Some inequalities for orderings of acyclic digraphs

Let $D=(V,A)$ be an acyclic digraph. For $x\in V$ define $e_{_{D}}(x)$ to be the difference of the indegree and the outdegree of $x$. An acyclic ordering of the vertices of $D$ is a one-to-one map $g: V \rightarrow [1,|V|]$ that has the property that for all $x,y\in V$ if $(x,y)\in A$, then g(x) < g(y).We prove that for every acyclic ordering $g$ of $D$ the following inequality holds:$$\sum_{x\in V} e_{_{D}}(x)\cdot g(x) ~\geq~ \frac{1}{2} \sum_{x\in V}[e_{_{D}}(x)]^2~.$$The class of acyclic digraphs for which equality holds is determined as the class of comparability digraphs of posets of order dimension two

A study of discrepancy results in partially ordered sets

In 2001, Fishburn, Tanenbaum, and Trenk published a pair of papers that introduced the notions of linear and weak discrepancy of a partially ordered set or poset. Linear discrepancy for a poset is the least k such that for any ordering of the points in the poset there is a pair of incomparable points at least distance k away in the ordering. Weak discrepancy is similar to linear discrepancy except that the distance is observed over weak labelings (i.e. two points can have the same label if they are incomparable, but order is still preserved). My thesis gives a variety of results pertaining to these properties and other forms of discrepancy in posets. The first chapter of my thesis partially answers a question of Fishburn, Tanenbaum, and Trenk that was to characterize those posets with linear discrepancy two. It makes the characterization for those posets with width two and references the paper where the full characterization is given. The second chapter introduces the notion of t-discrepancy which is similar to weak discrepancy except only the weak labelings with at most t copies of any label are considered. This chapter shows that determining a poset's t-discrepancy is NP-Complete. It also gives the t-discrepancy for the disjoint sum of chains and provides a polynomial time algorithm for determining t-discrepancy of semiorders. The third chapter presents another notion of discrepancy namely total discrepancy which minimizes the average distance between incomparable elements. This chapter proves that finding this value can be done in polynomial time unlike linear discrepancy and t-discrepancy. The final chapter answers another question of Fishburn, Tanenbaum, and Trenk that asked to characterize those posets that have equal linear and weak discrepancies. Though determining the answer of whether the weak discrepancy and linear discrepancy of a poset are equal is an NP-Complete problem, the set of minimal posets that have this property are given. At the end of the thesis I discuss two other open problems not mentioned in the previous chapters that relate to linear discrepancy. The first asks if there is a link between a poset's dimension and its linear discrepancy. The second refers to approximating linear discrepancy and possible ways to do it.Ph.D.Committee Chair: Trotter, William T.; Committee Member: Dieci, Luca; Committee Member: Duke, Richard; Committee Member: Randall, Dana; Committee Member: Tetali, Prasa

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

Partitions of combinatorial structures.

In this thesis we explore extremal, structural, and algorithmic problems involving the partitioning of combinatorial structures. We begin by considering problems from the theory of graph cuts. It is well known that every graph has a cut containing at least half its edges. We conjecture that (except for one example), given any two graphs on the same vertex set, we can partition the vertices so that at least half the edges of each graph go across the partition. We give a simple algorithm that comes close to proving this conjecture. We also prove, using probabilistic methods, that the conjecture holds for certain classes of graphs. We consider an analogue of the graph cut problem for posets and determine which graph cut results carry over to posets. We consider both extremal and algorithmic questions, and in particular, we show that the analogous maxcut problem for posets is polynomial-time solvable in contrast to the maxcut problem for graphs, which is NP-complete. Another partitioning problem we consider is that of obtaining a regular partition (in the sense of the Szemeredi Regularity Lemma) for posets, where the partition respects the order of the poset. We prove the existence of such order-preserving, regular partitions for both the comparability graph and the covering graph of a poset, and go on to derive further properties of such partitions. We give a new proof of an old result of Frankl and Furedi, which characterises all 3-uniform hypergraphs for which every set of 4 vertices spans exactly 0 or 2 edges. We use our new proof to derive a corresponding stability result. We also look at questions concerning an analogue of the graph linear extension problem for posets

