arXiv subjects

Uli Wagner

Publications and source records attributed to Uli Wagner.

At least 19 recordsLinked to original sources

arXiv2025

Hardness of 4-Colourings G-Colourable Graphs

We study the complexity of a class of promise graph homomorphism problems. For a fixed graph H, the H-colouring problem is to decide whether a given graph has a homomorphism to H. By a result of Hell and Ne\v{s}et\v{r}il, this problem is NP-hard for any non-bipartite loop-less graph H. Brakensiek and Guruswami [SODA 2018] conjectured the hardness extends to promise graph homomorphism problems as follows: fix a pair of non-bipartite loop-less graphs G, H such that there is a homomorphism from G to H, it is NP-hard to distinguish between graphs that are G-colourable and those that are not H-colourable. We confirm this conjecture in the cases when both G and H are 4-colourable. This is a common generalisation of previous results of Khanna, Linial, and Safra [Comb. 20(3): 393-415 (2000)] and of Krokhin and Opr\v{s}al [FOCS 2019]. The result is obtained by combining the algebraic approach to promise constraint satisfaction with methods of topological combinatorics and equivariant obstruction theory.

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arXiv2025

Linear relations between face numbers of levels in arrangements

We study linear relations between face numbers of levels in arrangements. Let $V = \{ v_1, \ldots, v_n \} \subset \mathbf{R}^{r}$ be a vector configuration in general position, and let $\mathcal{A}(V)$ be polar dual arrangement of hemispheres in the $d$-dimensional unit sphere $S^d$, where $d=r-1$. For $0\leq s \leq d$ and $0 \leq t \leq n$, let $f_{s,t}(V)$ denote the number of faces of \emph{level} $t$ and dimension $d-s$ in the arrangement $\mathcal{A}(V)$ (these correspond to partitions $V=V_-\sqcup V_0 \sqcup V_+$ by linear hyperplanes with $|V_0|=s$ and $|V_-|=t$). We call the matrix $f(V):=[f_{s,t}(V)]$ the \emph{$f$-matrix} of $V$. Completing a long line of research on linear relations between face numbers of levels in arrangements, we determine, for every $n\geq r \geq 1$, the affine space $\mathfrak{F}_{n,r}$ spanned by the $f$-matrices of configurations of $n$ vectors in general position in $\mathbf{R}^r$; moreover, we determine the subspace $\mathfrak{F}^0_{n,r} \subset \mathfrak{F}_{n,r}$ spanned by all \emph{pointed} vector configurations (i.e., such that $V$ is contained in some open linear halfspace), which correspond to point sets in $\mathbf{R}^d$. This generalizes the classical fact that the Dehn--Sommerville relations generate all linear relations between the face numbers of simple polytopes (the faces at level $0$) and answers a question posed by Andrzejak and Welzl in 2003. The key notion for the statements and the proofs of our results is the $g$-matrix of a vector configuration, which determines the $f$-matrix and generalizes the classical $g$-vector of a polytope. By Gale duality, we also obtain analogous results for partitions of vector configurations by sign patterns of nontrivial linear dependencies, and for \emph{Radon partitions} of point sets in $\mathbf{R}^d$.

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arXiv2025

Sublevels in arrangements and the spherical arc crossing number of complete graphs

Levels and sublevels in arrangements -- and, dually, $k$-sets and $(\leq k)$-sets -- are fundamental notions in discrete and computational geometry and natural generalizations of convex polytopes, which correspond to the $0$-level. A long-standing conjecture of Eckhoff, Linhart, and Welzl, which would generalize McMullen's Upper Bound Theorem for polytopes and provide an exact refinement of asymptotic bounds by Clarkson, asserts that for all $k\leq \lfloor \frac{n-d-2}{2}\rfloor$, the number of $(\leq k)$-sets of a set $S$ of $n$ points in $\mathbf{R}^d$ is maximized if $S$ is the vertex set of a neighborly polytope. As a new tool for studying this conjecture and related problems, we introduce the $g$-matrix, which generalizes both the $g$-vector of a simple polytope and a Gale dual version of the $g$-vector studied by Lee and Welzl. Our main result is that the $g$-matrix of every vector configuration in $\mathbf{R}^3$ is non-negative, which implies the Eckhoff--Linhart--Welzl conjecture in the case where $d=n-4$. As a corollary, we obtain the following result about crossing numbers: Consider a configuration $V\subset S^2 \subset \mathbf{R}^3$ of $n$ unit vectors, and connect every pair of vectors by the unique shortest geodesic arc between them in the unit sphere $S^2$. This yields a drawing of the complete graph $K_n$ in $S^2$, which we call a spherical arc drawing. Complementing previous results for rectilinear drawings, we show that the number of crossings in any spherical arc drawing of $K_n$ is at least $\frac{1}{4}\lfloor \frac{n}{2}\rfloor \lfloor \frac{n-1}{2}\rfloor \lfloor \frac{n-2}{2}\rfloor \lfloor \frac{n-3}{2}\rfloor$, which equals the conjectured value of the crossing number of $K_n$. Moreover, the lower bound is attained if $V$ is coneighborly, i.e., if every open linear halfspace contains at least $\lfloor (n-2)/2 \rfloor$ of the vectors in $V$.

