Constructing, sampling and counting graphical realizations of restricted degree sequences

TL;DR

This paper introduces a new problem of constructing and sampling graph realizations of degree sequences with forbidden edges, providing efficient algorithms and extending previous sampling results for special cases.

Contribution

It fully solves the restricted degree sequence problem for certain bipartite graphs and develops a fully polynomial sampler and approximation scheme for counting realizations.

Findings

Provides a complete solution for RDS when forbidden edges form a specific bipartite graph.

Develops an efficient uniform sampling algorithm for half-regular bipartite graphs.

Extends and simplifies proofs of existing sampling results for regular bipartite and directed graphs.

Abstract

With the current burst of network theory (especially in connection with social and biological networks) there is a renewed interest on realizations of given degree sequences. In this paper we propose an essentially new degree sequence problem: we want to find graphical realizations of a given degree sequence on labeled vertices, where certain would-be edges are {\em forbidden}. Then we want to sample uniformly and efficiently all these possible realizations. (This problem can be considered as a special case of Tutte's $f$-factor problem, however it has a favorable sampling speed.) We solve this {\em restricted degree sequence} (or RDS for short) problem completely if the forbidden edges form a bipartite graph, which consist of the union of a (not necessarily maximal) 1-factor and a (possible empty) star. Then we show how one can sample the space of all realizations of these RDSs uniformly and efficiently when the degree sequence describes a {\em half-regular} bipartite graph. Our result contains, as special cases, the well-known result of Kannan, Tetali and Vempala on sampling regular bipartite graphs and a recent result of Greenhill on sampling regular directed graphs (so it also provides new proofs of them). The RDS problem descried above is self-reducible, therefore our {\em fully polynomial almost uniform sampler} (a.k.a. FPAUS) on the space of all realizations also provides a {\em fully polynomial randomized approximation scheme} (a.k.a. FPRAS) for approximate counting of all realizations.