degree sequence . We are interested in sampling andcounting graphs whose degree sequences satisfy the degree interval constraints . We provide the first fully polynomial almost uniform sampler(FPAUS) for sampling and counting, respectively, graphs with near-regulardegree intervals, in which every node $i$ has a degree from an interval not toofar away from a given $d \in \N$. We also make use of the recent breakthrough of Anari et al. (2019)on sampling a base of a matroid under a strongly log-concave probabilitydistribution. As a more direct approach, we also study a natural Markov chain recently introduced by Rechner, Strowick and M\”uller-Hannemann (2018), based on threesimple local operations: Switches, hinge flips, and additions/deletions of asingle edge . We obtain the first theoretical results for this Markov Chain by showing it is rapidly mixing for the case of this MarkOV chain byshowing it was rapidly mixing

Author(s) : Georgios Amanatidis, Pieter Kleer