Related papers: Framework for $\exists \mathbb{R}$-Completeness of…
We study the two-dimensional geometric knapsack problem for convex polygons. Given a set of weighted convex polygons and a square knapsack, the goal is to select the most profitable subset of the given polygons that fits non-overlappingly…
Cutting and packing problems are present in many, at first glance unconnected, areas, therefore it's beneficial to have a good understanding of their underlying structure, to select proper techniques for finding solutions. Cutting and…
The first part of this paper is devoted to an analysis of moment problems in R^n with supports contained in a closed set defined by finitely many polynomial inequalities. The second part of the paper uses the representation results of…
Static packings of perfectly rigid particles are investigated theoretically and numerically. The problem of finding the contact forces in such packings is formulated mathematically. Letting the values of the contact forces define a vector…
The problem of packing smaller objects within a larger object has been of interest since decades. In these problems, in addition to the requirement that the smaller objects must lie completely inside the larger objects, they are expected to…
We study the problem of aggregating polygons by covering them with disjoint representative regions, thereby inducing a clustering of the polygons. Our objective is to minimize a weighted sum of the total area and the total perimeter of the…
The problem of packing of equal disks (or circles) into a rectangle is a fundamental geometric problem. (By a packing here we mean an arrangement of disks in a rectangle without overlapping.) We consider the following algorithmic…
Packing optimization is a prevalent problem that necessitates robust and efficient algorithms that are also simple to implement. One group of approaches is the raster methods, which rely on approximating the objects with pixelated…
This work puts forward a form finding problem of designing a least-volume vault that is a surface structure spanning over a plane region, which via pure compression transfers a vertically tracking load to the supporting boundary. Through a…
It is well-known that the convex and concave envelope of a multilinear polynomial over a box are polyhedral functions. Exponential-sized extended and projected formulations for these envelopes are also known. We consider the convexification…
We consider set covering problems where the underlying set system satisfies a particular replacement property w.r.t. a given partial order on the elements: Whenever a set is in the set system then a set stemming from it via the replacement…
In this paper the problem of maximizing the distance to a given fixed point over an intersection of balls is considered. It is known that this problem is NP complete in the general case, since any subset sum problem can be solved upon…
The Sequential Multiple Knapsack Problem is a special case of Multiple knapsack problem in which the items sizes are divisible. A characterization of the optimal solutions of the problem and a description of the convex hull of all the…
We study several consequences of the packing problem, a conjecture from combinatorial optimization, using algebraic invariants of square-free monomial ideals. While the packing problem is currently unresolved, we successfully settle the…
We prove that the art gallery problem is equivalent under polynomial time reductions to deciding whether a system of polynomial equations over the real numbers has a solution. The art gallery problem is a classical problem in computational…
We study dense packings of a large number of congruent non-overlapping circles inside a square by looking for configurations which maximize the packing density, defined as the ratio between the area occupied by the disks and the area of the…
The Orbit Problem consists of determining, given a matrix $A\in \mathbb{R}^{d\times d}$ and vectors $x,y\in \mathbb{R}^d$, whether there exists $n\in \mathbb{N}$ such that $A^n=y$. This problem was shown to be decidable in a seminal work of…
We consider the {\em Shaped Partition Problem} of partitioning $n$ given vectors in real $k$-space into $p$ parts so as to maximize an arbitrary objective function which is convex on the sum of vectors in each part, subject to arbitrary…
Cutting and packing problems arise in a large variety of industrial applications, where there is a need to cut pieces from a large object, or placing them inside a containers, without overlap. When the pieces or the containers have…
Many combinatorial optimization problems such as the bin packing and multiple knapsack problems involve assigning a set of discrete objects to multiple containers. These problems can be used to model task and resource allocation problems in…