The classical Stable Roommates problem asks whether it is possible to have a matching of an even number of agents such that no two agents which are not matched to each other would prefer to be with each other rather than with their assigned partners. We investigate Stable Roommates with complete (i.e., every agent can be matched with any other agent) or incomplete preferences, with ties (i.e., two agents are considered of equal value to some agent) or without ties. It is known that in general allowing ties makes the problem NP-complete. We provide algorithms for Stable Roommates that are, compared to those in the literature, more efficient when the input preferences are complete and have some structural property, such as being narcissistic, single-peaked, and single-crossing. However, when the preferences are incomplete and have ties, we show that being single-peaked and single-crossing does not reduce the computational complexity-Stable Roommates remains NP-complete. * A preliminary verion of this work appeared in the Proceedings of the 5th International Conference on Algorithmic Decision Theory (ADT '17) [8], volume 10576 of LNCS, pages 315-330, Springer, 2017. This full version contains proof details for Proposition 2.1, Observation 2.2, and Proposition 4.1. Furthermore, the reduction used for our main result (Theorem 4.4) was replaced by a completely new reduction showing NP-hardness for the case of narcissistic, single-peaked, and single-crossing preferences (the preferences in the previous reduction were not narcissistic).

We investigate preference profiles for a set V of voters, where each voter i has a preference order ≻ i on a finite set A of alternatives (that is, a linear order on A) such that for each two alternatives a, b ∈ A, voter i prefers a to b if a ≻ i b. Such a profile is narcissistic if each alternative a is preferred the most by at least one voter. It is single-peaked if there is a linear order ⊲ sp on the alternatives such that each voter's preferences on the alternatives along the order ⊲ sp are either strictly increasing, or strictly decreasing, or first strictly increasing and then strictly decreasing. It is single-crossing if there is a linear order ⊲ sc on the voters such that each pair of alternatives divides the order ⊲ sc into at most two suborders, where in each suborder, all voters have the same linear order on this pair. We show that for n voters and n alternatives, the number of single-peaked narcissistic profiles is n−1 i=2 n−1 i−1 while the number of single-crossing narcissistic profiles is 2 ( n−1 2 ) .When forming coalitions [14,8], building teams [5,11], or playing games, the individuals, who we jointly denote as voters, may have preferences over who is better than another as a potential coalition partner, a team member, or a player. In such situations, the voters and alternatives are identical, that is, A = V. Deriving from a simple psychological model, it seems natural to assume that each voter is narcissistic [5], meaning that she is her own ideal and, thus, most preferred alternative, that is, for each voter i ∈ V and each alternative b ∈ V \ {i}, it holds that i ≻ i b.Another well-studied property of voters preference orders on the set A of alternatives, the singlepeaked property, is characterized by a linear order ⊲ sp of the alternatives, where for each voter i, her preferences along the order ⊲ sp strictly increase until they reach the peak which is her most preferred alternative, and then strictly decrease, that is, for each alternative b ∈ A, the set {b} ∪ {a ∈ A | a ≻ i b} forms an interval in ⊲ sp . Black [6] introduced the concept of single-peakedness. He observed that voters' political interests over the parties are single-peaked, meaning that there is a left-to-right political spectrum of the parties such that each voter has a political ideal on this spectrum and the further away a party is from her ideal, the less she will like this party. Single-peaked preferences are also studied in psychology under the name unimodal orders [13,15].A third property, the single-crossing property, requires that there is a linear order of the voters such that the preference orders of the voters on each pair of alternatives along this order change at most once, that is, there is a linear order ⊲ sc of the voters where for each two distinct alternatives a, b ∈ A and for each three distinct voters i, j, k ∈ V with i ⊲ sc j ⊲ sc k, if a ≻ i b and a ≻ k b, then a ≻ j b. Mirrlees [23] and Roberts [24] introduced this concept in the field of economics. They observed that voters' preferences on income taxati...

Extending the ARAP energy with a term that depends on the face normal, energy minimization becomes an effective stylization tool for shapes represented as meshes. Our approach generalizes the possibilities of Cubic Stylization: the set of preferred normals can be chosen arbitrarily from the Gauss sphere, including semi‐discrete sets to model preference for cylinder‐ or cone‐like shapes. The optimization is designed to retain, similar to ARAP, the constant linear system in the global optimization. This leads to convergence behavior that enables interactive control over the parameters of the optimization. We provide various examples demonstrating the simplicity and versatility of the approach.

The classical Stable Roommates problem is to decide whether there exists a matching of an even number of agents such that no two agents which are not matched to each other would prefer to be with each other rather than with their respectively assigned partners. We investigate Stable Roommates with complete (i.e., every agent can be matched with any other agent) or incomplete preferences, with ties (i.e., two agents are considered of equal value to some agent) or without ties. It is known that in general allowing ties makes the problem NP-complete. We provide algorithms for Stable Roommates that are, compared to those in the literature, more efficient when the input preferences are complete and have some structural property, such as being narcissistic, single-peaked, and single-crossing. However, when the preferences are incomplete and have ties, we show that being single-peaked and single-crossing does not reduce the computational complexity—Stable Roommates remains NP-complete.

As‐rigid‐as‐possible (ARAP) surface modelling is widely used for interactive deformation of triangle meshes. We show that ARAP can be interpreted as minimizing a discretization of an elastic energy based on non‐conforming elements defined over dual orthogonal cells of the mesh. Using the intrinsic Voronoi cells rather than an orthogonal dual of the extrinsic mesh guarantees that the energy is non‐negative over each cell. We represent the intrinsic Delaunay edges extrinsically as polylines over the mesh, encoded in barycentric coordinates relative to the mesh vertices. This modification of the original ARAP energy, which we term iARAP, remedies problems stemming from non‐Delaunay edges in the original approach. Unlike the spokes‐and‐rims version of the ARAP approach it is less susceptible to the triangulation of the surface. We provide examples of deformations generated with iARAP and contrast them with other versions of ARAP. We also discuss the properties of the Laplace‐Beltrami operator implicitly introduced with the new discretization.