arXiv subjects

Martin Tancer

Publications and source records attributed to Martin Tancer.

At least 19 recordsLinked to original sources

arXiv2026

Non-acyclic spaces of line transversals

Cheong, Goaoc and Holmsen conjectured that every connected component of the space of line transversals to a family of pairwise disjoint open convex sets in $\mathbb{R}^d$ is acyclic. We disprove this conjecture by showing that the homology may be nontrivial in any fixed dimension provided that $d$ is large enough. More precisely, we show that for every $n \geq 1$ there is a finite family of pairwise disjoint open convex sets in $\mathbb{R}^{3n}$ such that the $(n-1)$st homology (over an arbitrary ring) of the space of line transversals to this family is nonzero.

math.CO↗

arXiv2026

Maximal matroids and counterexamples

Jackson and Tanigawa conjectured that the rigidity matroid $\mathcal{R}^d_n$ and the hyperconnectivity matroid $\mathcal{H}^d_n$ are the unique maximal matroids in the posets of $\{K_{d+2}, K_{d+2,d+2}\}$-matroids and $\{K_{d+2},K_{d+1,d+1}\}$-matroids, respectively. We disprove these conjectures by showing the existence of maximal matroids that are distinct from the proposed candidates.

math.CO↗

arXiv2025

A note on the computational complexity of weak saturation

We prove that determining the weak saturation number of a host graph $F$ with respect to a pattern graph $H$ is already a computationally hard problem when $H$ is the triangle. As our main tool we establish a connection between weak saturation and shellability of simplicial complexes.

math.CO↗

arXiv2024

Algorithms in 4-manifold topology

We show that there exists an algorithm that takes as input two closed, simply connected, topological 4-manifolds and decides whether or not these 4-manifolds are homeomorphic. In particular, we explain in detail how closed, simply connected, topological 4-manifolds can be naturally represented by a Kirby diagram consisting only of 2-handles. This representation is used as input for our algorithm. Along the way, we develop an algorithm to compute the Kirby-Siebenmann invariant of a closed, simply connected, topological 4-manifold from any of its Kirby diagrams and describe an algorithm that decides whether or not two intersection forms are isometric. In a slightly different direction, we discuss the decidability of the stable classification of smooth manifolds with more general fundamental groups. Here we show that there exists an algorithm that takes as input two closed, oriented, smooth 4-manifolds with fundamental groups isomorphic to a finite group with cyclic Sylow 2-subgroup, an infinite cyclic group, or a group of geometric dimension at most 3 (in the latter case we additionally assume that the universal covers of both 4-manifolds are not spin), and decides whether or not these two 4-manifolds are orientation-preserving stably diffeomorphic.

math.GT↗

arXiv2024

Pach's animal problem within the bounding box

A collection of unit cubes with integer coordinates in $\mathbb R^3$ is an animal if its union is homeomorphic to the 3-ball. Pach's animal problem asks whether any animal can be transformed to a single cube by adding or removing cubes one by one in such a way that any intermediate step is an animal as well. Here we provide an example of an animal that cannot be transformed to a single cube this way within its bounding box.

math.CO↗

arXiv2023

Simpler algorithmically unrecognizable 4-manifolds

Markov proved that there exists an unrecognizable 4-manifold, that is, a 4-manifold for which the homeomorphism problem is undecidable. In this paper we consider the question how close we can get to S^4 with an unrecognizable manifold. One of our achievements is that we show a way to remove so-called Markov's trick from the proof of existence of such a manifold. This trick contributes to the complexity of the resulting manifold. We also show how to decrease the deficiency (or the number of relations) in so-called Adian-Rabin set which is another ingredient that contributes to the complexity of the resulting manifold. Altogether, our approach allows to show that the connected sum #_9(S^2 x S^2) is unrecognizable while the previous best result is the unrecognizability of #_12(S^2 x S^2) due to Gordon.