Semantic networks

AbstractA semantic network is a graph of the structure of meaning. This article introduces semantic network systems and their importance in Artificial Intelligence, followed by I. the early background; II. a summary of the basic ideas and issues including link types, frame systems, case relations, link valence, abstraction, inheritance hierarchies and logic extensions; and III. a survey of ‘world-structuring’ systems including ontologies, causal link models, continuous models, relevance, formal dictionaries, semantic primitives and intersecting inference hierarchies. Speed and practical implementation are briefly discussed. The conclusion argues for a synthesis of relational graph theory, graph-grammar theory and order theory based on semantic primitives and multiple intersecting inference hierarchies

The Total Weak Discrepancy of a Partially Ordered Set

We define the total weak discrepancy of a poset P as the minimum nonnegative integer k for which there exists a function f : V → Z satisfying (i) if a \prec b then f(a) + 1 ≤ f(b) and (ii) Σ|f(a) − f(b)| ≤ k, where the sum is taken over all unordered pairs {a, b} of incomparable elements. If we allow k and f to take real values, we call the minimum k the fractional total weak discrepancy of P. These concepts are related to the notions of weak and fractional weak discrepancy, where (ii) must hold not for the sum but for each individual pair of incomparable elements of P. We prove that, unlike the latter, the total weak and fractional total weak discrepancy of P are always the same, and we give a polynomial-time algorithm to find their common value. We use linear programming duality and complementary slackness to obtain this result

Geoinformatic methodologies and quantitative tools for detecting hotspots and for multicriteria ranking and prioritization: application on biodiversity monitoring and conservation

