A continuous rating method for preferential voting. The complete case

A method is given for quantitatively rating the social acceptance of different options which are the matter of a complete preferential vote. Completeness means that every voter expresses a comparison (a preference or a tie) about each pair of options. The proposed method is proved to have certain desirable properties, which include: the continuity of the rates with respect to the data, a decomposition property that characterizes certain situations opposite to a tie, the Condorcet-Smith principle, and a property of clone consistency. One can view this rating method as a complement for the ranking method introduced in 1997 by Markus Schulze. It is also related to certain methods of one-dimensional scaling or cluster analysis.Comment: This is part one of a revised version of arxiv:0810.2263. Version 3 is the result of certain modifications, both in the statement of the problem and in the concluding remarks, that enhance the results of the paper; the results themselves remain unchange

Convexity in partial cubes: the hull number

We prove that the combinatorial optimization problem of determining the hull number of a partial cube is NP-complete. This makes partial cubes the minimal graph class for which NP-completeness of this problem is known and improves some earlier results in the literature. On the other hand we provide a polynomial-time algorithm to determine the hull number of planar partial cube quadrangulations. Instances of the hull number problem for partial cubes described include poset dimension and hitting sets for interiors of curves in the plane. To obtain the above results, we investigate convexity in partial cubes and characterize these graphs in terms of their lattice of convex subgraphs, improving a theorem of Handa. Furthermore we provide a topological representation theorem for planar partial cubes, generalizing a result of Fukuda and Handa about rank three oriented matroids.Comment: 19 pages, 4 figure

CD-independent subsets in meet-distributive lattices

A subset $X$ of a finite lattice $L$ is CD-independent if the meet of any two incomparable elements of $X$ equals 0. In 2009, Cz\'edli, Hartmann and Schmidt proved that any two maximal CD-independent subsets of a finite distributive lattice have the same number of elements. In this paper, we prove that if $L$ is a finite meet-distributive lattice, then the size of every CD-independent subset of $L$ is at most the number of atoms of $L$ plus the length of $L$. If, in addition, there is no three-element antichain of meet-irreducible elements, then we give a recursive description of maximal CD-independent subsets. Finally, to give an application of CD-independent subsets, we give a new approach to count islands on a rectangular board.Comment: 14 pages, 4 figure

Open questions in utility theory

Throughout this paper, our main idea is to explore different classical questions arising in Utility Theory, with a particular attention to those that lean on numerical representations of preference orderings. We intend to present a survey of open questions in that discipline, also showing the state-of-art of the corresponding literature.This work is partially supported by the research projects ECO2015-65031-R, MTM2015-63608-P (MINECO/ AEI-FEDER, UE), and TIN2016-77356-P (MINECO/ AEI-FEDER, UE)