Searcharxiv⌕ Search

arXiv subjects

Stefan Weltge

Publications and source records attributed to Stefan Weltge.

At least 37 records · Page 2Linked to original sources
arXiv2020

Minimum-cost integer circulations in given homology classes

Let $D$ be a directed graph cellularly embedded in a surface together with non-negative cost on its arcs. Given any integer circulation in $D$, we study the problem of finding a minimum-cost non-negative integer circulation in $D$ that is homologous over the integers to the given circulation. A special case of this problem arises in recent work on the stable set problem for graphs with bounded odd cycle packing number, in which the surface is non-orientable (Conforti et al., SODA'20). For orientable surfaces, polynomial-time algorithms have been obtained for different variants of this problem. We complement these results by showing that the convex hull of feasible solutions has a very simple polyhedral description. In contrast, only little seems to be known about the case of non-orientable surfaces. We show that the problem is strongly NP-hard for general non-orientable surfaces, and give the first polynomial-time algorithm for surfaces of fixed genus. For the latter, we provide a characterization of homology (over the integers) that allows us to recast the problem as a special integer program, which can be efficiently solved using proximity results and dynamic programming.

cs.DM↗
arXiv2020

Binary scalar products

Let $A,B \subseteq \mathbb{R}^d $ both span $\mathbb{R}^d$ such that $\langle a, b \rangle \in \{0,1\}$ holds for all $a \in A$, $b \in B$. We show that $ |A| \cdot |B| \le (d+1) 2^d $. This allows us to settle a conjecture by Bohn, Faenza, Fiorini, Fisikopoulos, Macchia, and Pashkovich (2015) concerning 2-level polytopes. Such polytopes have the property that for every facet-defining hyperplane $H$ there is a parallel hyperplane $H'$ such that $H \cup H'$ contain all vertices. The authors conjectured that for every $d$-dimensional 2-level polytope $P$ the product of the number of vertices of $P$ and the number of facets of $P$ is at most $d 2^{d+1}$, which we show to be true.

math.CO↗
arXiv2020

Improved lower bound on the dimension of the EU council's voting rules

Kurz and Napel (2015) proved that the voting system of the EU council (based on the 2014 population data) cannot be represented as the intersection of six weighted games, i.e., its dimension is at least 7. This set a new record for real-world voting rules and the authors posed the exact determination as a challenge. Recently, Chen, Cheung, and Ng (2019) showed that the dimension is at most 24. We provide the first improved lower bound and show that the dimension is at least 8.

math.OC↗
arXiv2019

The stable set problem in graphs with bounded genus and bounded odd cycle packing number

Consider the family of graphs without $ k $ node-disjoint odd cycles, where $ k $ is a constant. Determining the complexity of the stable set problem for such graphs $ G $ is a long-standing problem. We give a polynomial-time algorithm for the case that $ G $ can be further embedded in a (possibly non-orientable) surface of bounded genus. Moreover, we obtain polynomial-size extended formulations for the respective stable set polytopes. To this end, we show that $2$-sided odd cycles satisfy the Erdős-Pósa property in graphs embedded in a fixed surface. This extends the fact that odd cycles satisfy the Erdős-Pósa property in graphs embedded in a fixed orientable surface (Kawarabayashi & Nakamoto, 2007). Eventually, our findings allow us to reduce the original problem to the problem of finding a minimum-cost non-negative integer circulation of a certain homology class, which turns out to be efficiently solvable in our case.

cs.DM↗
arXiv2018

Strengthening Convex Relaxations of 0/1-Sets Using Boolean Formulas

In convex integer programming, various procedures have been developed to strengthen convex relaxations of sets of integer points. On the one hand, there exist several general-purpose methods that strengthen relaxations without specific knowledge of the set $ S $, such as popular linear programming or semi-definite programming hierarchies. On the other hand, various methods have been designed for obtaining strengthened relaxations for very specific sets that arise in combinatorial optimization. We propose a new efficient method that interpolates between these two approaches. Our procedure strengthens any convex set $ Q \subseteq \mathbb{R}^n $ containing a set $ S \subseteq \{0,1\}^n $ by exploiting certain additional information about $ S $. Namely, the required extra information will be in the form of a Boolean formula $ ϕ$ defining the target set $ S $. The aim of this work is to analyze various aspects regarding the strength of our procedure. As one result, interpreting an iterated application of our procedure as a hierarchy, our findings simplify, improve, and extend previous results by Bienstock and Zuckerberg on covering problems.

