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LP relaxation-based message passing algorithms provide an effective tool forMAP inference over Probabilistic Graphical Models. However, different LPrelaxations often have different objective functions and variables of differingdimensions, which presents a barrier to effective comparison and analysis. Inaddition, the computational complexity of LP relaxation-based methods growsquickly with the number of constraints. Reducing the number of constraintswithout sacrificing the quality of the solutions is thus desirable.  We propose a unified formulation under which existing MAP LP relaxations maybe compared and analysed. Furthermore, we propose a new tool called MarginalPolytope Diagrams. Some properties of Marginal Polytope Diagrams are exploitedsuch as node redundancy and edge equivalence. We show that using MarginalPolytope Diagrams allows the number of constraints to be reduced withoutloosening the LP relaxations. Then, using Marginal Polytope Diagrams andconstraint reduction, we develop three novel message passing algorithms, anddemonstrate that two of these show a significant improvement in speed overstate-of-art algorithms while delivering a competitive, and sometimes higher,quality of solution.

Constraint Reduction using Marginal Polytope Diagrams for MAP LP Relaxations