Related papers: Hardness of almost embedding simplicial complexes …

We show that it is $\mathsf{NP}$-hard to approximate the hyperspherical radius of a triangulated manifold up to an almost-polynomial factor.

We provide an irreducibility test in the ring K[[x]][y] whose complexity is quasi-linear with respect to the valuation of the discriminant, assuming the input polynomial F square-free and K a perfect field of characteristic zero or greater…

It is shown that any smooth closed orientable manifold of dimension $2k + 1$, $k \geq 2$, admits a smooth polynomially convex embedding into $\mathbb C^{3k}$. This improves by $1$ the previously known lower bound of $3k+1$ on the possible…

We consider the embeddability problem of a graph G into a two-dimensional simplicial complex C: Given G and C, decide whether G admits a topological embedding into C. The problem is NP-hard, even in the restricted case where C is…

We study the recognition complexity of subgraphs of k-connected planar cubic graphs for k = 1, 2, 3. We present polynomial-time algorithms to recognize subgraphs of 1- and 2-connected planar cubic graphs, both in the variable and fixed…

We consider 2-dimensional random simplicial complexes $Y$ in the multi-parameter model. We establish the multi-parameter threshold for the property that every 2-dimensional simplicial complex $S$ admits a topological embedding into $Y$…

A graph drawing in the plane is called an almost embedding if images of any two non-adjacent simplices (i.e. vertices or edges) are disjoint. We introduce integer invariants of almost embeddings: winding number, cyclic and triodic Wu…

$\newcommand{\eps}{\varepsilon} $We prove that for any $\eps > 0$ it is $\textsf{NP}$-hard to approximate the non-commutative Grothendieck problem to within a factor $1/2 + \eps$, which matches the approximation ratio of the algorithm of…

We consider the problem of computing the smallest possible distortion for embedding of a given n-point metric space into R^d, where d is fixed (and small). For d=1, it was known that approximating the minimum distortion with a factor better…

We consider the question of approximating Max 2-CSP where each variable appears in at most $d$ constraints (but with possibly arbitrarily large alphabet). There is a simple $(\frac{d+1}{2})$-approximation algorithm for the problem. We prove…

Let $K$ be a finite simplicial complex. We prove that the normalized expected Betti numbers of a random subcomplex in its $d$-th barycentric subdivision $\text{Sd}^d (K)$ converge to universal limits as $d$ grows to $+ \infty$. In…

We show that the problem of recognizing that a knot diagram represents a specific torus knot, or any torus knot at all, is in the complexity class ${\sf NP} \cap {\sf co\text{-}NP}$, assuming the generalized Riemann hypothesis. We also show…

We give a new, elementary proof of what we believe is the simplest known example of a ``natural'' problem in computational 3-dimensional topology that is $\mathsf{NP}$-hard -- namely, the \emph{Trivial Sublink Problem}: given a diagram $L$…

(1) We show that if a presentation of the trivial group is "hard to trivialize", in the sense that lots of Tietze moves are necessary to transform it into the trivial presentation, then the associated presentation complex (which is a…

We prove a combinatorial theorem on families of disjoint sub-boxes of a discrete cube, which implies that there are at most $2^{d+1}-2$ nearly neighbourly simplices in $\mathbb R^d$.

In this paper we extend the works of Tancer and of Malgouyres and Franc\'es, showing that $(d,k)$-collapsibility is NP-complete for $d\geq k+2$ except $(2,0)$. By $(d,k)$-collapsibility we mean the following problem: determine whether a…

We consider $d$-dimensional simplicial complexes which can be PL embedded in the $2d$-dimensional euclidean space. In short, we show that in any such complex, for any three vertices, the intersection of the link-complexes of the vertices is…

We prove a sufficient condition for a finite clique complex to collapse to a $k$-dimensional complex, and use this to exhibit thresholds for $(k+1)$-collapsibility in a sparse random clique complex. In particular, if every strongly…

We study the computational complexity of several problems connected with finding a maximal distance-$k$ matching of minimum cardinality or minimum weight in a given graph. We introduce the class of $k$-equimatchable graphs which is an edge…

Let $K\subset\mathbb{R}^d$ be a self-similar set generated by an iterated function system $\{\varphi_i\}_{i=1}^m$ satisfying the strong separation condition and let $f$ be a contracting similitude with $f(K)\subset K$. We show that $f(K)$…