THE WAIT-AND-SEE OPTION IN ASCENDING PRICE AUCTIONS

Cake-cutting protocols aim at dividing a ``cake'' (i.e., a divisible resource) and assigning the resulting portions to several players in a way that each of the players feels to have received a ``fair'' amount of the cake. An important notion of fairness is envy-freeness: No player wishes to switch the portion of the cake received with another player's portion. Despite intense efforts in the past, it is still an open question whether there is a \emph{finite bounded} envy-free cake-cutting protocol for an arbitrary number of players, and even for four players. We introduce the notion of degree of guaranteed envy-freeness (DGEF) as a measure of how good a cake-cutting protocol can approximate the ideal of envy-freeness while keeping the protocol finite bounded (trading being disregarded). We propose a new finite bounded proportional protocol for any number n \geq 3 of players, and show that this protocol has a DGEF of 1 + \lceil (n^2)/2 \rceil. This is the currently best DGEF among known finite bounded cake-cutting protocols for an arbitrary number of players. We will make the case that improving the DGEF even further is a tough challenge, and determine, for comparison, the DGEF of selected known finite bounded cake-cutting protocols.Comment: 37 pages, 4 figure

A Cryptographic Moving-Knife Cake-Cutting Protocol

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 fairness, all players must execute the protocol synchronously. Thus, the protocol cannot be executed on asynchronous networks such as the Internet. We show that the moving-knife protocol can be executed asynchronously by a discrete protocol using a secure auction protocol. The number of cuts is n-1 where n is the number of players, which is the minimum.Comment: In Proceedings IWIGP 2012, arXiv:1202.422

Solving kk-means on High-dimensional Big Data

In recent years, there have been major efforts to develop data stream algorithms that process inputs in one pass over the data with little memory requirement. For the $k$-means problem, this has led to the development of several $(1+\varepsilon)$-approximations (under the assumption that $k$ is a constant), but also to the design of algorithms that are extremely fast in practice and compute solutions of high accuracy. However, when not only the length of the stream is high but also the dimensionality of the input points, then current methods reach their limits. We propose two algorithms, piecy and piecy-mr that are based on the recently developed data stream algorithm BICO that can process high dimensional data in one pass and output a solution of high quality. While piecy is suited for high dimensional data with a medium number of points, piecy-mr is meant for high dimensional data that comes in a very long stream. We provide an extensive experimental study to evaluate piecy and piecy-mr that shows the strength of the new algorithms.Comment: 23 pages, 9 figures, published at the 14th International Symposium on Experimental Algorithms - SEA 201

Cutting the same fraction of several measures

We study some measure partition problems: Cut the same positive fraction of $d+1$ measures in $\mathbb R^d$ with a hyperplane or find a convex subset of $\mathbb R^d$ on which $d+1$ given measures have the same prescribed value. For both problems positive answers are given under some additional assumptions.Comment: 7 pages 2 figure

A Discrete and Bounded Envy-free Cake Cutting Protocol for Four Agents

We consider the well-studied cake cutting problem in which the goal is to identify a fair allocation based on a minimal number of queries from the agents. The problem has attracted considerable attention within various branches of computer science, mathematics, and economics. Although, the elegant Selfridge-Conway envy-free protocol for three agents has been known since 1960, it has been a major open problem for the last fifty years to obtain a bounded envy-free protocol for more than three agents. We propose a discrete and bounded envy-free protocol for four agents

Multiagent Negotiation for Fair and Unbiased Resource Allocation

This paper proposes a novel solution for the n agent cake cutting (resource allocation) problem. We propose a negotiation protocol for dividing a resource among n agents and then provide an algorithm for allotting portions of the resource. We prove that this protocol can enable distribution of the resource among n agents in a fair manner. The protocol enables agents to choose portions based on their internal utility function, which they do not have to reveal. In addition to being fair, the protocol has desirable features such as being unbiased and verifiable while allocating resources. In the case where the resource is two-dimensional (a circular cake) and uniform, it is shown that each agent can get close to l/n of the whole resource

Generalized Density-Functional Tight-Binding Repulsive Potentials from Unsupervised Machine Learning

We combine the approximate density-functional tight-binding (DFTB) method with unsupervised machine learning. This allows us to improve transferability and accuracy, make use of large quantum chemical data sets for the parametrization, and efficiently automatize the parametrization process of DFTB. For this purpose, generalized pair-potentials are introduced, where the chemical environment is included during the learning process, leading to more specific effective two-body potentials. We train on energies and forces of equilibrium and nonequilibrium structures of 2100 molecules, and test on ∼130 000 organic molecules containing O, N, C, H, and F atoms. Atomization energies of the reference method can be reproduced within an error of ∼2.6 kcal/mol, indicating drastic improvement over standard DFTB

The Role of Pressure in Inverse Design for Assembly

Isotropic pairwise interactions that promote the self assembly of complex particle morphologies have been discovered by inverse design strategies derived from the molecular coarse-graining literature. While such approaches provide an avenue to reproduce structural correlations, thermodynamic quantities such as the pressure have typically not been considered in self-assembly applications. In this work, we demonstrate that relative entropy optimization can be used to discover potentials that self-assemble into targeted cluster morphologies with a prescribed pressure when the iterative simulations are performed in the isothermal-isobaric ensemble. By tuning the pressure in the optimization, we generate a family of simple pair potentials that all self-assemble the same structure. Selecting an appropriate simulation ensemble to control the thermodynamic properties of interest is a general design strategy that could also be used to discover interaction potentials that self-assemble structures having, for example, a specified chemical potential.Comment: 29 pages, 8 figure