
Recognizing and Eliciting Weakly Single Crossing Profiles on Trees
The domain of single crossing preference profiles is a widely studied do...
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Computational Complexity of Testing Proportional Justified Representation
We consider a committee voting setting in which each voter approves of a...
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Approximation algorithm for the Multicovering Problem
Let H=(V,E) be a hypergraph with maximum edge size ℓ and maximum degree ...
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Recognizing SinglePeaked Preferences on an Arbitrary Graph: Complexity and Algorithms
This paper is devoted to a study of singlepeakedness on arbitrary graph...
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A ConstantFactor Approximation Algorithm for Vertex Guarding a WVPolygon
The problem of vertex guarding a simple polygon was first studied by Sub...
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Morphing tree drawings in a small 3D grid
We study crossingfree grid morphs for planar tree drawings using 3D. A ...
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New Characterizations of StrategyProofness under SinglePeakedness
We provide novel simple representations of strategyproof voting rules w...
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Proportional Representation under SingleCrossing Preferences Revisited
We study the complexity of determining a winning committee under the Chamberlin–Courant voting rule when voters' preferences are singlecrossing on a line, or, more generally, on a median graph (this class of graphs includes, e.g., trees and grids). For the line, Skowron et al. (2015) describe an O(n^2mk) algorithm (where n, m, k are the number of voters, the number of candidates and the committee size, respectively); we show that a simple tweak improves the time complexity to O(nmk). We then improve this bound for k=Ω(log n) by reducing our problem to the klink path problem for DAGs with concave Monge weights, obtaining a nm2^O(√(log kloglog n)) algorithm for the general case and a nearly linear algorithm for the Borda misrepresentation function. For trees, we point out an issue with the algorithm proposed by Clearwater, Puppe and Slinko (2015), and develop a O(nmk) algorithm for this case as well. For grids, we formulate a conjecture about the structure of optimal solutions, and describe a polynomialtime algorithm that finds a winning committee if this conjecture is true; we also explain how to convert this algorithm into a bicriterial approximation algorithm whose correctness does not depend on the conjecture.
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