arXiv subjects

Volker Kaibel

Publications and source records attributed to Volker Kaibel.

At least 37 records · Page 2Linked to original sources

arXiv2012

Combinatorial Bounds on Nonnegative Rank and Extended Formulations

An extended formulation of a polytope P is a polytope Q which can be projected onto P. Extended formulations of small size (i.e., number of facets) are of interest, as they allow to model corresponding optimization problems as linear programs of small sizes. The main known lower bounds on the minimum sizes of extended formulations for fixed polytope P (Yannakakis 1991) are closely related to the concept of nondeterministic communication complexity. We study the relative power and limitations of the bounds on several examples.

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arXiv2011

Orbitopal Fixing

The topic of this paper are integer programming models in which a subset of 0/1-variables encode a partitioning of a set of objects into disjoint subsets. Such models can be surprisingly hard to solve by branch-and-cut algorithms if the order of the subsets of the partition is irrelevant, since this kind of symmetry unnecessarily blows up the search tree. We present a general tool, called orbitopal fixing, for enhancing the capabilities of branch-and-cut algorithms in solving such symmetric integer programming models. We devise a linear time algorithm that, applied at each node of the search tree, removes redundant parts of the tree produced by the above mentioned symmetry. The method relies on certain polyhedra, called orbitopes, which have been introduced bei Kaibel and Pfetsch (Math. Programm. A, 114 (2008), 1-36). It does, however, not explicitly add inequalities to the model. Instead, it uses certain fixing rules for variables. We demonstrate the computational power of orbitopal fixing at the example of a graph partitioning problem.

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arXiv2011

Finding Descriptions of Polytopes via Extended Formulations and Liftings

We describe a technique to obtain linear descriptions for polytopes from extended formulations. The simple idea is to first define a suitable lifting function and then to find linear constraints that are valid for the polytope and guarantee lifted points to be contained in the extension. We explain the technique at an example from the literature (matching polytopes), obtain new simple proofs of results on path-set polytopes and small-cliques polytopes, and finally exploit the technique in order to derive linear descriptions of orbisacks, which are special Knapsack polytopes arising in the context of symmetry breaking in integer programming problems.

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arXiv2011

Extended Formulations in Combinatorial Optimization

The concept of representing a polytope that is associated with some combinatorial optimization problem as a linear projection of a higher-dimensional polyhedron has recently received increasing attention. In this paper (written for the newsletter Optima of the Mathematical Optimization Society), we provide a brief introduction to this topic and sketch some of the recent developments with respect to both tools for constructing such extended formulations as well as lower bounds on their sizes.

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arXiv2010

Constructing Extended Formulations from Reflection Relations

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).

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arXiv2010

Basic Polyhedral Theory

This is a chapter (planned to appear in Wiley's upcoming Encyclopedia of Operations Research and Management Science) describing parts of the theory of convex polyhedra that are particularly important for optimization. The topics include polyhedral and finitely generated cones, the Weyl-Minkowski Theorem, faces of polyhedra, projections of polyhedra, integral polyhedra, total dual integrality, and total unimodularity.

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arXiv2009

Another Proof of the Fact that Polyhedral Cones are Finitely Generated

In this note, we work out a simple inductive proof showing that every polyhedral cone K is the conic hull of a finite set X of vectors. The base cases of the induction are linear subspaces and linear halfspaces of linear subspaces. The proof also shows that the components of the vectors in X can be chosen (up to their sign) to be quotients of subdeterminants of the coefficient matrix of any inequality system defining K.

math.CO↗

arXiv2008

Two New Bounds on the Random-Edge Simplex Algorithm

We prove that the Random-Edge simplex algorithm requires an expected number of at most 13n/sqrt(d) pivot steps on any simple d-polytope with n vertices. This is the first nontrivial upper bound for general polytopes. We also describe a refined analysis that potentially yields much better bounds for specific classes of polytopes. As one application, we show that for combinatorial d-cubes, the trivial upper bound of 2^d on the performance of Random-Edge can asymptotically be improved by any desired polynomial factor in d.

math.CO↗

arXiv2008

Extended Formulations for Packing and Partitioning Orbitopes

We give compact extended formulations for the packing and partitioning orbitopes (with respect to the full symmetric group) described and analyzed in (Kaibel and Pfetsch, 2008). These polytopes are the convex hulls of all 0/1-matrices with lexicographically sorted columns and at most, resp. exactly, one 1-entry per row. They are important objects for symmetry reduction in certain integer programs. Using the extended formulations, we also derive a rather simple proof of the fact that basically shifted-column inequalities suffice in order to describe those orbitopes linearly.

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arXiv2007

A Short Proof of the VPN Tree Routing Conjecture on Ring Networks

The VPN Tree Routing Conjecture states that there always exists an optimal solution to the symmetric Virtual Private Network Design (sVPND) problem where the paths between all terminals form a tree. Only recently, Hurkens, Keijsper, and Stougie gave a proof of this conjecture for the special case of ring networks. Their proof is based on a dual pair of linear programs and is somewhat in- volved. We present a short proof of a slightly stronger conjecture which might also turn out to be useful for proving the VPN Tree Routing Conjecture for general networks.

