Figure 1.

(a) Problem graph of Eq. (7). Vertices and edges represent binary variables $x_i^{(q)}$ and the interactions between them, respectively. Although the penalty term generates fully connected vertical interactions between $x_i^{(q)}$ and $x_i^{(q\prime )}$, these are not shown for simplicity. Q binary variables ${\{x_i^{(q)}\}}_{q=1,2,...,Q}$ are assigned to each S_i, and the total number of binary variables is NQ. Although there exist 2^NQ configurations of the binary variables, only Q^N configurations satisfy the one-hot constraint. (b) Example of an undesirable partition, where binary variables enclosed by the green rectangle are extracted as a subproblem. The binary variables colored blue represent the tentatively selected components in the current solution. Superior feasible solutions cannot be explored by optimizing the extracted subproblem.