Preordering: A hybrid of correlation clustering and partial ordering

We discuss the preordering problem, a joint relaxation of the correlation clustering problem and the partial ordering problem. We show that preordering remains NP-hard even for values in $\{-1,0,1\}$. We introduce a linear-time $4$-approximation algorithm and a local search technique. For an integer linear program formulation, we establish a class of non-canonical facets of the associated preorder polytope. By solving a non-canonical linear program relaxation, we obtain non-trivial upper bounds on the objective value. We provide implementations of the algorithms we define, apply these to published social networks and compare the output and efficiency qualitatively and quantitatively.