Constructing Extended Formulations from Reflection Relations

TL;DR

This paper introduces a framework for constructing extended formulations of polytopes using reflection relations, enabling more efficient representations of complex polytopes like permutahedra and Huffman-polytopes.

Contribution

It develops a general polyhedral relations framework that extends inductive construction methods, particularly focusing on reflection relations for polynomial-size formulations.

Findings

Polynomial size extended formulations for polytopes from reflection groups

Efficient formulations for G-permutahedra of all finite reflection groups

Compact representations of Huffman-polytopes

Abstract

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 extended formulation of the permutahedron of size O(n log n) and Ben-Tal and Nemirovski's extended formulation with O(k) inequalities for the regular 2^k-gon) and for Huffman-polytopes (the convex hulls of the weight-vectors of Huffman codes).