Related papers: Slicing the hypercube is not easy

We show that we can obtain a reducible spherical curve from any non-trivial spherical curve by four or less inverse-half-twisted splices, i.e., the reductivity, which represents how reduced a spherical curve is, is four or less. We also…

We study hyperplane sections of smooth polarized $K3$-surfaces that split into unions of lines. We describe the dual adjacency graphs of such sections and find sharp upper bounds on their number. In most cases (starting from degree $6$), we…

We study the problem of learning a hypergraph via edge detecting queries. In this problem, a learner queries subsets of vertices of a hidden hypergraph and observes whether these subsets contain an edge or not. In general, learning a…

We study the balanced $k$-way hypergraph partitioning problem, with a special focus on its practical applications to manycore scheduling. Given a hypergraph on $n$ nodes, our goal is to partition the node set into $k$ parts of size at most…

Partial cubes are graphs isometrically embeddable into hypercubes. We analyze how isometric cycles in partial cubes behave and derive that every partial cube of girth more than 6 must have vertices of degree less than 3. As a direct…

We consider constructions of covering-radius-1 completely regular codes, or, equivalently, equitable 2-partitions (regular 2-partitions, perfect 2-colorings), of halved n-cubes. Keywords: completely regular code, equitable partition,…

Linear error-correcting codes can be used for constructing secret sharing schemes; however finding in general the access structures of these secret sharing schemes and, in particular, determining efficient access structures is difficult.…

We establish new exponential in dimension lower bounds for the Maximum Halfspace Discrepancy problem, which models linear classification. Both are fundamental problems in computational geometry and machine learning in their exact and…

We consider the problem of computing, given a set S of n points in the plane, which points of S are vertices of the convex hull of S. For certain variations of this problem, different proofs exist that the complexity of this problem in the…

Let $f: E \mapsto \{1,2,\dots,k\}$ be an edge coloring of the $n$ - dimensional hypercube $H_n$. By the palette at a vertex $v$ we mean the sequence $\left(f(e_1(v)), f(e_1(v)),\dots, f(e_n(v))\right)$, where $e_i(v)$ is the $i$ -…

We study the parameterized complexity of the following fundamental geometric problems with respect to the dimension $d$: i) Given $n$ points in $\Rd$, compute their minimum enclosing cylinder. ii) Given two $n$-point sets in $\Rd$, decide…

Let $M$ be a compact hypersurface with boundary $\partial M=\partial D_1 \cup \partial D_2$, $\partial D_1 \subset \Pi _1$, $\partial D_2 \subset \Pi _2$, $\Pi_1$ and $\Pi _2$ two parallel hyperplanes in $\mathbb{R}^{n+1}$ ($n \geq 2$).…

The Rubik's Cube is perhaps the world's most famous and iconic puzzle, well-known to have a rich underlying mathematical structure (group theory). In this paper, we show that the Rubik's Cube also has a rich underlying algorithmic…

In this short note we study two questions about the existence of subgraphs of the hypercube $Q_n$ with certain properties. The first question, due to Erd\H{o}s--Hamburger--Pippert--Weakley, asks whether there exists a bounded degree…

Current state-of-the-art methods for solving discrete optimization problems are usually restricted to convex settings. In this paper, we propose a general approach based on cutting planes for solving nonlinear, possibly nonconvex, binary…

We treat the boundary problem for complex varieties with isolated singularities, of complex dimension greater than or equal to 3, non necessarily compact, which are contained in strongly convex, open subsets of a complex Hilbert space H. We…

The $k$-cut complex was recently introduced by Bayer et al. as a generalization of earlier work of Fr{\"o}berg (1990) and Eagon and Reiner (1998), and was shown to be shellable for several classes of graphs. In this article, we prove that…

We consider hyperbolic manifolds with boundary, which admit an ideal triangulation with n ideal triangles and one edge. We prove that the number of these manifolds is $\exp(n\ln(n)+O(n))$.

Untangling is a process in which some vertices of a planar graph are moved to obtain a straight-line plane drawing. The aim is to move as few vertices as possible. We present an algorithm that untangles the cycle graph C_n while keeping at…

We show that for every cubic graph G with sufficiently large girth there exists a probability distribution on edge-cuts of G such that each edge is in a randomly chosen cut with probability at least 0.88672. This implies that G contains an…