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Volker Kaibel

Publications and source records attributed to Volker Kaibel.

At least 19 recordsLinked to original sources
arXiv2026

A Counterexample to Ziegler's Cross-Polytope Conjecture for Simplicial 0/1-Polytopes

Ziegler proved that every simplicial $d$-dimensional $0/1$-polytope has at most $2d$ vertices, and asked whether equality forces the polytope to be centrally symmetric and hence, equivalently, a $0/1$-realization of the $d$-dimensional cross polytope. In this note, we give a negative answer, exhibiting an explicit set of $14$ vertices in $\{0,1\}^7$ whose convex hull is a simplicial $7$-polytope and is not centrally symmetric. Moreover, via exhaustive enumeration we show that up to the symmetries of the cube, there are precisely five such polytopes in dimension $7$ (of two combinatorial types) that are not centrally symmetric.

math.CO↗
arXiv2024

Polytope Extensions with Linear Diameters

We describe constructions of extended formulations that establish a certain relaxed version of the Hirsch conjecture and prove that if there is a pivot rule for the simplex algorithm for which one can bound the number of steps by a polynomial in the diameter plus the number of facets of the polyhedron of feasible solutions then the general linear programming problem can be solved in strongly polynomial time.

math.CO↗
arXiv2024

Binary Cyclic Transversal Polytopes

With every family of finitely many subsets of a finite-dimensional vector space over the Galois-field with two elements we associate a cyclic transversal polytope. It turns out that those polytopes generalize several well-known polytopes that are relevant in combinatorial optimization, among them cut polytopes as well as stable set and matching polytopes. We introduce the class of lifted odd-set inequalities and prove results demonstrating their strength. In particular, we show that they suffice to describe cyclic transversal polytopes if the union of the sets in the family has rank at most two. We also describe extended formulations for cyclic transversal polytopes and introduce a special relaxation hierarchy for them.

math.CO↗
arXiv2024

Steiner Cut Dominants

For a subset T of nodes of an undirected graph G, a T-Steiner cut is a cut δ(S) where S intersects both T and the complement of T. The T-Steiner cut dominant} of G is the dominant CUT_+(G,T) of the convex hull of the incidence vectors of the T-Steiner cuts of G. For T={s,t}, this is the well-understood s-t-cut dominant. Choosing T as the set of all nodes of G, we obtain the \emph{cut dominant}, for which an outer description in the space of the original variables is still not known. We prove that, for each integer τ, there is a finite set of inequalities such that for every pair (G,T) with |T|\ <= τthe non-trivial facet-defining inequalities of CUT_+(G,T) are the inequalities that can be obtained via iterated applications of two simple operations, starting from that set. In particular, the absolute values of the coefficients and of the right-hand-sides in a description of CUT_+(G,T) by integral inequalities can be bounded from above by a function of |T|. For all |T| <= 5 we provide descriptions of CUT_+(G,T) by facet defining inequalities, extending the known descriptions of s-t-cut dominants.

math.CO↗
arXiv2024

Refined TSSOS

The moment-sum of squares hierarchy by Lasserre has become an established technique for solving polynomial optimization problems. It provides a monotonically increasing series of tight bounds, but has well-known scalability limitations. For structured optimization problems, the term-sparsity SOS (TSSOS) approach scales much better due to block-diagonal matrices, obtained by completing the connected components of adjacency graphs. This block structure can be exploited by semidefinite programming solvers, for which the overall runtime then depends heavily on the size of the largest block. However, already the first step of the TSSOS hierarchy may result in large diagonal blocks. We suggest a new approach that refines TSSOS iterations using combinatorial optimization and results in block-diagonal matrices with reduced maximum block sizes. Numerical results on a benchmark library show the large potential for computational speedup for unconstrained and constrained polynomial optimization problems, while obtaining almost identical bounds in comparison to established methods.