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arXiv2024

Eight-Partitioning Points in 3D, and Efficiently Too

An {\em eight-partition} of a finite set of points (respectively, of a continuous mass distribution) in $\mathbb{R}^3$ consists of three planes that divide the space into $8$ octants, such that each open octant contains at most $1/8$ of the points (respectively, of the mass). In 1966, Hadwiger showed that any mass distribution in $\mathbb{R}^3$ admits an eight-partition; moreover, one can prescribe the normal direction of one of the three planes. The analogous result for finite point sets follows by a standard limit argument. We prove the following variant of this result: Any mass distribution (or point set) in $\mathbb{R}^3$ admits an eight-partition for which the intersection of two of the planes is a line with a prescribed direction. Moreover, we present an efficient algorithm for calculating an eight-partition of a set of $n$ points in~$\mathbb{R}^3$ (with prescribed normal direction of one of the planes) in time $O^{*}(n^{7/3})$.

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arXiv2023

Hardness of linearly ordered 4-colouring of 3-colourable 3-uniform hypergraphs

A linearly ordered (LO) $k$-colouring of a hypergraph is a colouring of its vertices with colours $1, \dots, k$ such that each edge contains a unique maximal colour. Deciding whether an input hypergraph admits LO $k$-colouring with a fixed number of colours is NP-complete (and in the special case of graphs, LO colouring coincides with the usual graph colouring). Here, we investigate the complexity of approximating the `linearly ordered chromatic number' of a hypergraph. We prove that the following promise problem is NP-complete: Given a 3-uniform hypergraph, distinguish between the case that it is LO $3$-colourable, and the case that it is not even LO $4$-colourable. We prove this result by a combination of algebraic, topological, and combinatorial methods, building on and extending a topological approach for studying approximate graph colouring introduced by Krokhin, Opršal, Wrochna, and Živný (2023).

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arXiv2021

On the treewidth of triangulated 3-manifolds

In graph theory, as well as in 3-manifold topology, there exist several width-type parameters to describe how "simple" or "thin" a given graph or 3-manifold is. These parameters, such as pathwidth or treewidth for graphs, or the concept of thin position for 3-manifolds, play an important role when studying algorithmic problems; in particular, there is a variety of problems in computational 3-manifold topology - some of them known to be computationally hard in general - that become solvable in polynomial time as soon as the dual graph of the input triangulation has bounded treewidth. In view of these algorithmic results, it is natural to ask whether every 3-manifold admits a triangulation of bounded treewidth. We show that this is not the case, i.e., that there exists an infinite family of closed 3-manifolds not admitting triangulations of bounded pathwidth or treewidth (the latter implies the former, but we present two separate proofs). We derive these results from work of Agol, of Scharlemann and Thompson, and of Scharlemann, Schultens and Saito by exhibiting explicit connections between the topology of a 3-manifold M on the one hand and width-type parameters of the dual graphs of triangulations of M on the other hand, answering a question that had been raised repeatedly by researchers in computational 3-manifold topology. In particular, we show that if a closed, orientable, irreducible, non-Haken 3-manifold M has a triangulation of treewidth (resp. pathwidth) k then the Heegaard genus of M is at most 18(k+1) (resp. 4(3k+1)).

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arXiv2021

The Crossing Tverberg Theorem

Tverberg's theorem is one of the cornerstones of discrete geometry. It states that, given a set $X$ of at least $(d+1)(r-1)+1$ points in $\mathbb R^d$, one can find a partition $X=X_1\cup \ldots \cup X_r$ of $X$, such that the convex hulls of the $X_i$, $i=1,\ldots,r$, all share a common point. In this paper, we prove a strengthening of this theorem that guarantees a partition which, in addition to the above, has the property that the boundaries of full-dimensional convex hulls have pairwise nonempty intersections. Possible generalizations and algorithmic aspects are also discussed. As a concrete application, we show that any $n$ points in the plane in general position span $\lfloor n/3\rfloor$ vertex-disjoint triangles that are pairwise crossing, meaning that their boundaries have pairwise nonempty intersections; this number is clearly best possible. A previous result of Rebollar et al.\ guarantees $\lfloor n/6\rfloor$ pairwise crossing triangles. Our result generalizes to a result about simplices in $\mathbb R^d,d\ge2$.