math.GT↗

arXiv2022

Shellability is hard even for balls

The main goal of this paper is to show that shellability is NP-hard for triangulated d-balls (this also gives hardness for triangulated d-manifolds/d-pseudomanifolds with boundary) as soon as d is at least 3. This extends our earlier work with Goaoc, Pat\'akov\'a and Wagner on hardness of shellability of 2-complexes and answers some questions implicitly raised by Danaraj and Klee in 1978 and explicitly mentioned by Santamar\'ia-Galvis and Woodroofe. Together with the main goal, we also prove that collapsibility is NP-hard for 3-complexes embeddable in the 3-space, extending an earlier work of the second author and answering an open question mentioned by Cohen, Fasy, Miller, Nayyeri, Peng and Walkington; and that shellability is NP-hard for 2-complexes embeddable in the 3-space, answering another question of Santamar\'ia-Galvis and Woodroofe (in a slightly stronger form than what is given by the main result).

cs.CG↗

arXiv2022

NP-hardness of computing PL geometric category in dimension 2

The PL geometric category of a polyhedron $P$, denoted $\hbox{plgcat}(P)$, provides a natural upper bound for the Lusternik--Schnirelmann category and it is defined as the minimum number of PL collapsible subpolyhedra of $P$ that cover $P$. In dimension 2 the PL geometric category is at most~3. It is easy to characterize/recognize $2$-polyhedra $P$ with $\hbox{plgcat}(P) = 1$. Borghini provided a partial characterization of $2$-polyhedra with $\hbox{plgcat}(P) = 2$. We complement his result by showing that it is NP-hard to decide whether $\hbox{plgcat}(P)\leq 2$. Therefore, we should not expect much more than a partial characterization, at least in algorithmic sense. Our reduction is based on the observation that 2-dimensional polyhedra $P$ admitting a shellable subdivision satisfy $\hbox{plgcat}(P) \leq 2$ and a (nontrivial) modification of the reduction of Goaoc, Pat\'{a}k, Pat\'{a}kov\'{a}, Tancer and Wagner showing that shellability of $2$-complexes is NP-hard.

cs.CG↗

arXiv2021

Parameterized complexity of untangling knots

Deciding whether a diagram of a knot can be untangled with a given number of moves (as a part of the input) is known to be NP-complete. In this paper we determine the parameterized complexity of this problem with respect to a natural parameter called defect. Roughly speaking, it measures the efficiency of the moves used in the shortest untangling sequence of Reidemeister moves. We show that the II- moves in a shortest untangling sequence can be essentially performed greedily. Using that, we show that this problem belongs to W[P] when parameterized by the defect. We also show that this problem is W[P]-hard by a reduction from Minimum axiom set.

cs.CC↗

arXiv2021

Weak saturation of multipartite hypergraphs

Given $q$-uniform hypergraphs ($q$-graphs) $F,G$ and $H$, where $G$ is a spanning subgraph of $F$, $G$ is called weakly $H$-saturated in $F$ if the edges in $E(F)\setminus E(G)$ admit an ordering $e_1,\dots, e_k$ so that for all $i\in [k]$ the hypergraph $G\cup \{e_1,\dots,e_i\}$ contains an isomorphic copy of $H$ which in turn contains the edge $e_i$. The weak saturation number of $H$ in $F$ is the smallest size of an $H$-weakly saturated subgraph of $F$. Weak saturation was introduced by Bollob\'as in 1968, but despite decades of study our understanding of it is still limited. The main difficulty lies in proving lower bounds on weak saturation numbers, which typically withstands combinatorial methods and requires arguments of algebraic or geometrical nature. In our main contribution in this paper we determine exactly the weak saturation number of complete multipartite $q$-graphs in the directed setting, for any choice of parameters. This generalizes a theorem of Alon from 1985. Our proof combines the exterior algebra approach from the works of Kalai with the use of the colorful exterior algebra motivated by the recent work of Bulavka, Goodarzi and Tancer on the colorful fractional Helly theorem. In our second contribution answering a question of Kronenberg, Martins and Morrison, we establish a link between weak saturation numbers of bipartite graphs in the clique versus in a complete bipartite host graph. In a similar fashion we asymptotically determine the weak saturation number of any complete $q$-partite $q$-graph in the clique, generalizing another result of Kronenberg et al.