math.CO↗
arXiv2018

Extended Formulations for Radial Cones

This paper studies extended formulations for radial cones at vertices of polyhedra, where the radial cone of a polyhedron $ P $ at a vertex $ v \in P $ is the polyhedron defined by the constraints of $ P $ that are active at $ v $. Given an extended formulation for $ P $, it is easy to obtain an extended formulation of comparable size for each its radial cones. On the contrary, it is possible that radial cones of $ P $ admit much smaller extended formulations than $ P $ itself. A prominent example of this type is the perfect-matching polytope, which cannot be described by subexponential-size extended formulations (Rothvoß 2014). However, Ventura & Eisenbrand (2003) showed that its radial cones can be described by polynomial-size extended formulations. Moreover, they generalized their construction to $ V $-join polyhedra. In the same paper, the authors asked whether the same holds for the odd-cut polyhedron, the blocker of the $ V $-join polyhedron. We answer this question negatively. Precisely, we show that radial cones of odd-cut polyhedra cannot be described by subexponential-size extended formulations. To obtain our result, for a polyhedron $ P $ of blocking type, we establish a general relationship between its radial cones and certain faces of the blocker of $ P $.

cs.DM↗
arXiv2018

Distances of optimal solutions of mixed-integer programs

A classic result of Cook et al. (1986) bounds the distances between optimal solutions of mixed-integer linear programs and optimal solutions of the corresponding linear relaxations. Their bound is given in terms of the number of variables and a parameter $ Δ$, which quantifies sub-determinants of the underlying linear inequalities. We show that this distance can be bounded in terms of $ Δ$ and the number of integer variables rather than the total number of variables. To this end, we make use of a result by Olson (1969) in additive combinatorics and demonstrate how it implies feasibility of certain mixed-integer linear programs. We conjecture that our bound can be improved to a function that only depends on $ Δ$, in general.

math.OC↗
arXiv2017

Lifting Linear Extension Complexity Bounds to the Mixed-Integer Setting

Mixed-integer mathematical programs are among the most commonly used models for a wide set of problems in Operations Research and related fields. However, there is still very little known about what can be expressed by small mixed-integer programs. In particular, prior to this work, it was open whether some classical problems, like the minimum odd-cut problem, can be expressed by a compact mixed-integer program with few (even constantly many) integer variables. This is in stark contrast to linear formulations, where recent breakthroughs in the field of extended formulations have shown that many polytopes associated to classical combinatorial optimization problems do not even admit approximate extended formulations of sub-exponential size. We provide a general framework for lifting inapproximability results of extended formulations to the setting of mixed-integer extended formulations, and obtain almost tight lower bounds on the number of integer variables needed to describe a variety of classical combinatorial optimization problems. Among the implications we obtain, we show that any mixed-integer extended formulation of sub-exponential size for the matching polytope, cut polytope, traveling salesman polytope or dominant of the odd-cut polytope, needs $ Ω(n/\log n) $ many integer variables, where $ n $ is the number of vertices of the underlying graph. Conversely, the above-mentioned polyhedra admit polynomial-size mixed-integer formulations with only $ O(n) $ or $ O(n \log n) $ (for the traveling salesman polytope) many integer variables. Our results build upon a new decomposition technique that, for any convex set $ C $, allows for approximating any mixed-integer description of $ C $ by the intersection of $ C $ with the union of a small number of affine subspaces.

cs.DM↗
arXiv2017

Characterizing Polytopes Contained in the $0/1$-Cube with Bounded Chvátal-Gomory Rank

Let $S \subseteq \{0,1\}^n$ and $R$ be any polytope contained in $[0,1]^n$ with $R \cap \{0,1\}^n = S$. We prove that $R$ has bounded Chvátal-Gomory rank (CG-rank) provided that $S$ has bounded notch and bounded gap, where the notch is the minimum integer $p$ such that all $p$-dimensional faces of the $0/1$-cube have a nonempty intersection with $S$, and the gap is a measure of the size of the facet coefficients of $\mathsf{conv}(S)$. Let $H[\bar{S}]$ denote the subgraph of the $n$-cube induced by the vertices not in $S$. We prove that if $H[\bar{S}]$ does not contain a subdivision of a large complete graph, then both the notch and the gap are bounded. By our main result, this implies that the CG-rank of $R$ is bounded as a function of the treewidth of $H[\bar{S}]$. We also prove that if $S$ has notch $3$, then the CG-rank of $R$ is always bounded. Both results generalize a recent theorem of Cornuéjols and Lee, who proved that the CG-rank is bounded by a constant if the treewidth of $H[\bar{S}]$ is at most $2$.