math.OC↗
arXiv2023

Optimal Sufficient Requirements on the Embedded Ising Problem in Polynomial Time

One of the central applications for quantum annealers is to find the solutions of Ising problems. Suitable Ising problems, however, need to be formulated such that they, on the one hand, respect the specific restrictions of the hardware and, on the other hand, represent the original problems which shall actually be solved. We evaluate sufficient requirements on such an embedded Ising problem analytically and transform them into a linear optimization problem. With an objective function aiming to minimize the maximal absolute problem parameter, the precision issues of the annealers are addressed. Due to the redundancy of several constraints, we can show that the formally exponentially large optimization problem can be reduced and finally solved in polynomial time for the standard embedding setting where the embedded vertices induce trees. This allows to formulate provably equivalent embedded Ising problems in a practical setup.

quant-ph↗
arXiv2020

Scale-free spanning trees: complexity, bounds and algorithms

We introduce and study the general problem of finding a most "scale-free-like" spanning tree of a connected graph. It is motivated by a particular problem in epidemiology, and may be useful in studies of various dynamical processes in networks. We employ two possible objective functions for this problem and introduce the corresponding algorithmic problems termed $m$-SF and $s$-SF Spanning Tree problems. We prove that those problems are APX- and NP-hard, respectively, even in the classes of cubic, bipartite and split graphs. We study the relations between scale-free spanning tree problems and the max-leaf spanning tree problem, which is the classical algorithmic problem closest to ours. For split graphs, we explicitly describe the structure of optimal spanning trees and graphs with extremal solutions. Finally, we propose two Integer Linear Programming formulations and two fast heuristics for the $s$-SF Spanning Tree problem, and experimentally assess their performance using simulated and real data.

math.CO↗
arXiv2017

A Note on Matchings Constructed during Edmonds' Weighted Perfect Matching Algorithm

We reprove that all the matchings constructed during Edmonds' weighted perfect matching algorithm are optimal among those of the same cardinality (provided that certain mild restrictions are obeyed on the choices the algorithm makes). We conclude that in order to solve a weighted matching problem it is not needed to solve a weighted perfect matching problem in an auxiliary graph of doubled size. This result was known before, e.g., posed as an exercise in see Lawler's book from 1976, but is not present in several modern books on combinatorial optimization.

cs.DM↗
arXiv2016

Maximum Semidefinite and Linear Extension Complexity of Families of Polytopes

We relate the maximum semidefinite and linear extension complexity of a family of polytopes to the cardinality of this family and the minimum pairwise Hausdorff distance of its members. This result directly implies a known lower bound on the maximum semidefinite extension complexity of 0/1-polytopes. We further show how our result can be used to improve on the corresponding bounds known for polygons with integer vertices. Our geometric proof builds upon nothing else than a simple well-known property of maximum volume inscribed ellipsoids of convex bodies. In particular, it does not rely on factorizations over the semidefinite cone and thus avoids involved procedures of balancing them as required, e.g., in [Briet, Dadush & Pokutta 2015]. We hope that revealing the geometry behind the phenomenon opens doors for further results. Moreover, we show that the linear extension complexity of every d-dimensional 0/1-polytope is bounded from above by O(2^d / d).

math.OC↗
arXiv2016

A Short Proof that the Extension Complexity of the Correlation Polytope Grows Exponentially

We establish that the extension complexity of the nXn correlation polytope is at least 1.5^n by a short proof that is self-contained except for using the fact that every face of a polyhedron is the intersection of all facets it is contained in. The main innovative aspect of the proof is a simple combinatorial argument showing that the rectangle covering number of the unique-disjointness matrix is at least 1.5^n, and thus the nondeterministic communication complexity of the unique-disjointness predicate is at least .58n. We thereby slightly improve on the previously best known lower bounds 1.24^n and .31n, respectively.

math.CO↗
arXiv2015

The Unimodular Intersection Problem

We show that finding minimally intersecting $n$ paths from $s$ to $t$ in a directed graph or $n$ perfect matchings in a bipartite graph can be done in polynomial time. This holds more generally for unimodular set systems.

