scholarly journals Contiguous Cake Cutting: Hardness Results and Approximation Algorithms

2020 ◽  
Vol 69 ◽  
pp. 109-141
Author(s):  
Paul Goldberg ◽  
Alexandros Hollender ◽  
Warut Suksompong
Keyword(s):  
Approximation Algorithms ◽  
Efficient Algorithms ◽  
Decision Problems ◽  
Fair Allocation ◽  
Prior Work ◽  
Cake Cutting ◽  
Np Hardness

We study the fair allocation of a cake, which serves as a metaphor for a divisible resource, under the requirement that each agent should receive a contiguous piece of the cake. While it is known that no finite envy-free algorithm exists in this setting, we exhibit efficient algorithms that produce allocations with low envy among the agents. We then establish NP-hardness results for various decision problems on the existence of envy-free allocations, such as when we fix the ordering of the agents or constrain the positions of certain cuts. In addition, we consider a discretized setting where indivisible items lie on a line and show a number of hardness results extending and strengthening those from prior work. Finally, we investigate connections between approximate and exact envy-freeness, as well as between continuous and discrete cake cutting.

scholarly journals Contiguous Cake Cutting: Hardness Results and Approximation Algorithms

2020 ◽  
Vol 34 (02) ◽  
pp. 1990-1997
Author(s):  
Paul W. Goldberg ◽  
Alexandros Hollender ◽  
Warut Suksompong
Keyword(s):  
Approximation Algorithms ◽  
Efficient Algorithms ◽  
Decision Problems ◽  
Fair Allocation ◽  
Prior Work ◽  
Cake Cutting ◽  
Np Hardness

We study the fair allocation of a cake, which serves as a metaphor for a divisible resource, under the requirement that each agent should receive a contiguous piece of the cake. While it is known that no finite envy-free algorithm exists in this setting, we exhibit efficient algorithms that produce allocations with low envy among the agents. We then establish NP-hardness results for various decision problems on the existence of envy-free allocations, such as when we fix the ordering of the agents or constrain the positions of certain cuts. In addition, we consider a discretized setting where indivisible items lie on a line and show a number of hardness results strengthening those from prior work.

scholarly journals APPROXIMATING SMALLEST ENCLOSING BALLS WITH APPLICATIONS TO MACHINE LEARNING

2009 ◽  
Vol 19 (05) ◽  
pp. 389-414 ◽  
Author(s):  
FRANK NIELSEN ◽  
RICHARD NOCK
Keyword(s):  
Machine Learning ◽  
Hybrid Algorithms ◽  
Decision Problems ◽  
Dual Method ◽  
Euclidean Ball ◽  
Prior Work ◽  
Euclidean Spaces ◽  
Information Theoretic ◽  
Primal Dual ◽  
Covering Set

In this paper, we first survey prior work for computing exactly or approximately the smallest enclosing balls of point or ball sets in Euclidean spaces. We classify previous work into three categories: (1) purely combinatorial, (2) purely numerical, and (3) recent mixed hybrid algorithms based on coresets. We then describe two novel tailored algorithms for computing arbitrary close approximations of the smallest enclosing Euclidean ball. These deterministic heuristics are based on solving relaxed decision problems using a primal-dual method. The primal-dual method is interpreted geometrically as solving for a minimum covering set, or dually as seeking for a minimum piercing set. Finally, we present some applications in machine learning of the exact and approximate smallest enclosing ball procedure, and discuss about its extension to non-Euclidean information-theoretic spaces.

Automatic Composition of Encoding Scheme and Search Operators in System Architecture Optimization

2021 ◽  
Author(s):  
Gabriel Apaza ◽  
Daniel Selva
Keyword(s):  
System Architecture ◽  
Decision Problems ◽  
Search Performance ◽  
Prior Work ◽  
User Models ◽  
Encoding Scheme ◽  
Navigation And Control ◽  
Automatic Composition ◽  
And Control ◽  
Architecture Optimization

Abstract The purpose of this paper is to propose a new method for the automatic composition of both encoding schemes and search operators for system architecture optimization. The method leverages prior work that identified a set of six patterns that appear often in system architecture decision problems (down-selecting, combining, assigning, partitioning, permuting, and connecting). First, the user models the architecture space as a directed graph, where nodes are decisions belonging to one of the aforementioned patterns, and edges are dependencies between decisions that affect architecture enumeration (e.g., the outcome of decision 1 affects the number of alternatives available for decision 2). Then, based on a library of encoding scheme and operator fragments that are appropriate for each pattern, an algorithm automatically composes an encoding scheme and corresponding search operators by traversing the graph. The method is demonstrated in two case studies: a study integrating three architectural decisions for constructing a portfolio of earth observing satellite missions, and a study integrating eight architectural decisions for designing a guidance navigation and control system. Results suggest that this method has comparable search performance to hand-crafted formulations from experts. Furthermore, the proposed method drastically reducing the need for practitioners to write new code.

