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Matthias Brugger

Publications and source records attributed to Matthias Brugger.

5 recordsLinked to original sources
arXiv2023

On the Circuit Diameter Conjecture for Counterexamples to the Hirsch Conjecture

Circuit diameters of polyhedra are a fundamental tool for studying the complexity of circuit augmentation schemes for linear programming and for finding lower bounds on combinatorial diameters. The main open problem in this area is the circuit diameter conjecture, the analogue of the Hirsch conjecture in the circuit setting. A natural question is whether the well-known counterexamples to the Hirsch conjecture carry over. Previously, Stephen and Yusun showed that the Klee-Walkup counterexample to the unbounded Hirsch conjecture does not transfer to the circuit setting. Our main contribution is to show that the original counterexamples for the other variants, for bounded polytopes and using monotone walks, also do not transfer. Our results rely on new observations on structural properties of these counterexamples. To resolve the bounded case, we exploit the geometry of certain $2$-faces of the polytopes underlying all known bounded Hirsch counterexamples in Santos' work. For Todd's monotone Hirsch counterexample, we provide two alternative approaches. The first one uses sign-compatible circuit walks, and the second one uses the observation that Todd's polytope is anti-blocking. Along the way, we enumerate all linear programs over the polytope and find four new orientations that contradict the monotone Hirsch conjecture, while the remaining $7107$ satisfy the bound.

math.CO↗
arXiv2022

Circuits in Extended Formulations

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.

math.OC↗
arXiv2021

On the Complexity of Recognizing Integrality and Total Dual Integrality of the $\{0,1/2\}$-Closure

The $\{0,\frac{1}{2}\}$-closure of a rational polyhedron $\{ x \colon Ax \le b \}$ is obtained by adding all Gomory-Chvátal cuts that can be derived from the linear system $Ax \le b$ using multipliers in $\{0,\frac{1}{2}\}$. We show that deciding whether the $\{0,\frac{1}{2}\}$-closure coincides with the integer hull is strongly NP-hard. A direct consequence of our proof is that, testing whether the linear description of the $\{0,\frac{1}{2}\}$-closure derived from $Ax \le b$ is totally dual integral, is strongly NP-hard.

cs.DM↗
arXiv2020

Limitations of the Hyperplane Separation Technique for Bounding the Extension Complexity of Polytopes

We illustrate the limitations of the hyperplane separation bound, a non-combinatorial lower bound on the extension complexity of a polytope. Most notably, this bounding technique is used by Rothvoß (J ACM 64.6:41, 2017) to establish an exponential lower bound for the perfect matching polytope. We point out that the technique is sensitive to the particular choice of slack matrix. For the canonical slack matrices of the spanning tree polytope and the completion time polytope, we show that the lower bounds produced by the hyperplane separation method are trivial. These bounds may, however, be strengthened by normalizing rows and columns of the slack matrices.

math.CO↗