Fig. 3

Illustration of the forward optimal evaluation and traceback algorithms (by the running example of graph coloring; Fig. 1). We elaborate steps of the computation guided by the gentle tree decomposition corresponding to Fig. 2B (top). The indices of variables in the difference set are underlined. On the left, we sketch the computation of the messages \(m_{w\rightarrow u}\) and \(m_{u\rightarrow v}\): For every assignment of the separator set, the algorithm maximizes over assignments of the difference variable (it dismisses invalid assignments); in the computation of \(m_{u\rightarrow v}\), it used the already computed message \(m_{w\rightarrow u}\). On the right, we show the corresponding computations to assign values to the underlined variables during traceback: given an optimal assignment to the variables in v, we first infer that \(X_5=2\) is an optimal continuation, and finally \(X_3=2\)