scholarly journals Computational Aspects of Conditional Minisum Approval Voting in Elections with Interdependent Issues

2020 ◽  
Author(s):  
Evangelos Markakis ◽  
Georgios Papasotiropoulos
Keyword(s):  
Approximation Algorithms ◽  
Structural Properties ◽  
Upper Bounds ◽  
Efficient Algorithms ◽  
Approval Voting ◽  
Voting Rule ◽  
Special Cases ◽  
Computational Aspects

Approval voting provides a simple, practical framework for multi-issue elections, and the most representative example among such election rules is the classic Minisum approval voting rule. We consider a generalization of Minisum, introduced by the work of Barrot and Lang [2016], referred to as Conditional Minisum, where voters are also allowed to express dependencies between issues. The price we have to pay when we move to this higher level of expressiveness is that we end up with a computationally hard rule. Motivated by this, we focus on the computational aspects of Conditional Minisum, where progress has been rather scarce so far. We identify restrictions to every voter's dependencies, under which we provide the first multiplicative approximation algorithms for the problem. The restrictions involve upper bounds on the number of dependencies an issue can have on the others. At the same time, by additionally requiring certain structural properties for the union of dependencies cast by the whole electorate, we obtain optimal efficient algorithms for well-motivated special cases. Overall, our work provides a better understanding on the complexity implications introduced by conditional voting.

scholarly journals Networked Fairness in Cake Cutting

2017 ◽  
Author(s):  
Xiaohui Bei ◽  
Youming Qiao ◽  
Shengyu Zhang
Keyword(s):  
Network Structure ◽  
Transitive Closure ◽  
Rooted Tree ◽  
Fair Division ◽  
Efficient Algorithms ◽  
Research Directions ◽  
Proportional Allocation ◽  
Cake Cutting ◽  
Underlying Network ◽  
New Research

We introduce a graphical framework for fair division in cake cutting, where comparisons between agents are limited by an underlying network structure. We generalize the classical fairness notions of envy-freeness and proportionality in this graphical setting. An allocation is called envy-free on a graph if no agent envies any of her neighbor's share, and is called proportional on a graph if every agent values her own share no less than the average among her neighbors, with respect to her own measure. These generalizations enable new research directions in developing simple and efficient algorithms that can produce fair allocations under specific graph structures. On the algorithmic frontier, we first propose a moving-knife algorithm that outputs an envy-free allocation on trees. The algorithm is significantly simpler than the discrete and bounded envy-free algorithm introduced in [Aziz and Mackenzie, 2016] for compete graphs. Next, we give a discrete and bounded algorithm for computing a proportional allocation on transitive closure of trees, a class of graphs by taking a rooted tree and connecting all its ancestor-descendant pairs.

Constraints in fair division

2021 ◽  
Vol 19 (2) ◽  
pp. 46-61
Author(s):  
Warut Suksompong
Keyword(s):  
Fundamental Problem ◽  
Fair Division ◽  
Fair Allocation ◽  
Allocation Of Resources ◽  
Open Questions ◽  
Cake Cutting

The fair allocation of resources to interested agents is a fundamental problem in society. While the majority of the fair division literature assumes that all allocations are feasible, in practice there are often constraints on the allocation that can be chosen. In this survey, we discuss fairness guarantees for both divisible (cake cutting) and indivisible resources under several common types of constraints, including connectivity, cardinality, matroid, geometric, separation, budget, and conflict constraints. We also outline a number of open questions and directions.

Deals or No Deals: Contract Design for Online Advertising

2021 ◽  
Author(s):  
Anthony Kim ◽  
Vahab Mirrokni ◽  
Hamid Nazerzadeh
Keyword(s):  
Approximation Algorithms ◽  
Real Time ◽  
Online Advertising ◽  
Contract Design ◽  
Constant Factor ◽  
Common Value ◽  
Distributional Information ◽  
Using Data ◽  
Np Hardness ◽  
Better Than

We present a formal study of first-look and preferred deals that are a recently introduced generation of contracts for selling online advertisements, which generalize traditional reservation contracts and are suitable for advertisers with advanced targeting capabilities. Under these deals, one or more advertisers gain priority access to an inventory of impressions before others and can choose to purchase in real time. In particular, we propose constant-factor approximation algorithms for maximizing the revenue that can be obtained from these deals when offered to all or a subset of the advertisers, whose valuation distributions can be independent or correlated through a common value component. We evaluate our algorithms using data from Google’s ad exchange platform and show they perform better than the approximation guarantees and obtain significantly higher revenue than auctions; in certain cases, the observed revenue is 85%–96% of the optimal revenue achievable. We also prove the NP-hardness of designing deals when advertisers’ valuations are arbitrarily correlated and the optimality of menus of deals among a certain class of selling mechanisms in an incomplete distributional information setting.

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