math.CO↗

arXiv2020

Optimal bounds for the colorful fractional Helly theorem

The well known fractional Helly theorem and colorful Helly theorem can be merged into the so called colorful fractional Helly theorem. It states: For every $\alpha \in (0, 1]$ and every non-negative integer $d$, there is $\beta_{col} = \beta_{col}(\alpha, d) \in (0, 1]$ with the following property. Let $\mathcal{F}_1, \dots, \mathcal{F}_{d+1}$ be finite nonempty families of convex sets in $\mathbb{R}^d$ of sizes $n_1, \dots, n_{d+1}$ respectively. If at least $\alpha n_1 n_2 \cdots n_{d+1}$ of the colorful $(d+1)$-tuples have a nonempty intersection, then there is $i \in [d+1]$ such that $\mathcal{F}_i$ contains a subfamily of size at least $\beta_{col} n_i$ with a nonempty intersection. (A colorful $(d+1)$-tuple is a $(d+1)$-tuple $(F_1, \dots , F_{d+1})$ such that $F_i$ belongs to $\mathcal{F}_i$ for every $i$.) The colorful fractional Helly theorem was first stated and proved by B\'ar\'any, Fodor, Montejano, Oliveros, and P\'or in 2014 with $\beta_{col} = \alpha/(d+1)$. In 2017 Kim proved the theorem with better function $\beta_{col}$, which in particular tends to $1$ when $\alpha$ tends to $1$. Kim also conjectured what is the optimal bound for $\beta_{col}(\alpha, d)$ and provided the upper bound example for the optimal bound. The conjectured bound coincides with the optimal bounds for the (non-colorful) fractional Helly theorem proved independently by Eckhoff and Kalai around 1984. We verify Kim's conjecture by extending Kalai's approach to the colorful scenario. Moreover, we obtain optimal bounds also in more general setting when we allow several sets of the same color.

math.CO↗

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, Gr\"{u}nbaum 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 Gr\"unbaum'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 $n \geq 3$.

math.CO↗

arXiv2019

Shellings and sheddings induced by collapses

We say that a pure simplicial complex ${\mathbf K}$ of dimension $d$ satisfies the removal-collapsibility condition if ${\mathbf K}$ is either empty or ${\mathbf K}$ becomes collapsible after removing $\tilde \beta_d ({\mathbf K}; {\mathbb Z}_2)$ facets, where $\tilde \beta_d ({\mathbf K}; {\mathbb Z}_2)$ denotes the $d$th reduced Betti number. In this paper, we show that if the link of each face of a pure simplicial complex ${\mathbf K}$ (including the link of the empty face which is the whole ${\mathbf K}$) satisfy the removal-collapsibility condition, then the second barycentric subdivision of ${\mathbf K}$ is vertex decomposable and in particular shellable. This is a higher dimensional generalization of a result of Hachimori, who proved that that if the link of each vertex of a pure 2-dimensional simplicial complex ${\mathbf K}$ is connected, and ${\mathbf K}$ becomes simplicially collapsible after removing $\tilde{\chi}({\mathbf K})$ facets, where $\tilde \chi ({\mathbf K})$ denotes the reduced Euler characteristic, then the second barycentric subdivision of ${\mathbf K}$ is shellable. For the proof, we introduce a new variant of decomposability of a simplicial complex, stronger than vertex decomposability, which we call star decomposability. This notion may be of independent interest.

