相关论文: Packing a cake into a box
The design of algorithms for political redistricting generally takes one of two approaches: optimize an objective such as compactness or, drawing on fair division, construct a protocol whose outcomes guarantee partisan fairness. We aim to…
The problem of fair division known as "cake cutting" has been the focus of multiple papers spanning several decades. The most prominent problem in this line of work has been to bound the query complexity of computing an envy-free outcome in…
This paper proposes a cake-cutting protocol using cryptography when the cake is a heterogeneous good that is represented by an interval on a real line. Although the Dubins-Spanier moving-knife protocol with one knife achieves simple…
Alice and Bob want to cut a cake; however, in contrast to the usual problems of fair division, they want to cut it unfairly. More precisely, they want to cut it in ratio $(a:b)$. (We can assume gcd(a,b)=1.) Let f(a,b) be the number of cuts…
Cake cutting is a classic model for studying fair division of a heterogeneous, divisible resource among agents with individual preferences. Addressing cake division under a typical requirement that each agent must receive a connected piece…
Pillow boxes are surfaces used for gift boxes, packaging, and even architectural applications. By definition, a pillow box is isometric to a double rectangle consisting of two copies of a rectangle. If the crease pattern is allowed to…
We address the problem of fair division, or cake cutting, with the goal of finding truthful mechanisms. In the case of a general measure space ("cake") and non-atomic, additive individual preference measures - or utilities - we show that…
We study the geometric aspects of the magic trick called Gozinta Boxes. We generalize Gozinta Boxes to other dimensions, and we show that in three and higher dimensions, the maximum number of boxes is 3, and in two dimensions, the maximum…
Given a set of $p$ players we consider problems concerning envy-free allocation of collections of $k$ pieces from a given set of goods or chores. We show that if $p\le n$ and each player can choose $k$ pieces out of $n$ pieces of a cake,…
In cake-cutting, strategy-proofness is a very costly requirement in terms of fairness: for n=2 it implies a dictatorial allocation, whereas for n > 2 it requires that one agent receives no cake. We show that a weaker version of this…
We study the proportional chore division problem where a protocol wants to divide an undesirable object, called chore, among $n$ different players. The goal is to find an allocation such that the cost of the chore assigned to each player be…
Consider n straight line cuts of a circular pizza made so as to maximize the number of pieces. We investigate how fair such a maximal division may be and how many slices are obtained if the cuts are successfully made with a certain…
Suppose that your mother gave you n candies. You have to eat at least one candy each day. One possibility is to eat all n of them the first day. The other extreme is to make them last n days, and only eat one candy a day. Altogether, you…
In short geometrization conjecture of W.\,Thurston (finally proved by G.~Perelman) says that any oriented $3$-manifold can be canonically partitioned into pieces, which have a geometric structure of one of the eight types. In the seminal…
In the classic circle packing problem, one asks whether a given set of circles can be packed into a given container. Packing problems like this have been shown to be $\mathsf{NP}$-hard. In this paper, we present new sufficient conditions…
The classical cake cutting problem studies how to find fair allocations of a heterogeneous and divisible resource among multiple agents. Two of the most commonly studied fairness concepts in cake cutting are proportionality and…
We prove several results addressing the envy-free division problem in the presence of an unpredictable (secretive) player, called the "dragon". There are two basic scenarios. 1. There are $r-1$ players and a dragon. Once the "cake" is…
We study the problem of whether rectangular polyominoes with holes are cube-foldable, that is, whether they can be folded into a cube, if creases are only allowed along grid lines. It is known that holes of sufficient size guarantee that…
When can a polyomino piece of paper be folded into a unit cube? Prior work studied tree-like polyominoes, but polyominoes with holes remain an intriguing open problem. We present sufficient conditions for a polyomino with one or several…
A knot in the three-sphere is doubly slice if it is the cross-section of an unknotted two-sphere in the four-sphere. For low-crossing knots, the most complete work to date gives a classification of doubly slice knots through 9 crossings. We…