arXiv subjects

Martin Tancer

Publications and source records attributed to Martin Tancer.

At least 37 records · Page 2Linked to original sources

arXiv2018

Embeddability in $\mathbb{R}^3$ is NP-hard

We prove that the problem of deciding whether a 2- or 3-dimensional simplicial complex embeds into $\mathbb{R}^3$ is NP-hard. Our construction also shows that deciding whether a 3-manifold with boundary tori admits an $\mathbb{S}^{3}$ filling is NP-hard. The former stands in contrast with the lower dimensional cases which can be solved in linear time,and the latter with a variety of computational problems in 3-manifold topology (for example, unknot or 3-sphere recognition, which are in NP and co-NP assuming the Generalized Riemann Hypothesis). Our reduction encodes a satisfiability instance into the embeddability problem of a 3-manifold with boundary tori, and relies extensively on techniques from low-dimensional topology, most importantly Dehn fillings on link complements.

math.GT↗

arXiv2018

On Betti numbers of flag complexes with forbidden induced subgraphs

We analyze the asymptotic extremal growth rate of the Betti numbers of clique complexes of graphs on n vertices not containing a fixed forbidden induced subgraph H. In particular, we prove a theorem of the alternative: for any H the growth rate achieves exactly one of five possible exponentials, that is, independent of the field of coefficients, the nth root of the maximal total Betti number over n-vertex graphs with no induced copy of H has a limit, as n tends to infinity, and, ranging over all H, exactly five different limits are attained. For the interesting case where H is the 4-cycle, the above limit is 1, and we prove a slightly superpolynomial upper bound.

math.CO↗

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.

math.CO↗

arXiv2017

Hardness of almost embedding simplicial complexes in $\mathbb R^d$

A map $f\colon K\to \mathbb R^d$ of a simplicial complex is an almost embedding if $f(σ)\cap f(τ)=\emptyset$ whenever $σ,τ$ are disjoint simplices of $K$. Theorem. Fix integers $d,k\ge2$ such that $d=\frac{3k}2+1$. (a) Assume that $P\ne NP$. Then there exists a finite $k$-dimensional complex $K$ that does not admit an almost embedding in $\mathbb R^d$ but for which there exists an equivariant map $\tilde K\to S^{d-1}$. (b) The algorithmic problem of recognition almost embeddability of finite $k$-dimensional complexes in $\mathbb R^d$ is NP hard. The proof is based on the technique from the Matoušek-Tancer-Wagner paper (proving an analogous result for embeddings), and on singular versions of the higher-dimensional Borromean rings lemma and a generalized van Kampen--Flores theorem.

math.GT↗

arXiv2017

Shortest path embeddings of graphs on surfaces

The classical theorem of Fáry states that every planar graph can be represented by an embedding in which every edge is represented by a straight line segment. We consider generalizations of Fáry's theorem to surfaces equipped with Riemannian metrics. In this setting, we require that every edge is drawn as a shortest path between its two endpoints and we call an embedding with this property a shortest path embedding. The main question addressed in this paper is whether given a closed surface S, there exists a Riemannian metric for which every topologically embeddable graph admits a shortest path embedding. This question is also motivated by various problems regarding crossing numbers on surfaces. We observe that the round metrics on the sphere and the projective plane have this property. We provide flat metrics on the torus and the Klein bottle which also have this property. Then we show that for the unit square flat metric on the Klein bottle there exists a graph without shortest path embeddings. We show, moreover, that for large g, there exist graphs G embeddable into the orientable surface of genus g, such that with large probability a random hyperbolic metric does not admit a shortest path embedding of G, where the probability measure is proportional to the Weil-Petersson volume on moduli space. Finally, we construct a hyperbolic metric on every orientable surface S of genus g, such that every graph embeddable into S can be embedded so that every edge is a concatenation of at most O(g) shortest paths.

cs.CG↗

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 $C_*(K) \to C_*(\mathbb R^d)$.

math.CO↗

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.

math.CO↗

arXiv2016

A Direct Proof of the Strong Hanani-Tutte Theorem on the Projective Plane

We reprove the strong Hanani-Tutte theorem on the projective plane. In contrast to the previous proof by Pelsmajer, Schaefer and Stasi, our method is constructive and does not rely on the characterization of forbidden minors, which gives hope to extend it to other surfaces. Moreover, our approach can be used to provide an efficient algorithm turning a Hanani-Tutte drawing on the projective plane into an embedding.

cs.CG↗

arXiv2015

Recognition of collapsible complexes is NP-complete

We prove that it is NP-complete to decide whether a given (3-dimensional) simplicial complex is collapsible. This work extends a result of Malgouyres and Francés showing that it is NP-complete to decide whether a given simplicial complex collapses to a 1-complex.

