Related papers: Consequences of the packing problem
Symmetry is an important problem in many combinatorial problems. One way of dealing with symmetry is to add constraints that eliminate symmetric solutions. We survey recent results in this area, focusing especially on two common and useful…
This paper presents a theory of non-linear integer/real arithmetic and algorithms for reasoning about this theory. The theory can be conceived as an extension of linear integer/real arithmetic with a weakly-axiomatized multiplication…
We give a new combinatorial characterization of the big height of a squarefree monomial ideal leading to a new bound for the projective dimension of a monomial ideal.
We study an extension of the classical Bin Packing Problem, where each item consumes the bin capacity during a given time window that depends on the item itself. The problem asks for finding the minimum number of bins to pack all the items…
We survey research relating algebraic properties of powers of squarefree monomial ideals to combinatorial structures. In particular, we describe how to detect important properties of (hyper)graphs by solving ideal membership problems and…
We give an overview of the 2024 Computational Geometry Challenge targeting the problem \textsc{Maximum Polygon Packing}: Given a convex region $P$ in the plane, and a collection of simple polygons $Q_1, \ldots, Q_n$, each $Q_i$ with a…
The problem of packing boxes into a large box is often a part of a larger problem. For example in furniture supply chain applications, one needs to decide what trucks to use to transport furniture between production sites and distribution…
Can you decide if there is a coincidence in the numbers counting two different combinatorial objects? For example, can you decide if two regions in $\mathbb{R}^3$ have the same number of domino tilings? There are two versions of the…
The Bin Packing Problem (BPP) is a well-established combinatorial optimization (CO) problem. Since it has many applications in our daily life, e.g. logistics and resource allocation, people are seeking efficient bin packing algorithms. On…
Keller proposed a combinatorial conjecture on construction of an n-by-infinite matrix, which comes from showing the existence of many orbits of different sizes in certain linear group actions. He proved it for the case n=4, and we show that…
We announce misere-play solutions to several previously-unsolved combinatorial games. The solutions are described in terms of misere quotients--commutative monoids that encode the additive structure of specific misere-play games. We also…
We consider a package assignment problem with multiple units of indivisible items. The seller can specify preferences over partitions of their supply between buyers as packaging costs. We propose incremental costs together with a graph that…
For every list of integers x_1, ..., x_m there is some j such that x_1 + ... + x_j - x_{j+1} - ... - x_m \approx 0. So the list can be nearly balanced and for this we only need one alternation between addition and subtraction. But what if…
A tight continuous relaxation is a crucial factor in solving mixed integer formulations of many NP-hard combinatorial optimization problems. The (weighted) max $k$-cut problem is a fundamental combinatorial optimization problem with…
This is a survey and research note on the modified Orlik conjecture derived from the division theorem introduced in [2]. The division theorem is a generalization of classical addition-deletion theorems for free arrangements. The division…
Extension problems for polynomial valuations on different cones of convex functions are investigated. It is shown that for the classes of functions under consideration, the extension problem reduces to a simple geometric obstruction on the…
The matching problem plays a basic role in combinatorial optimization and in statistical mechanics. In its stochastic variants, optimization decisions have to be taken given only some probabilistic information about the instance. While the…
We investigate finite-dimensional constrained structured optimization problems, featuring composite objective functions and set-membership constraints. Offering an expressive yet simple language, this problem class provides a modeling…
If a collection of identical particles is poured into a container, different shapes will fill to different densities. But what is the shape that fills a container as close as possible to a pre-specified, desired density? We demonstrate a…
The purpose of this note is to provide an overview of the containment problem for symbolic and ordinary powers of homogeneous ideals, related conjectures and examples. We focus here on ideals with zero dimensional support. This is an area…