by assuming the network $\mathcal{G}$ is bipartite or, equivalently, that $\chi(\mathcal{G}) = 2$, i.e., it can be colored with two colors. An example of a bipartite network is a grid. We start with this assumption for two reasons: the exposition is simpler, and we prove that our algorithm converges for this case. Without loss of generality, assume the nodes are ordered such that $\mathcal{C}_1 = \{1,2,\ldots,C_1\}$ and $\mathcal{C}_2 = \{C_1+1,C_1+2,\ldots,C\}$.

A common technique to decouple problem  is to assign copies of the global variable $x$ to each node and then constrain all copies to be equal. Denoting the copy held by node $p$ with $x_p \in \mathbb{R}^n$, problem  is written equivalently as $$\label{Eq:BipClonedProb}
    \begin{array}{cl}
      \underset{\bar{x} = (x_1,\ldots,x_P)}{\text{minimize}} & f_1(x_1) + f_2(x_2) +\cdots + f_P(x_P) \\
      \text{subject to} & x_p \in X_p\,,\quad p = 1,\ldots,P \\
                        & x_i = x_j \,,\quad \{i,j\} \in \mathcal{E}\,,
    \end{array}$$ where $\bar{x}= (x_1,\ldots,x_P) \in (\mathbb{R}^n)^P$ is the optimization variable. Problem  is no longer coupled by the intersection of the sets $X_p$ or by a common variable in all $f_p$’s, as , but instead by the new equations $x_i=x_j$, for all pairs of edges in the network $\{i,j\} \in \mathcal{E}$. These equations enforce all copies to be 