Extended Formulations for Combinatorial Polytopes

i Abstract Typically polytopes arising from real world problems have a lot of facets. In some cases even no linear descriptions for them are known. On the other hand many of these polytopes can be described much nicer and with less facets using extended formulations, i.e. as a projection of simpler higher dimensional polytopes. The presented work studies extended formulations for polytopes: the possibilities to construct extended formulations and limitations of them. In the first part, some known techniques for constructions of extended formulations are reviewed and the new framework of polyhedral relations (see Kaibel and Pashkovich [2011]) is presented. We in particular elaborate on the special case of reflection relations. Reflection relations provide extended formulations for several polytopes that can be constructed by iteratively taking convex hulls of polytopes and their refelections at hyperplanes. Using this framework we are able to derive small extended formulations for the G-permutahedra of all finite reflection groups G. The second part deals with extended formulations which use special structures of graphs involved in combinatorial problems. Here we present some known extended formulations and apply a few changes to the extended formulation of Gerards for the perfect matching polytope in graphs with small genus in order to reduce its size. Furthermore a new compact proof of an extended formulation of Rivin for the subtour elimination polytope is provided. The third part (partly based on joint work with Volker Kaibel, Samuel Fiorini and Dirk Oliver Theis, see Fiorini, Kaibel, Pashkovich, and Theis [2011a]) involves general questions on the extended formulations of polytopes. The primal interest here are lower bounds for extended formulations. We study different techniques to obtain these lower bounds, all of which could be derived from so called non-negative factorizations of the slack matrix of the initial polytope. The minimal such factorization provides the minimal number of inequalities needed in an extended formulation. We compare different techniques, find their limitations and provide examples of the polytopes for which they give tight lower bounds on the complexity of extensions. The fourth part studies the impact of symmetry on the sizes of extended formulations. In joint work with Volker Kaibel and Dirk Oliver Theis we showed that for certain constrained cardinality matching and cycle polytopes there exist no polynomial symmetric extended formulations, but there are polynomial non-symmetric ones (for further details see Kaibel, Pashkovich, and Theis [2010]). Beyond these results the thesis also contains a proof showing that the well known symmetric extended formulation for the permutahedron via the Birkhoff polytope is the best (up to a constant factor) one among symmetric extended formulations (see Pashkovich [2009]). Typically polytopes arising from real world problems have a lot of facets. In some cases even no linear descriptions for them are known. On the other hand many of these polytopes can be described much nicer and with less facets using extended formulations, i.e. as a projection of simpler higher dimensional polytopes. The presented work studies extended formulations for polytopes: the possibilities to construct extended formulations and limitations of them. In the first part, some known techniques for constructions of extended formulations are reviewed and the new framework of polyhedral relations (see Kaibel and Pashkovich [2011]) is presented. We in particular elaborate on the special case of reflection relations. Reflection relations provide extended formulations for several polytopes that can be constructed by iteratively taking convex hulls of polytopes and their refelections at hyperplanes. Using this framework we are able to derive small extended formulations for the G-permutahedra of all finite reflection groups G. The second part deals with extended formulations which use special structures of graphs involved in combinatorial problems. Here we present some known extended formulations and apply a few changes to the extended formulation of Gerards for the perfect matching polytope in graphs with small genus in order to reduce its size. Furthermore a new compact proof of an extended formulation of Rivin for the subtour elimination polytope is provided. The third part (partly based on joint work with Volker Kaibel, Samuel Fiorini and Dirk Oliver Theis, see Fiorini, Kaibel, Pashkovich, and Theis [2011a]) involves general questions on the extended formulations of polytopes. The primal interest here are lower bounds for extended formulations. We study different techniques to obtain these lower bounds, all of which could be derived from so called non-negative factorizations of the slack matrix of the initial polytope. The minimal such factorization provides the minimal number of inequalities needed in an extended formulation. We compare different techniques, find their limitations and provide examples of the polytopes for which they give tight lower bounds on the complexity of extensions. The fourth part studies the impact of symmetry on the sizes of extended formulations. In joint work with Volker Kaibel and Dirk Oliver Theis we showed that for certain constrained cardinality matching and cycle polytopes there exist no polynomial symmetric extended formulations, but there are polynomial non-symmetric ones (for further details see Kaibel, Pashkovich, and Theis [2010]). Beyond these results the thesis also contains a proof showing that the well known symmetric extended formulation for the permutahedron via the Birkhoff polytope is the best (up to a constant factor) one among symmetric extended formulations (see Pashkovich [2009]).