  We consider the problem of decentralized composite optimization over a
symmetric connected graph, in which each node holds its own agent-specific
private convex functions, and communications are only allowed between nodes
with direct links. A variety of algorithms have been proposed to solve such a
problem in an alternating direction method of multiplier (ADMM) framework. Many
of these algorithms, however, need to include some proximal term in the
augmented Lagrangian function such that the resulting algorithm can be
implemented in a decentralized manner. The use of the proximal term slows down
the convergence speed because it forces the current solution to stay close to
the solution obtained in the previous iteration. To address this issue, in this
paper, we first introduce the notion of simplest bipartite graph, which is
defined as a bipartite graph that has a minimum number of edges to keep the
graph connected. A simple two-step message passing-based procedure is proposed
to find a simplest bipartite graph associated with the original graph. We show
that the simplest bipartite graph has some interesting properties. By utilizing
these properties, an ADMM without involving any proximal terms can be developed
to perform decentralized composite optimization over the simplest bipartite
graph. Simulation results show that our proposed method achieves a much faster
convergence speed than existing state-of-the-art decentralized algorithms.