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arXiv2007

On cardinality constrained cycle and path polytopes

Given a directed graph D = (N, A) and a sequence of positive integers 1 <= c_1 < c_2 < ... < c_m <= |N|, we consider those path and cycle polytopes that are defined as the convex hulls of simple paths and cycles of D of cardinality c_p for some p, respectively. We present integer characterizations of these polytopes by facet defining linear inequalities for which the separation problem can be solved in polynomial time. These inequalities can simply be transformed into inequalities that characterize the integer points of the undirected counterparts of cardinality constrained path and cycle polytopes. Beyond we investigate some further inequalities, in particular inequalities that are specific to odd/even paths and cycles.

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arXiv2006

Packing and Partitioning Orbitopes

We introduce orbitopes as the convex hulls of 0/1-matrices that are lexicographically maximal subject to a group acting on the columns. Special cases are packing and partitioning orbitopes, which arise from restrictions to matrices with at most or exactly one 1-entry in each row, respectively. The goal of investigating these polytopes is to gain insight into ways of breaking certain symmetries in integer programs by adding constraints, e.g., for a well-known formulation of the graph coloring problem. We provide a thorough polyhedral investigation of packing and partitioning orbitopes for the cases in which the group acting on the columns is the cyclic group or the symmetric group. Our main results are complete linear inequality descriptions of these polytopes by facet-defining inequalities. For the cyclic group case, the descriptions turn out to be totally unimodular, while for the symmetric group case both the description and the proof are more involved. Nevertheless, the associated separation problem can be solved in linear time also in this case.

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arXiv2005

Revlex-Initial 0/1-Polytopes

We introduce revlex-initial 0/1-polytopes as the convex hulls of reverse-lexicographically initial subsets of 0/1-vectors. These polytopes are special knapsack-polytopes. It turns out that they have remarkable extremal properties. In particular, we use these polytopes in order to prove that the minimum numbers f(d, n) of facets and the minimum average degree a(d, n) of the graph of a d-dimensional 0/1-polytope with n vertices satisfy f(d, n) <= 3d and a(d, n) <= d + 4. We furthermore show that, despite the sparsity of their graphs, revlex-initial 0/1-polytopes satisfy a conjecture due to Mihail and Vazirani, claiming that the graphs of 0/1-polytopes have edge-expansion at least one.

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arXiv2004

The Simplex Algorithm in Dimension Three

We investigate the worst-case behavior of the simplex algorithm on linear programs with three variables, that is, on 3-dimensional simple polytopes. Among the pivot rules that we consider, the ``random edge'' rule yields the best asymptotic behavior as well as the most complicated analysis. All other rules turn out to be much easier to study, but also produce worse results: Most of them show essentially worst-possible behavior; this includes both Kalai's ``random-facet'' rule, which without dimension restriction is known to be subexponential, as well as Zadeh's deterministic history dependent rule, for which no non-polynomial instances in general dimensions have been found so far.

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arXiv2004

Low-dimensional faces of random 0/1-polytopes

Let P be a random $d$-dimensional 0/1-polytope with $n(d)$ vertices, and denote by $ϕ_k(P)$ the \emph{$k$-face density} of $P$, i.e., the quotient of the number of $k$-dimensional faces of $P$ and $\binom{n(d)}{k+1}$. For each $k\ge 2$, we establish the existence of a sharp threshold for the $k$-face density and determine the values of the threshold numbers $τ_k$ such that, for all $ε>0$, $$ E(ϕ_k(P)) = \begin{cases} 1-o(1) & \text{if $n(d)\le 2^{(τ_k-ε)d}$ for all $d$} o(1) & \text{if $n(d)\ge 2^{(τ_k+ε)d}$ for all $d$} \end{cases} $$ holds for the expected value of $ϕ_k(P)$. The threshold for $k=1$ has recently been determined in \texttt{math.CO/0306246}. In particular, these results indicate that the high face densities often encountered in polyhedral combinatorics (e.g., for the cut-polytopes of complete graphs) should be considered more as a phenomenon of the general geometry of 0/1-polytopes than as a feature of the special combinatorics of the underlying problems.

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arXiv2003

On the graph-density of random 0/1-polytopes

Let X_{d,n} be an n-element subset of {0,1}^d chosen uniformly at random, and denote by P_{d,n} := conv X_{d,n} its convex hull. Let D_{d,n} be the density of the graph of P_{d,n} (i.e., the number of one-dimensional faces of P_{d,n} divided by n(n-1)/2). Our main result is that, for any function n(d), the expected value of D_{d,n(d)} converges (with d tending to infinity) to one if, for some arbitrary e > 0, n(d) <= (\sqrt{2}-e)^d holds for all large d, while it converges to zero if n(d) >= (\sqrt{2}+e)^d holds for all large d.

math.CO↗

arXiv2003

The Random Edge Rule on Three-Dimensional Linear Programs

The worst-case expected length f(n) of the path taken by the simplex algorithm with the Random Edge pivot rule on a 3-dimensional linear program with n constraints is shown to be bounded by 1.3445 n <= f(n) <= 1.4943 n for large enough n.

math.CO↗

arXiv2002

Counting Lattice Triangulations

We discuss the problem to count, or, more modestly, to estimate the number f(m,n) of unimodular triangulations of the planar grid of size $m\times n$. Among other tools, we employ recursions that allow one to compute the (huge) number of triangulations for small m and rather large n by dynamic programming; we show that this computation can be done in polynomial time if m is fixed, and present computational results from our implementation of this approach. We also present new upper and lower bounds for large m and n, and we report about results obtained from a computer simulation of the random walk that is generated by flips.

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