Chi ha la responsabilità di gestire un’area protetta non solo deve essere consapevole dei problemi ambientali dell’area ma dovrebbe anche avere a disposizione dati aggiornati e appropriati strumenti metodologici per esaminare accuratamente ogni singolo problema. In effetti, il decisore ambientale deve organizzare in anticipo le fasi necessarie a fronteggiare le prevedibili variazioni che subirà la pressione antropica sulle aree protette. L’obiettivo principale della Tesi è di natura metodologica e riguarda il confronto tra differenti metodi statistici multivariati utili per l’individuazione di punti critici nello spazio e per l’ordinamento degli “oggetti ambientali” di studio e quindi per l’individuazione delle priorità di intervento ambientale. L’obiettivo ambientale generale è la conservazione del patrimonio di biodiversità. L’individuazione, tramite strumenti statistici multivariati, degli habitat aventi priorità ecologica è solamente il primo fondamentale passo per raggiungere tale obiettivo. L’informazione ecologica, integrata nel contesto antropico, è un successivo essenziale passo per effettuare valutazioni ambientali e per pianificare correttamente le azioni volte alla conservazione. Un’ampia serie di dati ed informazioni è stata necessaria per raggiungere questi obiettivi di gestione ambientale. I dati ecologici sono forniti dal Ministero dell’Ambiente Italiano e provengono al Progetto “Carta della Natura” del Paese. I dati demografici sono invece forniti dall’Istituto Italiano di Statistica (ISTAT). I dati si riferiscono a due aree geografiche italiane: la Val Baganza (Parma) e l’Oltrepò Pavese e Appennino Ligure-Emiliano. L’analisi è stata condotta a due differenti livelli spaziali: ecologico-naturalistico (l’habitat) e amministrativo (il Comune). Corrispondentemente, i risultati più significativi ottenuti sono: 1. Livello habitat: il confronto tra due metodi di ordinamento e determinazione delle priorità, il metodo del Vettore Ideale e quello della Preminenza, tramite l’utilizzo di importanti metriche ecologiche come il Valore Ecologico (E.V.) e la Sensibilità Ecologica (E.S.), fornisce dei risultati non direttamente comparabili. Il Vettore Ideale, non essendo un procedimento basato sulla ranghizzazione dei valori originali, sembra essere preferibile nel caso di paesaggi molto eterogenei in senso spaziale. Invece, il metodo della Preminenza probabilmente è da preferire in paesaggi ecologici aventi un basso grado di eterogeneità intesa nel senso di differenze non troppo grandi nel E.V. ed E.S. degli habitat. 2. Livello comunale: Al fine di prendere delle decisioni gestionali ed essendo gli habitat solo delle suddivisioni naturalistiche di un dato territorio, è necessario spostare l’attenzione sulle corrispondenti unità amministrative territoriali (i Comuni). Da questo punto di vista, l’introduzione della demografia risulta essere un elemento centrale oltre che di novità nelle analisi ecologico-ambientali. In effetti, l’analisi demografica rende il risultato di cui al punto 1 molto più realistico introducendo altre dimensioni (la pressione antropica attuale e le sue tendenze) che permettono l’individuazione di aree ecologicamente fragili. Inoltre, tale approccio individua chiaramente le responsabilità ambientali di ogni singolo ente territoriale nei riguardi della difesa della biodiversità. In effetti un ordinamento dei Comuni sulla base delle caratteristiche ambientali e demografiche, chiarisce le responsabilità gestionali di ognuno di essi. Un’applicazione concreta di questa necessaria quanto utile integrazione di dati ecologici e demografici viene discussa progettando una Rete Ecologica (E.N.). La Rete cosi ottenuta infatti presenta come elemento di novità il fatto di non essere “statica” bensì “dinamica” nel senso che la sua pianificazione tiene in considerazione il trend di pressione antropica al fine di individuare i probabili punti di futura fragilità e quindi di più critica gestione.Who has the responsibility to manage a conservation zone, not only must be aware of environmental problems but should have at his disposal updated databases and appropriate methodological instruments to examine carefully each individual case. In effect he has to arrange, in advance, the necessary steps to withstand the foreseeable variations in the trends of human pressure on conservation zones. The essential objective of this Thesis is methodological that is to compare different multivariate statistical methods useful for environmental hotspot detection and for environmental prioritization and ranking. The general environmental goal is the conservation of the biodiversity patrimony. The individuation, through multidimensional statistical tools, of habitats having top ecological priority, is only the first basic step to accomplish this aim. Ecological information integrated in the human context is an essential further step to make environmental evaluations and to plan correct conservation actions. A wide series of data and information has been necessary to accomplish environmental management tasks. Ecological data are provided by the Italian Ministry of the Environment and they refer to the Map of Italian Nature Project database. The demographic data derives from the Italian Institute of Statistics (ISTAT). The data utilized regards two Italian areas: Baganza Valley and Oltrepò Pavese and Ligurian-Emilian Apennine. The analysis has been carried out at two different spatial/scale levels: ecological-naturalistic (habitat level) and administrative (Commune level). Correspondingly, the main obtained results are: 1. Habitat level: comparing two ranking and prioritization methods, Ideal Vector and Salience, through important ecological metrics like Ecological Value (E.V.) and Ecological Sensitivity (E.S.), gives results not directly comparable. Being not based on a ranking process, Ideal Vector method seems to be used preferentially in landscapes characterized by high spatial heterogeneity. On the contrary, Salience method is probably to be preferred in ecological landscapes characterized by a low degree of heterogeneity in terms of not large differences concerning habitat E.V. and E.S.. 2. Commune level: Being habitat only a naturalistic partition of a given territory, it is necessary, for management decisions, to move towards the corresponding administrative units (Communes). From this point of view, the introduction of demography is an essential element of novelty in environmental analysis. In effect, demographic analysis makes the goal at point 1 more realistic introducing other dimensions (actual human pressure and its trend) which allows the individuation of environmentally fragile areas. Furthermore this approach individuates clearly the environmental responsibility of each administrative body for what concerns the biodiversity conservation. In effect communes’ ranking, according to environmental/demographic features, clarify the responsibilities of each administrative body. A concrete application of this necessary and useful integration of ecological and demographic data has been developed in designing an Ecological Network (E.N.).The obtained E.N. has the novelty to be not “static” but “dynamic” that is the network planning take into account the demographic pressure trends in the individuation of the probable future fragile points