17 research outputs found
On the structure of acyclic binary relations
We investigate the structure of acyclic binary relations from different points of view. On the one hand, given a nonempty set we study real-valued bivariate maps that satisfy suitable functional equations, in a way that their associated binary relation is acyclic. On the other hand, we consider acyclic directed graphs as well as their representation by means of incidence matrices. Acyclic binary relations can be extended to the asymmetric part of a linear order, so that, in particular, any directed acyclic graph has a topological sorting.This work has been partially supported by the research projects MTM2012-37894-C02-02, TIN2013-47605-P, ECO2015-65031-R, MTM2015-63608-P (MINECO/FEDER), TIN2016-77356-P and the Research Services of the Public University of Navarre (Spain)
Classification of preordered spaces in terms of monotones -- Filling in the gaps
Following the recent introduction of new classes of monotones, like injective
monotones or strict monotone multi-utilities, we present the classification of
preordered spaces in terms of both the existence and cardinality of real-valued
monotones and the cardinality of the quotient space. In particular, we take
advantage of a characterization of real-valued monotones in terms of separating
families of increasing sets in order to obtain a more complete classification
consisting of classes that are strictly different from each other
A topological study for the existence of lower-semicontinuous Richter-Peleg multi-utilities
In the present paper we study necessary and sufficient conditions for the
existence of a semicontinuous and finite Richter-Peleg multi-utility for a
preorder. It is well know that, given a preorder on a topological space, if
there is a lower (upper) semicontinuous Richter-Peleg multi-utility, then the
topology of the space must be finer than the Upper (resp. Lower) topology.
However, this condition does not guarantee the existence of a semicontinuous
representation.
We search for finer topologies which are necessary for semicontinuity, as
well as that they could guarantee the existence of a semicontinuous
representation. As a result, we prove that Scott topology (that refines the
Upper one) must be contained in the topology of the space in case there exists
a finite lower semicontinuous Richter-Peleg multi-utility. However, as it is
shown, the existence of this representation cannot be guaranteed
Absolute Expressiveness of Subgraph-Based Centrality Measures
In graph-based applications, a common task is to pinpoint the most important or "central" vertex in a (directed or undirected) graph, or rank the vertices of a graph according to their importance. To this end, a plethora of so-called centrality measures have been proposed in the literature. Such measures assess which vertices in a graph are the most important ones by analyzing the structure of the underlying graph. A family of centrality measures that are suited for graph databases has been recently proposed by relying on the following simple principle: the importance of a vertex in a graph is relative to the number of "relevant" connected subgraphs surrounding it; we refer to the members of this family as subgraph-based centrality measures. Although it has been shown that such measures enjoy several favourable properties, their absolute expressiveness remains largely unexplored. The goal of this work is to precisely characterize the absolute expressiveness of the family of subgraph-based centrality measures by considering both directed and undirected graphs. To this end, we characterize when an arbitrary centrality measure is a subgraph-based one, or a subgraph-based measure relative to the induced ranking. These characterizations provide us with technical tools that allow us to determine whether well-established centrality measures are subgraph-based. Such a classification, apart from being interesting in its own right, gives useful insights on the structural similarities and differences among existing centrality measures
Matching with Incomplete Preferences
I study a two-sided marriage market in which agents have incomplete
preferences -- i.e., they find some alternatives incomparable. The strong
(weak) core consists of matchings wherein no coalition wants to form a new
match between themselves, leaving some (all) agents better off without harming
anyone. The strong core may be empty, while the weak core can be too large. I
propose the concept of the "compromise core" -- a nonempty set that sits
between the weak and the strong cores. Similarly, I define the men-(women-)
optimal core and illustrate its benefit in an application to India's
engineering college admissions system
Labeling Methods for Partially Ordered Paths
The landscape of applications and subroutines relying on shortest path
computations continues to grow steadily. This growth is driven by the
undeniable success of shortest path algorithms in theory and practice. It also
introduces new challenges as the models and assessing the optimality of paths
become more complicated. Hence, multiple recent publications in the field adapt
existing labeling methods in an ad-hoc fashion to their specific problem
variant without considering the underlying general structure: they always deal
with multi-criteria scenarios and those criteria define different partial
orders on the paths. In this paper, we introduce the partial order shortest
path problem (POSP), a generalization of the multi-objective shortest path
problem (MOSP) and in turn also of the classical shortest path problem. POSP
captures the particular structure of many shortest path applications as special
cases. In this generality, we study optimality conditions or the lack of them,
depending on the objective functions' properties. Our final contribution is a
big lookup table summarizing our findings and providing the reader an easy way
to choose among the most recent multicriteria shortest path algorithms
depending on their problem's weight structure. Examples range from
time-dependent shortest path and bottleneck path problems to the fuzzy shortest
path problem and complex financial weight functions studied in the public
transportation community. Our results hold for general digraphs and therefore
surpass previous generalizations that were limited to acyclic graphs
Amalgamation in classes of involutive commutative residuated lattices
The amalgamation property and its variants are in strong relationship with
various syntactic interpolation properties of substructural logics, hence its
investigation in varieties of residuated lattices is of particular interest.
The amalgamation property is investigated in some classes of non-divisible,
non-integral, and non-idempotent involutive commutative residuated lattices in
this paper. It is proved that the classes of odd and even totally ordered,
involutive, commutative residuated lattices fail the amalgamation property. It
is also proved that their subclasses formed by their idempotent-symmetric
algebras have the amalgamation property but fail the strong amalgamation
property. Finally, it is shown that the variety of semilinear,
idempotent-symmetric, odd, involutive, commutative residuated lattices has the
amalgamation property, and hence also the transferable injections property
Policy-making and policy assessments with partially ordered alternatives
The present work collects three essays on social choice and decision-making in the presence of multiple objectives and severe informational limitations. When feasible alternatives must be ordered according to their performance under various criteria, it is typically necessary to make use of a specific functional relation and assume the implied rates of substitution between scores in different criteria. In the special case of collective choice and voting, rather than having proper rates of substitution, each individually preferred ordering of the alternatives is usually weighted according to its frequency in the population. Both decision frameworks imply the availability of extensive information about such functional relation and the proper weights of each criterion or must acknowledge a vast and arbitrary discretion to those in charge of resolving the decision process. The alternative approach herein discussed consists in applying the Pareto criterion to identify Pareto-superior alternatives in each pairwise comparison, a procedure that easily produces an incomplete ordering. Then, applying a tool of Order Theory, a complete ordering is identified from the linear extensions of the partially ordered set derived from the Pareto criterion. The claim is that this method highlights conflicts in value judgements and in incomparable criteria, allowing to search for a conflict-mitigating solution that doesn\u2019t make assumptions on the reciprocal importance of criteria or judgements. The method is actually a combination of existing but unrelated approaches in Social Choice Theory and in Order Theory and provides outcomes with interesting properties. The essays present, respectively, an axiomatic discussion of the properties of this approach and two applications to policy issues