cs.CG↗

arXiv2015

Bounds for Pach's selection theorem and for the minimum solid angle in a simplex

We estimate the selection constant in the following geometric selection theorem by Pach: For every positive integer $d$ there is a constant $c_d > 0$ such that whenever $X_1,..., X_{d+1}$ are $n$-element subsets of $\mathbb{R}^d$, then we can find a point $\mathbf{p} \in \mathbb{R}^d$ and subsets $Y_i \subseteq X_i$ for every $i \in [d+1]$, each of size at least $c_d n$, such that $\mathbf{p}$ belongs to all {\em rainbow} $d$-simplices determined by $Y_1,..., Y_{d+1}$, that is, simplices with one vertex in each $Y_i$. We show a super-exponentially decreasing upper bound $c_d\leq e^{-(1/2-o(1))(d \ln d)}$. The ideas used in the proof of the upper bound also help us prove Pach's theorem with $c_d \geq 2^{-2^{d^2 + O(d)}}$, which is a lower bound doubly exponentially decreasing in $d$ (up to some polynomial in the exponent). For comparison, Pach's original approach yields a triply exponentially decreasing lower bound. On the other hand, Fox, Pach, and Suk recently obtained a hypergraph density result implying a proof of Pach's theorem with $c_d \geq2^{-O(d^2\log d)}$. In our construction for the upper bound, we use the fact that the minimum solid angle of every $d$-simplex is super-exponentially small. This fact was previously unknown and might be of independent interest. For the lower bound, we improve the "separation" part of the argument by showing that in one of the key steps only $d+1$ separations are necessary, compared to $2^d$ separations in the original proof. We also provide a measure version of Pach's theorem.

math.MG↗

arXiv2014

Simplifying inclusion-exclusion formulas

Let $\mathcal{F}=\{F_1,F_2, \ldots,F_n\}$ be a family of $n$ sets on a ground set $S$, such as a family of balls in $\mathbb{R}^d$. For every finite measure $μ$ on $S$, such that the sets of $\mathcal{F}$ are measurable, the classical inclusion-exclusion formula asserts that $μ(F_1\cup F_2\cup\cdots\cup F_n)=\sum_{I:\emptyset\ne I\subseteq[n]} (-1)^{|I|+1}μ\Bigl(\bigcap_{i\in I} F_i\Bigr)$; that is, the measure of the union is expressed using measures of various intersections. The number of terms in this formula is exponential in $n$, and a significant amount of research, originating in applied areas, has been devoted to constructing simpler formulas for particular families $\mathcal{F}$. We provide an upper bound valid for an arbitrary $\mathcal{F}$: we show that every system $\mathcal{F}$ of $n$ sets with $m$ nonempty fields in the Venn diagram admits an inclusion-exclusion formula with $m^{O(\log^2n)}$ terms and with $\pm1$ coefficients, and that such a formula can be computed in $m^{O(\log^2n)}$ expected time. For every $\varepsilon>0$ we also construct systems with Venn diagram of size $m$ for which every valid inclusion-exclusion formula has the sum of absolute values of the coefficients at least $Ω(m^{2-\varepsilon})$.

math.CO↗

arXiv2014

Untangling two systems of noncrossing curves

We consider two systems of curves $(α_1,...,α_m)$ and $(β_1,...,β_n)$ drawn on a compact two-dimensional surface $M$ with boundary. Each $α_i$ and each $β_j$ is either an arc meeting the boundary of $M$ at its two endpoints, or a closed curve. The $α_i$ are pairwise disjoint except for possibly sharing endpoints, and similarly for the $β_j$. We want to "untangle" the $β_j$ from the $α_i$ by a self-homeomorphism of $M$; more precisely, we seek a homeomorphism $ϕ:M\rightarrow M$ fixing the boundary of $M$ pointwise such that the total number of crossings of the $α_i$ with the $ϕ(β_j)$ is as small as possible. This problem is motivated by an application in the algorithmic theory of embeddings and 3-manifolds. We prove that if $M$ is planar, i.e., a sphere with $h\geq 0$ boundary components ("holes"), then $O(mn)$ crossings can be achieved (independently of $h$), which is asymptotically tight, as an easy lower bound shows. In general, for an arbitrary (orientable or nonorientable) surface $M$ with $h$ holes and of (orientable or nonorientable) genus $g$, we obtain an $O((m+n)^4)$ upper bound, again independent of $h$ and $g$. The proofs rely, among others, on a result concerning simultaneous planar drawings of graphs by Erten and Kobourov.

