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We study the set of all decompositions (clusterings) of a graph through itscharacterization as a set of lifted multicuts. This leads us to practicallyrelevant insights related to the definition of a class of decompositions bymust-join and must-cut constraints and related to the comparison of clusteringsby metrics. To find optimal decompositions defined by minimum cost liftedmulticuts, we establish some properties of some facets of lifted multicutpolytopes, define efficient separation procedures and apply these in abranch-and-cut algorithm.

Analysis and Optimization of Graph Decompositions by Lifted Multicuts