Related papers: Complexity of Interlocking Polyominoes
Tightness is a generalisation of the notion of convexity: a space is tight if and only if it is "as convex as possible", given its topological constraints. For a simplicial complex, deciding tightness has a straightforward exponential time…
Given a set of disjoint simple polygons $\sigma_1, \ldots, \sigma_n$, of total complexity $N$, consider a convexification process that repeatedly replaces a polygon by its convex hull, and any two (by now convex) polygons that intersect by…
Given a set $S \subseteq \mathbb{R}^d$, a hollow polytope has vertices in $S$ but contains no other point of $S$ in its interior. We prove upper and lower bounds on the maximum number of vertices of hollow polytopes whose facets are…
Counting Euclidean triangulations with vertices in a finite set $\C$ of the convex hull $\conv(\C)$ of $\C$ is difficult in general, both algorithmically and theoretically. The aim of this paper is to describe nearly convex polygons, a…
The aim in packing problems is to decide if a given set of pieces can be placed inside a given container. A packing problem is defined by the types of pieces and containers to be handled, and the motions that are allowed to move the pieces.…
We study the problems of covering or partitioning a polygon $P$ (possibly with holes) using a minimum number of small pieces, where a small piece is a connected sub-polygon contained in an axis-aligned unit square. For covering, we seek to…
We estimate the frequency of polynomial iterations which falls in a given multiplicative subgroup of a finite field of $p$ elements. We also give a lower bound on the size of the subgroup which is multiplicatively generated by the first $N$…
We study the packing of a large number of congruent and non--overlapping circles inside a regular polygon. We have devised efficient algorithms that allow one to generate configurations of $N$ densely packed circles inside a regular polygon…
This paper investigated the problem of embedding a simple Hamiltonian Cycle with n vertices on n points inside a simple polygon. This problem seeks to embed a straight-line cycle (without bends), which does not intersect either itself or…
We introduce the notions of de Bruijn polyominoes and prismatic polyominoes, which generalize the notions of de Bruijn sequences and arrays. Given a small fixed polyomino $p$ and a set of colors $[n]$, a de Bruijn polyomino for $(p,n)$ is a…
This work is divided into three parts. The first part concerns polynomials in one variable with all real roots. We consider linear transformations that preserve real rootedness, as well as matrices that preserve interlacing. The second part…
We classify all convex polyomino ideals which are linearly related or have a linear resolution. Convex stack polyominoes whose ideals are extremal Gorenstein are also classified. In addition, we characterize, in combinatorial terms, the…
We introduce the notion of interlacing log-concavity of a polynomial sequence $\{P_m(x)\}_{m\geq 0}$, where $P_m(x)$ is a polynomial of degree m with positive coefficients $a_{i}(m)$. This sequence of polynomials is said to be interlacing…
We study the problem of generating monomials of a polynomial in the context of enumeration complexity. In this setting, the complexity measure is the delay between two solutions and the total time. We present two new algorithms for…
Polynomial zonotopes, a non-convex set representation, have a wide range of applications from real-time motion planning and control in robotics, to reachability analysis of nonlinear systems and safety shielding in reinforcement learning.…
Given a permutation polynomial of a large finite field, finding its inverse is usually a hard problem. Based on a piecewise interpolation formula, we construct the inverses of cyclotomic mapping permutation polynomials of arbitrary finite…
Two polygons are amicable if the perimeter of one is equal to the area of the other and vice versa. A polygon is a lattice polygon if its vertices are on the integer lattice $\Z^2$. We show that there is one pair of amicable lattice…
Orthogonal surfaces are nice mathematical objects which have interesting connections to various fields, e.g., integer programming, monomial ideals and order dimension. While orthogonal surfaces in one or two dimensions are rather trivial…
It is well-known that every isosceles tetrahedron (disphenoid) admits infinitely many simple closed geodesics on its surface. They can be naturally enumerated by pairs of co-prime integers $n > m > 1$ with two additional cases $(1,0)$ and…
In this paper, we study movements of simple polygonal chains in 3D. We say that an open, simple polygonal chain can be straightened if it can be continuously reconfigured to a straight sequence of segments in such a manner that both the…