Given a graph \(G = (V,\!E)\) with \(n\) vertices and \(m\) edges, the Dominating Set problem asks for a set \(\mathcal{D} \subseteq V\) of minimal cardinality such that every vertex either is in \(D\) or adjacent to a member of \(D\). Although there is little hope for a kernelization algorithm on general graphs due to the W[2]-hardness of Dominating Set, data reduction rules are extensively used in practice.

We consider a maximum entropy edge weight model that allows for negative weights. Given a graph $G$ and possible weights $\mathcal{W}$ typically consisting of positive and negative values, the model selects edge weights $w \in \mathcal{W}^m$ uniformly at random from all weights that do not introduce a negative cycle. We propose an MCMC process and show that it converges to the required distribution. We then engineer an implementation of the process using a dynamic version of Johnson's algorithm in connection with a bidirectional Dijkstra search as well as an innovative resampling method. We empirically study the performance characteristics of these novel sampling algorithms as well as the output produced by the model.