Computing Socially-Efficient Cake Divisions

We consider a setting in which a single divisible good ("cake") needs to be divided between n players, each with a possibly different valuation function over pieces of the cake. For this setting, we address the problem of finding divisions that maximize the social welfare, focusing on divisions where each player needs to get one contiguous piece of the cake. We show that for both the utilitarian and the egalitarian social welfare functions it is NP-hard to find the optimal division. For the utilitarian welfare, we provide a constant factor approximation algorithm, and prove that no FPTAS is possible unless P=NP. For egalitarian welfare, we prove that it is NP-hard to approximate the optimum to any factor smaller than 2. For the case where the number of players is small, we provide an FPT (fixed parameter tractable) FPTAS for both the utilitarian and the egalitarian welfare objectives

Deterministic, Strategyproof, and Fair Cake Cutting

We study the classic cake cutting problem from a mechanism design perspective, in particular focusing on deterministic mechanisms that are strategyproof and fair. We begin by looking at mechanisms that are non-wasteful and primarily show that for even the restricted class of piecewise constant valuations there exists no direct-revelation mechanism that is strategyproof and even approximately proportional. Subsequently, we remove the non-wasteful constraint and show another impossibility result stating that there is no strategyproof and approximately proportional direct-revelation mechanism that outputs contiguous allocations, again, for even the restricted class of piecewise constant valuations. In addition to the above results, we also present some negative results when considering an approximate notion of strategyproofness, show a connection between direct-revelation mechanisms and mechanisms in the Robertson-Webb model when agents have piecewise constant valuations, and finally also present a (minor) modification to the well-known Even-Paz algorithm that has better incentive-compatible properties for the cases when there are two or three agents.Comment: A shorter version of this paper will appear at IJCAI 201

07261 Abstracts Collection -- Fair Division

From 24.06. to 29.06.2007, the Dagstuhl Seminar 07261 % generate automatically ``Fair Division\u27\u27 % generate automatically was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

Monotonicity and Competitive Equilibrium in Cake-cutting

We study the monotonicity properties of solutions in the classic problem of fair cake-cutting --- dividing a heterogeneous resource among agents with different preferences. Resource- and population-monotonicity relate to scenarios where the cake, or the number of participants who divide the cake, changes. It is required that the utility of all participants change in the same direction: either all of them are better-off (if there is more to share or fewer to share among) or all are worse-off (if there is less to share or more to share among). We formally introduce these concepts to the cake-cutting problem and examine whether they are satisfied by various common division rules. We prove that the Nash-optimal rule, which maximizes the product of utilities, is resource-monotonic and population-monotonic, in addition to being Pareto-optimal, envy-free and satisfying a strong competitive-equilibrium condition. Moreover, we prove that it is the only rule among a natural family of welfare-maximizing rules that is both proportional and resource-monotonic.Comment: Revised versio

Discontinuity preserving image registration for breathing induced sliding organ motion

Image registration is a powerful tool in medical image analysis and facilitates the clinical routine in several aspects. It became an indispensable device for many medical applications including image-guided therapy systems. The basic goal of image registration is to spatially align two images that show a similar region of interest. More speci�cally, a displacement �eld respectively a transformation is estimated, that relates the positions of the pixels or feature points in one image to the corresponding positions in the other one. The so gained alignment of the images assists the doctor in comparing and diagnosing them. There exist di�erent kinds of image registration methods, those which are capable to estimate a rigid transformation or more generally an a�ne transformation between the images and those which are able to capture a more complex motion by estimating a non-rigid transformation. There are many well established non-rigid registration methods, but those which are able to preserve discontinuities in the displacement �eld are rather rare. These discontinuities appear in particular at organ boundaries during the breathing induced organ motion. In this thesis, we make use of the idea to combine motion segmentation with registration to tackle the problem of preserving the discontinuities in the resulting displacement �eld. We introduce a binary function to represent the motion segmentation and the proposed discontinuity preserving non-rigid registration method is then formulated in a variational framework. Thus, an energy functional is de�ned and its minimisation with respect to the displacement �eld and the motion segmentation will lead to the desired result. In theory, one can prove that for the motion segmentation a global minimiser of the energy functional can be found, if the displacement �eld is given. The overall minimisation problem, however, is non-convex and a suitable optimisation strategy has to be considered. Furthermore, depending on whether we use the pure L1-norm or an approximation of it in the formulation of the energy functional, we use di�erent numerical methods to solve the minimisation problem. More speci�cally, when using an approximation of the L1-norm, the minimisation of the energy functional with respect to the displacement �eld is performed through Brox et al.'s �xed point iteration scheme, and the minimisation with respect to the motion segmentation with the dual algorithm of Chambolle. On the other hand, when we make use of the pure L1-norm in the energy functional, the primal-dual algorithm of Chambolle and Pock is used for both, the minimisation with respect to the displacement �eld and the motion segmentation. This approach is clearly faster compared to the one using the approximation of the L1-norm and also theoretically more appealing. Finally, to support the registration method during the minimisation process, we incorporate additionally in a later approach the information of certain landmark positions into the formulation of the energy functional, that makes use of the pure L1-norm. Similarly as before, the primal-dual algorithm of Chambolle and Pock is then used for both, the minimisation with respect to the displacement �eld and the motion segmentation. All the proposed non-rigid discontinuity preserving registration methods delivered promising results for experiments with synthetic images and real MR images of breathing induced liver motion

Evaporation of the pancake-vortex lattice in weakly-coupled layered superconductors

We calculate the melting line of the pancake-vortex system in a layered superconductor, interpolating between two-dimensional (2D) melting at high fields and the zero-field limit of single-stack evaporation. Long-range interactions between pancake vortices in different layers permit a mean-field approach, the ``substrate model'', where each 2D crystal fluctuates in a substrate potential due to the vortices in other layers. We find the thermal stability limit of the 3D solid, and compare the free energy to a 2D liquid to determine the first-order melting transition and its jump in entropy.Comment: 4 pages, RevTeX, two postscript figures incorporated using eps

Fair Division of a Graph

We consider fair allocation of indivisible items under an additional constraint: there is an undirected graph describing the relationship between the items, and each agent's share must form a connected subgraph of this graph. This framework captures, e.g., fair allocation of land plots, where the graph describes the accessibility relation among the plots. We focus on agents that have additive utilities for the items, and consider several common fair division solution concepts, such as proportionality, envy-freeness and maximin share guarantee. While finding good allocations according to these solution concepts is computationally hard in general, we design efficient algorithms for special cases where the underlying graph has simple structure, and/or the number of agents -or, less restrictively, the number of agent types- is small. In particular, despite non-existence results in the general case, we prove that for acyclic graphs a maximin share allocation always exists and can be found efficiently.Comment: 9 pages, long version of accepted IJCAI-17 pape