  In this paper, we propose several graph-based extensions of the
Douglas-Rachford splitting (DRS) method to solve monotone inclusion problems
involving the sum of $N$ maximal monotone operators. Our construction is based
on a two-layer architecture that we refer to as bilevel graphs, to which we
associate a generalization of the DRS algorithm that presents the prescribed
structure. The resulting schemes can be understood as unconditionally stable
frugal resolvent splitting methods with a minimal lifting in the sense of Ryu
[Math Program 182(1):233-273, 2020], as well as instances of the (degenerate)
Preconditioned Proximal Point method, which provides robust convergence
guarantees. We further describe how the graph-based extensions of the DRS
method can be leveraged to design new fully distributed protocols. Applications
to a congested optimal transport problem and to distributed Support Vector
Machines show interesting connections with the underlying graph topology and
highly competitive performances with state-of-the-art distributed optimization
approaches.