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arXiv2020

Connectivity of Triangulation Flip Graphs in the Plane

Given a finite point set P in general position in the plane, a full triangulation is a maximal straight-line embedded plane graph on P. A partial triangulation is a full triangulation of some subset P' of P containing all extreme points in P. A bistellar flip on a partial triangulation flips an edge (an edge flip), removes a non-extreme point of degree 3, or adds a point in P \ P' as vertex of degree 3. The bistellar flip graph has all partial triangulations as vertices, and a pair of partial triangulations is adjacent if they can be obtained from one another by a bistellar flip. The edge flip graph is defined with full triangulations as vertices, and edge flips determining the adjacencies. Lawson showed in the early 70s that these graphs are connected. Our goal is to investigate these graphs, with emphasis on vertex connectivity. For sets of n points in the plane in general position, we show that the edge flip graph is (n/2-2)-connected, and the bistellar flip graph is (n-3)-connected; both results are tight. The latter bound matches the situation for the subfamily of regular triangulations, ie. partial triangulations obtained by lifting the points to 3-space and projecting back the lower convex hull. Here (n-3)-connectivity has been known since the late 80s via the secondary polytope due to Gelfand, Kapranov & Zelevinsky and Balinski's Theorem. For the edge flip-graphs, the vertex connectivity can be shown to be at least as large as (and hence equal to) the minimum degree, provided n is large enough. Our methods yield several other results.

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arXiv2020

Barycentric cuts through a convex body

Let $K$ be a convex body in $\mathbb{R}^n$ (i.e., a compact convex set with nonempty interior). Given a point $p$ in the interior of $K$, a hyperplane $h$ passing through $p$ is called barycentric if $p$ is the barycenter of $K \cap h$. In 1961, Grünbaum raised the question whether, for every $K$, there exists an interior point $p$ through which there are at least $n+1$ distinct barycentric hyperplanes. Two years later, this was seemingly resolved affirmatively by showing that this is the case if $p=p_0$ is the point of maximal depth in $K$. However, while working on a related question, we noticed that one of the auxiliary claims in the proof is incorrect. Here, we provide a counterexample; this re-opens Grünbaum's question. It follows from known results that for $n \geq 2$, there are always at least three distinct barycentric cuts through the point $p_0 \in K$ of maximal depth. Using tools related to Morse theory we are able to improve this bound: four distinct barycentric cuts through $p_0$ are guaranteed if $n \geq 3$.

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arXiv2018

Shellability is NP-complete

We prove that for every $d\geq 2$, deciding if a pure, $d$-dimensional, simplicial complex is shellable is NP-hard, hence NP-complete. This resolves a question raised, e.g., by Danaraj and Klee in 1978. Our reduction also yields that for every $d \ge 2$ and $k \ge 0$, deciding if a pure, $d$-dimensional, simplicial complex is $k$-decomposable is NP-hard. For $d \ge 3$, both problems remain NP-hard when restricted to contractible pure $d$-dimensional complexes. Another simple corollary of our result is that it is NP-hard to decide whether a given poset is CL-shellable.

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arXiv2017

A Proof of the Orbit Conjecture for Flipping Edge-Labelled Triangulations

Given a triangulation of a point set in the plane, a \emph{flip} deletes an edge $e$ whose removal leaves a convex quadrilateral, and replaces $e$ by the opposite diagonal of the quadrilateral. It is well known that any triangulation of a point set can be reconfigured to any other triangulation by some sequence of flips. We explore this question in the setting where each edge of a triangulation has a label, and a flip transfers the label of the removed edge to the new edge. It is not true that every labelled triangulation of a point set can be reconfigured to every other labelled triangulation via a sequence of flips. We characterize when this is possible by proving the \emph{Orbit Conjecture} of Bose, Lubiw, Pathak and Verdonschot which states that \emph{all} labels can be simultaneously mapped to their destination if and only if \emph{each} label individually can be mapped to its destination. Furthermore, we give a polynomial-time algorithm to find a sequence of flips to reconfigure one labelled triangulation to another, if such a sequence exists, and we prove an upper bound of $O(n^7)$ on the length of the flip sequence. Our proof uses the topological result that the sets of pairwise non-crossing edges on a planar point set form a simplicial complex that is homeomorphic to a high-dimensional ball (this follows from a result of Orden and Santos; we give a different proof based on a shelling argument). The dual cell complex of this simplicial ball, called the \emph{flip complex}, has the usual flip graph as its $1$-skeleton. We use properties of the $2$-skeleton of the flip complex to prove the Orbit Conjecture.

