Set Of Linear Orders Research Articles

On Finding Two Posets that Cover Given Linear Orders

The Poset Cover Problem is an optimization problem where the goal is to determine a minimum set of posets that covers a given set of linear orders. This problem is relevant in the field of data mining, specifically in determining directed networks or models that explain the ordering of objects in a large sequential dataset. It is already known that the decision version of the problem is NP-Hard while its variation where the goal is to determine only a single poset that covers the input is in P. In this study, we investigate the variation, which we call the 2-Poset Cover Problem, where the goal is to determine two posets, if they exist, that cover the given linear orders. We derive properties on posets, which leads to an exact solution for the 2-Poset Cover Problem. Although the algorithm runs in exponential-time, it is still significantly faster than a brute-force solution. Moreover, we show that when the posets being considered are tree-posets, the running-time of the algorithm becomes polynomial, which proves that the more restricted variation, which we called the 2-Tree-Poset Cover Problem, is also in P.

Condorcet domains satisfying Arrow’s single-peakedness

Condorcet domains are sets of linear orders with the property that, whenever the preferences of all voters belong to this set, the majority relation of any profile with an odd number of voters is transitive. Maximal Condorcet domains historically have attracted a special attention. We study maximal Condorcet domains that satisfy Arrow’s single-peakedness which is more general than Black’s single-peakedness. We show that all maximal Black’s single-peaked domains on the set of m alternatives are isomorphic but we found a rich variety of maximal Arrow’s single-peaked domains. We discover their recursive structure, prove that all of them have cardinality 2m−1, and characterise them by two conditions: connectedness and minimal richness. We also classify Arrow’s single-peaked Condorcet domains for m≤5 alternatives.

Condorcet domains, median graphs and the single-crossing property

Condorcet domains are sets of linear orders with the property that, whenever the preferences of all voters of a society belong to this set, their majority relation has no cycles. We observe that, without loss of generality, every such domain can be assumed to be closed in the sense that it contains the majority relation of every profile with an odd number of voters whose preferences belong to this domain. We show that every closed Condorcet domain can be endowed with the structure of a median graph and that, conversely, every median graph is associated with a closed Condorcet domain (in general, not uniquely). Condorcet domains that have linear graphs (chains) associated with them are exactly the preference domains with the classical single-crossing property. As a corollary, we obtain that a domain with the so-called ‘representative voter property’ is either a single-crossing domain or a very special domain containing exactly four different preference orders whose associated median graph is a 4-cycle. Maximality of a Condorcet domain imposes additional restrictions on the associated median graph. We prove that among all trees only (some) chains can be associated graphs of maximal Condorcet domains, and we characterize those single-crossing domains which are maximal Condorcet domains. Finally, using the characterization of Nehring and Puppe (J Econ Theory 135:269–305, 2007) of monotone Arrovian aggregation, we show that any closed Condorcet domain admits a rich class of strategy-proof social choice functions.

Subcyclic sets of linear orders

Subcyclic sets of linear orders

A theory of coalition formation in committees

Let G = ( N, W) be a strong weighted majority game and let A be a set of alternatives. Denote by L the set of linear orders on A. A social choice function F: L N → A is a representation of G if the simple game G ∗( F) associated with F equals G. A coalition S is determining in G if it satisfies the following condition. Let F be a representation of G and let R N ϵ L N . Then, if a simple majority of the members of S consider an alternative x to be their best choice, then S can ‘enforce’ x to be a Nash equilibrium payoff in the resulting non-cooperative voting game g( F, R N ). In this paper we generalize the above notion of a determining coalition to committees (i.e., proper and monotonic simple games), and give a complete characterization of the set of determining coalitions of a committee. Furthermore, we discuss our notion of a determining coalition in the light of some real-life data on formation of coalitions in town councils in Israel.