Approximate Sampling and Counting of Graphs with NearRegular Degree Intervals
Abstract
The approximate uniform sampling of graphs with a given degree sequence is a wellknown, extensively studied problem in theoretical computer science and has significant applications, e.g., in the analysis of social networks. 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 and counting graphs whose degree sequences satisfy the degree interval constraints. A natural scenario where this problem arises is in hypothesis testing on social networks that are only partially observed. In this work, we provide the first fully polynomial almost uniform sampler (FPAUS) as well as the first fully polynomial randomized approximation scheme (FPRAS) for sampling and counting, respectively, graphs with nearregular degree intervals, in which every node $i$ has a degree from an interval not too far away from a given $d \in \N$. In order to design our FPAUS, we rely on various stateoftheart tools from Markov chain theory and combinatorics. In particular, we provide the first nontrivial algorithmic application of a breakthrough result of Liebenau and Wormald (2017) regarding an asymptotic formula for the number of graphs with a given nearregular degree sequence. Furthermore, we also make use of the recent breakthrough of Anari et al. (2019) on sampling a base of a matroid under a strongly logconcave probability distribution. As a more direct approach, we also study a natural Markov chain recently introduced by Rechner, Strowick and MüllerHannemann (2018), based on three simple local operations: Switches, hinge flips, and additions/deletions of a single edge. We obtain the first theoretical results for this Markov chain by showing it is rapidly mixing for the case of nearregular degree intervals of size at most one.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.09068
 Bibcode:
 2021arXiv211009068A
 Keywords:

 Computer Science  Discrete Mathematics;
 Computer Science  Data Structures and Algorithms;
 Mathematics  Combinatorics