math.CO↗

arXiv2014

Shellability of the higher pinched Veronese posets

The pinched Veronese poset $V^*_n$ is the poset with ground set consisting of all non-negative integer vectors of length n such that the sum of their coordinates is divisible by $n$ with exception of the vector $(1,...,1)$. For two vectors $a$ and $b$ in $V^*_n$ we have $a \leq b$ if and only if $b - a$ belongs to the ground set of $V^*_n$. We show that every interval in $V^*_n$ is shellable for $n$ at least 4. In order to obtain the result, we develop a new method for showing that a poset is shellable. This method differs from classical lexicographic shellability. Shellability of intervals in $V^*_n$ has consequences in commutative algebra. As a corollary we obtain a combinatorial proof of the fact that the pinched Veronese ring is Koszul for $n \geq 4$. (This also follows from a result by Conca, Herzog, Trung and Valla.)

math.CO↗

arXiv2014

Embeddability in the 3-sphere is decidable

We show that the following algorithmic problem is decidable: given a $2$-dimensional simplicial complex, can it be embedded (topologically, or equivalently, piecewise linearly) in $\mathbf{R}^3$? By a known reduction, it suffices to decide the embeddability of a given triangulated 3-manifold $X$ into the 3-sphere $S^3$. The main step, which allows us to simplify $X$ and recurse, is in proving that if $X$ can be embedded in $S^3$, then there is also an embedding in which $X$ has a short meridian, i.e., an essential curve in the boundary of $X$ bounding a disk in $S^3\setminus X$ with length bounded by a computable function of the number of tetrahedra of $X$.

math.GT↗

arXiv2012

Non-embeddability of geometric lattices and buildings

A fundamental question for simplicial complexes is to find the lowest dimensional Euclidean space in which they can be embedded. We investigate this question for order complexes of posets. We show that order complexes of thick geometric lattices as well as several classes of finite buildings, all of which are order complexes, are hard to embed. That means that such d-dimensional complexes require (2d + 1)-dimensional Euclidean space for an embedding. (This dimension is in general always sufficient for any d-complex.) We develop a method to show non-embeddability for general order complexes of posets which builds on properties of the van Kampen obstruction.

math.CO↗

arXiv2012

Good covers are algorithmically unrecognizable

A good cover in R^d is a collection of open contractible sets in R^d such that the intersection of any subcollection is either contractible or empty. Motivated by an analogy with convex sets, intersection patterns of good covers were studied intensively. Our main result is that intersection patterns of good covers are algorithmically unrecognizable. More precisely, the intersection pattern of a good cover can be stored in a simplicial complex called nerve which records which subfamilies of the good cover intersect. A simplicial complex is topologically d-representable if it is isomorphic to the nerve of a good cover in R^d. We prove that it is algorithmically undecidable whether a given simplicial complex is topologically d-representable for any fixed d \geq 5. The result remains also valid if we replace good covers with acyclic covers or with covers by open d-balls. As an auxiliary result we prove that if a simplicial complex is PL embeddable into R^d, then it is topologically d-representable. We also supply this result with showing that if a "sufficiently fine" subdivision of a k-dimensional complex is d-representable and k \leq (2d-3)/3, then the complex is PL embeddable into R^d.

cs.CG↗

arXiv2011

Intersection patterns of convex sets via simplicial complexes, a survey

The task of this survey is to present various results on intersection patterns of convex sets. One of main tools for studying intersection patterns is a point of view via simplicial complexes. We recall the definitions of so called $d$-representable, $d$-collapsible and $d$-Leray simplicial complexes which are very useful for this study. We study the differences among these notions and we also focus on computational complexity for recognizing them. A list of Helly-type theorems is presented in the survey and it is also discussed how (important) role play the above mentioned notions for the theorems. We also consider intersection patterns of good covers which generalize collections of convex sets (the sets may be `curvy'; however, their intersections cannot be too complicated). We mainly focus on new results.

math.CO↗

arXiv2011

On the Complexity of Planar Covering of Small Graphs

The problem Cover(H) asks whether an input graph G covers a fixed graph H (i.e., whether there exists a homomorphism G to H which locally preserves the structure of the graphs). Complexity of this problem has been intensively studied. In this paper, we consider the problem PlanarCover(H) which restricts the input graph G to be planar. PlanarCover(H) is polynomially solvable if Cover(H) belongs to P, and it is even trivially solvable if H has no planar cover. Thus the interesting cases are when H admits a planar cover, but Cover(H) is NP-complete. This also relates the problem to the long-standing Negami Conjecture which aims to describe all graphs having a planar cover. Kratochvil asked whether there are non-trivial graphs for which Cover(H) is NP-complete but PlanarCover(H) belongs to P. We examine the first nontrivial cases of graphs H for which Cover(H) is NP-complete and which admit a planar cover. We prove NP-completeness of PlanarCover(H) in these cases.

math.CO↗