Circuits in Extended Formulations

TL;DR

This paper investigates how circuits of a polyhedron relate to those of its extended formulations, revealing that circuit inheritance under projection is limited and can differ exponentially, with characterizations of when inheritance occurs.

Contribution

It demonstrates that circuits are not generally inherited through projections, provides minimal counterexamples, and characterizes polyhedra with universal circuit inheritance.

Findings

Circuit inheritance under projection can be exponentially large in difference.

Counterexamples exist for any fixed projection unless the map is injective.

Characterization of polyhedra with inherited circuits from all projections.

Abstract

Circuits and extended formulations are classical concepts in linear programming theory. The circuits of a polyhedron are the elementary difference vectors between feasible points and include all edge directions. We study the connection between the circuits of a polyhedron $P$ and those of an extended formulation of $P$, i.e., a description of a polyhedron $Q$ that linearly projects onto $P$. It is well known that the edge directions of $P$ are images of edge directions of $Q$. We show that this `inheritance' under taking projections does not extend to the set of circuits. We provide counterexamples with a provably minimal number of facets, vertices, and extreme rays, including relevant polytopes from clustering, and show that the difference in the number of circuits that are inherited and those that are not can be exponentially large in the dimension. We further prove that counterexamples exist for any fixed linear projection map, unless the map is injective. Finally, we characterize those polyhedra $P$ whose circuits are inherited from all polyhedra $Q$ that linearly project onto $P$. Conversely, we prove that every polyhedron $Q$ satisfying mild assumptions can be projected in such a way that the image polyhedron $P$ has a circuit with no preimage among the circuits of $Q$. Our proofs build on standard constructions such as homogenization and disjunctive programming.