math.OC↗
arXiv2017

Extension complexities of Cartesian products involving a pyramid

It is an open question whether the linear extension complexity of the Cartesian product of two polytopes P, Q is the sum of the extension complexities of P and Q. We give an affirmative answer to this question for the case that one of the two polytopes is a pyramid.

math.OC↗
arXiv2016

Optimality certificates for convex minimization and Helly numbers

We consider the problem of minimizing a convex function over a subset of R^n that is not necessarily convex (minimization of a convex function over the integer points in a polytope is a special case). We define a family of duals for this problem and show that, under some natural conditions, strong duality holds for a dual problem in this family that is more restrictive than previously considered duals.

math.OC↗
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

Tight bounds on discrete quantitative Helly numbers

Given a subset S of R^n, let c(S,k) be the smallest number t such that whenever finitely many convex sets have exactly k common points in S, there exist at most t of these sets that already have exactly k common points in S. For S = Z^n, this number was introduced by Aliev et al. [2014] who gave an explicit bound showing that c(Z^n,k) = O(k) holds for every fixed n. Recently, Chestnut et al. [2015] improved this to c(Z^n,k) = O(k (log log k)(log k)^{-1/3} ) and provided the lower bound c(Z^n,k) = Omega(k^{(n-1)/(n+1)}). We provide a combinatorial description of c(S,k) in terms of polytopes with vertices in S and use it to improve the previously known bounds as follows: We strengthen the bound of Aliev et al. [2014] by a constant factor and extend it to general discrete sets S. We close the gap for Z^n by showing that c(Z^n,k) = Theta(k^{(n-1)/(n+1)}) holds for every fixed n. Finally, we determine the exact values of c(Z^n,k) for all k <= 4.

math.CO↗
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

Notions of maximality for integral lattice-free polyhedra: the case of dimension three

Lattice-free sets (convex subsets of $\mathbb{R}^d$ without interior integer points) and their applications for cutting-plane methods in mixed-integer optimization have been studied in recent literature. Notably, the family of all integral lattice-free polyhedra which are not properly contained in another integral lattice-free polyhedron has been of particular interest. We call these polyhedra $\mathbb{Z}^d$-maximal. It is known that, for fixed $d$, the family $\mathbb{Z}^d$-maximal integral lattice-free polyhedra is finite up to unimodular equivalence. In view of possible applications in cutting-plane theory, one would like to have a classification of this family. However, this turns out to be a challenging task already for small dimensions. In contrast, the subfamily of all integral lattice-free polyhedra which are not properly contained in any other lattice-free set, which we call $\mathbb{R}^d$-maximal lattice-free polyhedra, allow a rather simple geometric characterization. Hence, the question was raised for which dimensions the notions of $\mathbb{Z}^d$-maximality and $\mathbb{R}^d$-maximality are equivalent. This was known to be the case for dimensions one and two. On the other hand, Nill and Ziegler (2011) showed that for dimension $d \ge 4$, there exist polyhedra which are $\mathbb{Z}^d$-maximal but not $\mathbb{R}^d$-maximal. In this article, we consider the remaining case $d = 3$ and prove that for integral polyhedra the notions of $\mathbb{R}^3$-maximality and $\mathbb{Z}^3$-maximality are equivalent. As a consequence, the classification of all $\mathbb{R}^3$-maximal integral polyhedra by Averkov, Wagner and Weismantel (2011) contains all $\mathbb{Z}^3$-maximal integral polyhedra.

math.CO↗
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

Hidden Vertices in Extensions of Polytopes

Some widely known compact extended formulations have the property that each vertex of the corresponding extension polytope is projected onto a vertex of the target polytope. In this paper, we prove that for heptagons with vertices in general position none of the minimum size extensions has this property. Additionally, for any d >= 2 we construct a family of d-polytopes such that at least 1/9 of all vertices of any of their minimum size extensions is not projected onto vertices.

math.CO↗