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arXiv2017

Computing simplicial representatives of homotopy group elements

A central problem of algebraic topology is to understand the homotopy groups $π_d(X)$ of a topological space $X$. For the computational version of the problem, it is well known that there is no algorithm to decide whether the fundamental group $π_1(X)$ of a given finite simplicial complex $X$ is trivial. On the other hand, there are several algorithms that, given a finite simplicial complex $X$ that is simply connected (i.e., with $π_1(X)$ trivial), compute the higher homotopy group $π_d(X)$ for any given $d\geq 2$. %The first such algorithm was given by Brown, and more recently, Čadek et al. However, these algorithms come with a caveat: They compute the isomorphism type of $π_d(X)$, $d\geq 2$ as an \emph{abstract} finitely generated abelian group given by generators and relations, but they work with very implicit representations of the elements of $π_d(X)$. Converting elements of this abstract group into explicit geometric maps from the $d$-dimensional sphere $S^d$ to $X$ has been one of the main unsolved problems in the emerging field of computational homotopy theory. Here we present an algorithm that, given a~simply connected space $X$, computes $π_d(X)$ and represents its elements as simplicial maps from a suitable triangulation of the $d$-sphere $S^d$ to $X$. For fixed $d$, the algorithm runs in time exponential in $size(X)$, the number of simplices of $X$. Moreover, we prove that this is optimal: For every fixed $d\geq 2$, we construct a family of simply connected spaces $X$ such that for any simplicial map representing a generator of $π_d(X)$, the size of the triangulation of $S^d$ on which the map is defined, is exponential in $size(X)$.

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arXiv2016

Bounding Helly numbers via Betti numbers

We show that very weak topological assumptions are enough to ensure the existence of a Helly-type theorem. More precisely, we show that for any non-negative integers $b$ and $d$ there exists an integer $h(b,d)$ such that the following holds. If $\mathcal F$ is a finite family of subsets of $\mathbb R^d$ such that $\tildeβ_i\left(\bigcap\mathcal G\right) \le b$ for any $\mathcal G \subsetneq \mathcal F$ and every $0 \le i \le \lceil d/2 \rceil-1$ then $\mathcal F$ has Helly number at most $h(b,d)$. Here $\tildeβ_i$ denotes the reduced $\mathbb Z_2$-Betti numbers (with singular homology). These topological conditions are sharp: not controlling any of these $\lceil d/2 \rceil$ first Betti numbers allow for families with unbounded Helly number. Our proofs combine homological non-embeddability results with a Ramsey-based approach to build, given an arbitrary simplicial complex $K$, some well-behaved chain map $C_*(K) \to C_*(\mathbb R^d)$.

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arXiv2016

On Generalized Heawood Inequalities for Manifolds: a van Kampen--Flores-type Nonembeddability Result

The fact that the complete graph $K_5$ does not embed in the plane has been generalized in two independent directions. On the one hand, the solution of the classical Heawood problem for graphs on surfaces established that the complete graph $K_n$ embeds in a closed surface $M$ (other than the Klein bottle) if and only if $(n-3)(n-4)\leq 6b_1(M)$, where $b_1(M)$ is the first $\mathbb Z_2$-Betti number of $M$. On the other hand, van Kampen and Flores proved that the $k$-skeleton of the $n$-dimensional simplex (the higher-dimensional analogue of $K_{n+1}$) embeds in $\mathbb R^{2k}$ if and only if~$n \le 2k+1$. Two decades ago, Kühnel conjectured that the $k$-skeleton of the $n$-simplex embeds in a compact, $(k-1)$-connected $2k$-manifold with $k$th $\mathbb Z_2$-Betti number $b_k$ only if the following generalized Heawood inequality holds: $\binom{n-k-1}{k+1} \le \binom{2k+1}{k+1}b_k$. This is a common generalization of the case of graphs on surfaces as well as the van Kampen--Flores theorem (the special cases $k=1$ and $b_k=0$, respectively), and also closely related to the theory of face numbers of triangulated manifolds. In the spirit of Kühnel's conjecture, we prove that if the $k$-skeleton of the $n$-simplex embeds in a $2k$-manifold with $k$th $\mathbb Z_2$-Betti number $b_k$, then $n \le 2b_k\binom{2k+2}{k} + 2k + 4$. This bound is weaker than the generalized Heawood inequality, but does not require the assumption that $M$ is $(k-1)$-connected. Our results generalize to maps without $q$-covered points, in the spirit of Tverberg's theorem, for $q$ a prime power. Our proof uses a result of Volovikov about maps that satisfy a certain homological triviality condition.

