Related papers: Packing a cake into a box
In how many ways can you place n chocolate pieces all of different sizes in an n by n chocolate box, in such a way that when you go from left to right and from top to bottom, there are no gaps AND the sizes increase along each row and each…
Assume you have a pizza consisting of four ingredients (e.g., bread, tomatoes, cheese and olives) that you want to share with your friend. You want to do this fairly, meaning that you and your friend should get the same amount of each…
We study the query complexity of cake cutting and give lower and upper bounds for computing approximately envy-free, perfect, and equitable allocations with the minimum number of cuts. The lower bounds are tight for computing connected…
Assume you have a 2-dimensional pizza with $2n$ ingredients that you want to share with your friend. For this you are allowed to cut the pizza using several straight cuts, and then give every second piece to your friend. You want to do this…
This paper extends the classic cake-cutting problem to a situation in which the "cake" is divided among families. Each piece of cake is owned and used simultaneously by all members of the family. A typical example of such a cake is land. We…
We consider the classic problem of envy-free division of a heterogeneous good ("cake") among several agents. It is known that, when the allotted pieces must be connected, the problem cannot be solved by a finite algorithm for 3 or more…
The paper considers fair allocation of resources that are already allocated in an unfair way. This setting requires a careful balance between the fairness considerations and the rights of the present owners. The paper presents re-division…
We study classic cake-cutting problems, but in discrete models rather than using infinite-precision real values, specifically, focusing on their communication complexity. Using general discrete simulations of classical infinite-precision…
Austin's moving knife procedure was originally introduced to find a consensus division of an interval/circular cake between two agents, each of whom believes that they receive exactly half of the cake. We generalise this in two ways: we…
In 1907, Henry Ernest Dudeney posed a puzzle: ``cut any equilateral triangle \dots\ into as few pieces as possible that will fit together and form a perfect square'' (without overlap, via translation and rotation). Four weeks later, Dudeney…
We give a new proof of the following interesting fact recently proved by Bower and Michael: if a d-dimensional rectangular box can be tiled using translates of two types of rectangular bricks, then it can also be tiled in the following way.…
Modern 3D printing technologies and the upcoming mass-customization paradigm call for efficient methods to produce and distribute arbitrarily-shaped 3D objects. This paper introduces an original algorithm to split a 3D model in parts that…
Cake-cutting is a fundamental model of dividing a heterogeneous resource, such as land, broadcast time, and advertisement space. In this study, we consider the problem of dividing a discrete cake fairly in which the indivisible goods are…
We study the problem of fairly allocating a divisible resource, also known as cake cutting, with an additional requirement that the shares that different agents receive should be sufficiently separated from one another. This captures, for…
We explore an instance of the question of partitioning a polygon into pieces, each of which is as ``circular'' as possible, in the sense of having an aspect ratio close to 1. The aspect ratio of a polygon is the ratio of the diameters of…
In this paper we study some cube packing problems. In particular we are interested in compact subsets of $\mathbb{R}^n,n\geq 2$, which contain boundaries of cubes with all side lengths in $(0,1)$. We show here that such sets must have lower…
In this paper, we consider a game played on a rectangular $m \times n$ gridded chocolate bar. Each move, a player breaks the bar along a grid line. Each move after that consists of taking any piece of chocolate and breaking it again along…
We propose an online form of the cake cutting problem. This models situations where players arrive and depart during the process of dividing a resource. We show that well known fair division procedures like cut-and-choose and the…
We study the discrete variation of the classical cake-cutting problem where n players divide a 1-dimensional cake with exactly (n-1) cuts, replacing the continuous, infinitely divisible "cake" with a necklace of discrete, indivisible…
Alon, Bohman, Holzman and Kleitman proved that any partition of a $d$-dimensional discrete box into proper sub-boxes must consist of at least $2^d$ sub-boxes. Recently, Leader, Mili\'{c}evi\'{c} and Tan considered the question of how many…