Deriving compact extended formulations via LP-based separation techniques

The best formulations for some combinatorial optimization problems are integer linear programming models with an exponential number of rows and/or columns, which are solved incrementally by generating missing rows and columns only when needed. As an alternative to row generation, some exponential formulations can be rewritten in a compact extended form, which have only a polynomial number of constraints and a polynomial, although larger, number of variables. As an alternative to column generation, there are compact extended formulations for the dual problems, which lead to compact equivalent primal formulations, again with only a polynomial number of constraints and variables. In this this paper we introduce a tool to derive compact extended formulations and survey many combinatorial optimization problems for which it can be applied. The tool is based on the possibility of formulating the separation procedure by an LP model. It can be seen as one further method to generate compact extended formulations besides other tools of geometric and combinatorial nature present in the literature.