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arXiv2016

Eliminating Higher-Multiplicity Intersections, II. The Deleted Product Criterion in the $r$-Metastable Range

Motivated by Tverberg-type problems in topological combinatorics and by classical results about embeddings (maps without double points), we study the question whether a finite simplicial complex K can be mapped into R^d without higher-multiplicity intersections. We focus on conditions for the existence of almost r-embeddings, i.e., maps from K to R^d without r-intersection points among any set of r pairwise disjoint simplices of K. Generalizing the classical Haefliger-Weber embeddability criterion, we show that a well-known necessary deleted product condition for the existence of almost r-embeddings is sufficient in a suitable r-metastable range of dimensions (r d > (r+1) dim K +2). This significantly extends one of the main results of our previous paper (which treated the special case where d=rk and dim K=(r-1)k, for some k> 3).

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arXiv2016

On Expansion and Topological Overlap

We give a detailed and easily accessible proof of Gromov's Topological Overlap Theorem. Let $X$ be a finite simplicial complex or, more generally, a finite polyhedral cell complex of dimension $d$. Informally, the theorem states that if $X$ has sufficiently strong higher-dimensional expansion properties (which generalize edge expansion of graphs and are defined in terms of cellular cochains of $X$) then $X$ has the following topological overlap property: for every continuous map $X\rightarrow \mathbf{R}^d$ there exists a point $p\in \mathbf{R}^d$ that is contained in the images of a positive fraction $μ>0$ of the $d$-cells of $X$. More generally, the conclusion holds if $\mathbf{R}^d$ is replaced by any $d$-dimensional piecewise-linear (PL) manifold $M$, with a constant $μ$ that depends only on $d$ and on the expansion properties of $X$, but not on $M$.

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arXiv2016

Finding non-orientable surfaces in 3-manifolds

We investigate the complexity of finding an embedded non-orientable surface of Euler genus $g$ in a triangulated $3$-manifold. This problem occurs both as a natural question in low-dimensional topology, and as a first non-trivial instance of embeddability of complexes into $3$-manifolds. We prove that the problem is NP-hard, thus adding to the relatively few hardness results that are currently known in 3-manifold topology. In addition, we show that the problem lies in NP when the Euler genus g is odd, and we give an explicit algorithm in this case.

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arXiv2015

On Eigenvalues of Random Complexes

We consider higher-dimensional generalizations of the normalized Laplacian and the adjacency matrix of graphs and study their eigenvalues for the Linial-Meshulam model $X^k(n,p)$ of random $k$-dimensional simplicial complexes on $n$ vertices. We show that for $p=Ω(\log n/n)$, the eigenvalues of these matrices are a.a.s. concentrated around two values. The main tool, which goes back to the work of Garland, are arguments that relate the eigenvalues of these matrices to those of graphs that arise as links of $(k-2)$-dimensional faces. Garland's result concerns the Laplacian; we develop an analogous result for the adjacency matrix. The same arguments apply to other models of random complexes which allow for dependencies between the choices of $k$-dimensional simplices. In the second part of the paper, we apply this to the question of possible higher-dimensional analogues of the discrete Cheeger inequality, which in the classical case of graphs relates the eigenvalues of a graph and its edge expansion. It is very natural to ask whether this generalizes to higher dimensions and, in particular, whether the higher-dimensional Laplacian spectra capture the notion of coboundary expansion - a generalization of edge expansion that arose in recent work of Linial and Meshulam and of Gromov. We show that this most straightforward version of a higher-dimensional discrete Cheeger inequality fails, in quite a strong way: For every $k\geq 2$ and $n\in \mathbb{N}$, there is a $k$-dimensional complex $Y^k_n$ on $n$ vertices that has strong spectral expansion properties (all nontrivial eigenvalues of the normalised $k$-dimensional Laplacian lie in the interval $[1-O(1/\sqrt{n}),1+O(1/\sqrt{n})]$) but whose coboundary expansion is bounded from above by $O(\log n/n)$ and so tends to zero as $n\rightarrow \infty$; moreover, $Y^k_n$ can be taken to have vanishing integer homology in dimension less than $k$.

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