Sorting in c log n parallel steps
Combinatorica
Polyhedral characterization of discrete dynamic programming
Operations Research
Structure of a simple scheduling polyhedron
Mathematical Programming: Series A and B
On Polyhedral Approximations of the Second-Order Cone
Mathematics of Operations Research
Symmetry matters for the sizes of extended formulations
IPCO'10 Proceedings of the 14th international conference on Integer Programming and Combinatorial Optimization
IPCO'10 Proceedings of the 14th international conference on Integer Programming and Combinatorial Optimization
An algebraic approach to symmetric extended formulations
ISCO'12 Proceedings of the Second international conference on Combinatorial Optimization
On the linear description of the Huffman trees polytope
Discrete Applied Mathematics
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There are many examples of optimization problems whose associated polyhedra can be described much nicer, and with way less inequalities, by projections of higher dimensional polyhedra than this would be possible in the original space. However, currently not many general tools to construct such extended formulations are available. In this paper, we develop a framework of polyhedral relations that generalizes inductive constructions of extended formulations via projections, and we particularly elaborate on the special case of reflection relations. The latter ones provide polynomial size extended formulations for several polytopes that can be constructed as convex hulls of the unions of (exponentially) many copies of an input polytope obtained via sequences of reflections at hyperplanes. We demonstrate the use of the framework by deriving small extended formulations for the G-permutahedra of all finite reflection groups G (generalizing both Goeman's [6] extended formulation of the permutahedron of size O(n log n) and Ben-Tal and Nemirovski's [2] extended formulation with O(k) inequalities for the regular 2k-gon) and for Huffman-polytopes (the convex hulls of the weight-vectors of Huffman codes).