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==Total dual integrality==
Total dual integrality has probably been the most powerful tool to prove integrality of polyhedra arising from combinatorial problems. The reason is that very often the natural linear description of the integer polyhedron turns out to be totally dual integral. Several conjectures suggest that this is true for large classes on integer polyhedra. One of the most striking is the [[Conforti-Cornuéjols conjecture on [[clutters with the MFMC property]]:{{IncludeOpenProblem|Clutters with Conforti-Cornuéjols conjecture on the MFMC property}}
A well-studied class is the system of clique inequalities of graphs: the system is TDI if and only if it defines an integer polyhedron (i.e. the graph is perfect). It is an exciting question how far this can be extended. One particular question concerns [[Are t-perfect graphs strongly t-perfect?|t-perfect graphs]]:
==Integer preimages==
The integer decomposition property can be considered as part of a larger framework: we say that a polytope ''P'' and a linear transformation ''f'' have the '''integer preimage property''' if every integer point in ''f(P)'' has an integer preimage in ''P''. The task is to find classes of polytope-transformation pairs which have this property. For example:
* ''P'' is a polymatroid and ''f'' is a projection,
* The integer decomposition property is obtained by considering <math>f:{\mathbb R}^{nk}\to {\mathbb R}^n</math> defined by <math>f(x_1,\dots,x_k)=x_1+\dots+x_k</math>.