math.CO↗

arXiv2019

Even maps, the Colin de~Verdi\`ere number and representations of graphs

Van der Holst and Pendavingh introduced a graph parameter $\sigma$, which coincides with the more famous Colin de Verdi\`{e}re graph parameter $\mu$ for small values. However, the definition of $\sigma$ is much more geometric/topological directly reflecting embeddability properties of the graph. They proved $\mu(G) \leq \sigma(G) + 2$ and conjectured $\mu(G) \leq \sigma(G)$ for any graph $G$. We confirm this conjecture. As far as we know, this is the first topological upper bound on $\mu(G)$ which is, in general, tight. Equality between $\mu$ and $\sigma$ does not hold in general as van der Holst and Pendavingh showed that there is a graph $G$ with $\mu(G) \leq 18$ and $\sigma(G)\geq 20$. We show that the gap appears on much smaller values, namely, we exhibit a graph $H$ for which $\mu(H)\leq 7$ and $\sigma(H)\geq 8$. We also prove that, in general, the gap can be large: The incidence graphs $H_q$ of finite projective planes of order $q$ satisfy $\mu(H_q) \in O(q^{3/2})$ and $\sigma(H_q) \geq q^2$.

math.CO↗

arXiv2019

Embeddings of $k$-complexes into $2k$-manifolds

We improve the bound on K\"uhnel's problem to determine the smallest $n$ such that the $k$-skeleton of an $n$-simplex $\Delta_n^{(k)}$ does not embed into a compact PL $2k$-manifold $M$ by showing that if $\Delta_n^{(k)}$ embeds into $M$, then $n\leq (2k+1)+(k+1)\beta_k(M;\mathbb Z_2)$. As a consequence we obtain improved Radon and Helly type results for set systems in such manifolds. Our main tool is a new description of an obstruction for embeddability of a $k$-complex $K$ into a compact PL $2k$-manifold $M$ via the intersection form on $M$. In our approach we need that for every map $f\colon K\to M$ the restriction to the $(k-1)$-skeleton of $K$ is nullhomotopic. In particular, this condition is satisfied in interesting cases if $K$ is $(k-1)$-connected, for example a $k$-skeleton of $n$-simplex, or if $M$ is $(k-1)$-connected. In addition, if $M$ is $(k-1)$-connected and $k\geq 3$, the obstruction is complete, meaning that a $k$-complex $K$ embeds into $M$ if and only if the obstruction vanishes. For trivial intersection forms, our obstruction coincides with the standard van Kampen obstruction. However, if the form is non-trivial, the obstruction is not linear but rather 'quadratic' in a sense that it vanishes if and only if certain system of quadratic diophantine equations is solvable. This may potentially be useful in attacking algorithmic decidability of embeddability of $k$-complexes into PL $2k$-manifolds.

math.AT↗

arXiv2018

The unbearable hardness of unknotting

We prove that deciding if a diagram of the unknot can be untangled using at most $k$ Riedemeister moves (where $k$ is part of the input) is NP-hard. We also prove that several natural questions regarding links in the $3$-sphere are NP-hard, including detecting whether a link contains a trivial sublink with $n$ components, computing the unlinking number of a link, and computing a variety of link invariants related to four-dimensional topology (such as the $4$-ball Euler characteristic, the slicing number, and the $4$-dimensional clasp number).

math.GT↗

arXiv2018

On the growth of the M\"obius function of permutations

We study the values of the M\"obius function $\mu$ of intervals in the containment poset of permutations. We construct a sequence of permutations $\pi_n$ of size $2n-2$ for which $\mu(1,\pi_n)$ is given by a polynomial in $n$ of degree 7. This construction provides the fastest known growth of $|\mu(1,\pi)|$ in terms of $|\pi|$, improving a previous quadratic bound by Smith. Our approach is based on a formula expressing the M\"obius function of an arbitrary permutation interval $[\alpha,\beta]$ in terms of the number of embeddings of the elements of the interval into $\beta$.

math.CO↗

arXiv2017

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↗