math.OC↗
arXiv2015

Subgraph Polytopes and Independence Polytopes of Count Matroids

Given an undirected graph, the non-empty subgraph polytope is the convex hull of the characteristic vectors of pairs (F, S) where S is a non-empty subset of nodes and F is a subset of the edges with both endnodes in S. We obtain a strong relationship between the non-empty subgraph polytope and the spanning forest polytope. We further show that these polytopes provide polynomial size extended formulations for independence polytopes of count matroids, which generalizes recent results obtained by Iwata et al. referring to sparsity matroids. As a byproduct, we obtain new lower bounds on the extension complexity of the spanning forest polytope in terms of extension complexities of independence polytopes of these matroids.

cs.DM↗
arXiv2015

Simple Extensions of Polytopes

We introduce the simple extension complexity of a polytope P as the smallest number of facets of any simple (i.e., non-degenerate in the sense of linear programming) polytope which can be projected onto P. We devise a combinatorial method to establish lower bounds on the simple extension complexity and show for several polytopes that they have large simple extension complexities. These examples include both the spanning tree and the perfect matching polytopes of complete graphs, uncapacitated flow polytopes for non-trivially decomposable directed acyclic graphs, hypersimplices, and random 0/1-polytopes with vertex numbers within a certain range. On our way to obtain the result on perfect matching polytopes we generalize a result of Padberg and Rao's on the adjacency structures of those polytopes. To complement the lower bounding techniques we characterize in which cases known construction techniques yield simple extensions.

math.CO↗
arXiv2014

Lower Bounds on the Sizes of Integer Programs Without Additional Variables

Let X be the set of integer points in some polyhedron. We investigate the smallest number of facets of any polyhedron whose set of integer points is X. This quantity, which we call the relaxation complexity of X, corresponds to the smallest number of linear inequalities of any integer program having X as the set of feasible solutions that does not use auxiliary variables. We show that the use of auxiliary variables is essential for constructing polynomial size integer programming formulations in many relevant cases. In particular, we provide asymptotically tight exponential lower bounds on the relaxation complexity of the integer points of several well-known combinatorial polytopes, including the traveling salesman polytope and the spanning tree polytope.

math.CO↗
arXiv2014

Forbidden vertices

In this work, we introduce and study the forbidden-vertices problem. Given a polytope P and a subset X of its vertices, we study the complexity of linear optimization over the subset of vertices of P that are not contained in X. This problem is closely related to finding the k-best basic solutions to a linear problem. We show that the complexity of the problem changes significantly depending on the encoding of both P and X. We provide additional tractability results and extended formulations when P has binary vertices only. Some applications and extensions to integral polytopes are discussed.

math.OC↗
arXiv2013

Which Nonnegative Matrices Are Slack Matrices?

In this paper we characterize the slack matrices of cones and polytopes among all nonnegative matrices. This leads to an algorithm for deciding whether a given matrix is a slack matrix. The underlying decision problem is equivalent to the polyhedral verification problem whose complexity is unknown.

math.OC↗
arXiv2012

Symmetry Matters for Sizes of Extended Formulations

In 1991, Yannakakis (J. Comput. System Sci., 1991) proved that no symmetric extended formulation for the matching polytope of the complete graph K_n with n nodes has a number of variables and constraints that is bounded subexponentially in n. Here, symmetric means that the formulation remains invariant under all permutations of the nodes of K_n. It was also conjectured in the paper mentioned above that "asymmetry does not help much," but no corresponding result for general extended formulations has been found so far. In this paper we show that for the polytopes associated with the matchings in K_n with log(n) (rounded down) edges there are non-symmetric extended formulations of polynomial size, while nevertheless no symmetric extended formulations of polynomial size exist. We furthermore prove similar statements for the polytopes associated with cycles of length log(n) (rounded down). Thus, with respect to the question for smallest possible extended formulations, in general symmetry requirements may matter a lot. Compared to the extended abtract that has appeared in the Proceedings of IPCO XIV at Lausanne, this paper does not only contain proofs that had been ommitted there, but it also presents slightly generalized and sharpened lower bounds.

math.CO↗