We examine the problem of counting interval graphs. We answer the question posed by Hanlon, of whether the formal power series generating function of the number of interval graphs on n vertices has a positive radius of convergence. We have found that it is zero. We have obtained a lower bound and an upper bound on the number of interval graphs on n vertices. We also study the application of interval graphs to the dynamic storage allocation problem. Dynamic storage allocation has been shown to be NP-complete by Stockmeyer. Coloring interval graphs on-line has applications to dynamic storage allocation. The most colors used by Kierstead's algorithm is 3 ω -2, where ω is the size of the largest clique in the graph. We determine a lower bound on the colors used. One such lower bound is 2 ω -1.

A (convex) polytope P is said to be 2-level if every hyperplane H that is facet-defining for P has a parallel hyperplane H' that contains all the vertices of P which are not contained in H.Two level polytopes appear in different areas of mathematics, in particular in contexts related to discrete geometry and optimization. We study the problem of enumerating all combinatorial types of 2-level polytopes of a fixed dimension d. We describe the first algorithm to achieve this. We ran it to produce the complete database for d <= 8. Our results show that the number of combinatorial types of 2-level d-polytopes is surprisingly small for low dimensions d.We provide an upper bound for the number of combinatorially inequivalent 2-level d-polytopes. We phrase this counting problem in terms of counting some objects called 2-level configurations, that capture the class of "maximal" rank d 0/1-matrices, including (maximal) slack matrices of 2-level cones and 2-level polytopes. We provide a proof that the number of d-dimensional 2-level configurations coming from cones and polytopes, up to linear equivalence, is at most 2^{O(d^2 log d)}.Finally, we prove that the extension complexity of every stable set polytope of a bipartite graph with n nodes is O(n^2 log n) and that there exists an infinite class of bipartite graphs such that, for every n-node graph in this class, its stable set polytope has extension complexity equal to Omega(n log n). / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

Unter dem Evolutionsprozeß eines Objekts, das aus einer gegebenen Klasse zufällig ausgewählt wird, versteht man das folgende Gedankenexperiment. Zu einem geeigneten Parameter der Objekte der Klasse betrachtet man die Teilklasse derjenigen Objekte, bei denen dieser Parameter einen bestimmten Wert x annimmt. Dadurch stellen sich die folgenden Fragen: Wie sieht ein typisches Objekt dieser Teilklasse aus? Wieviele Objekte gibt es in der Teilklasse? Und: Wie verändern sich die Antworten auf die ersten beiden Fragen, wenn sich x verändert? Die vorliegende Dissertation behandelt Phasenübergänge im Evolutionsprozeß teilweiser Ordnungen und bestimmt die Anzahl teilweiser Ordnungen mit einer gegebenen Anzahl vergleichbarer Paare. Wir bezeichnen durch Pn,d die Klasse aller teilweisen Ordnungen mit n Punkten und dn2 vergleichbaren Paaren. 1978 bestimmte Dhar |Pn,d| im Intervall 1/8 < d < 3/16 und zeigte, daß hier eine typische Ordnung aus drei "Ebenen" besteht. 1979 bestimmten Kleitman und Rothschild |Pn,d| im Intervall 0 < d < 1/8 und zeigten, daß hier eine typische Ordnung aus zwei Ebenen besteht, also bipartit ist. Das Hauptergebnis der Dissertation ist es, ein vollständiges Bild des Evolutionsprozesses zu geben. Wir bestimmen |Pn,d| im gesamten Intervall 0 < d < 1/2 und zeigen, daß es unendlich viele Phasenübergänge gibt. Abschließend beschreiben wir, wie sich die Struktur einer typischen Ordnung während dieser Phasen verändert. / The evolution process of a random structure from a certain class denotes the following "experiment". Choose a parameter of the objects in the class under consideration and consider only the subclass of those objects where the parameter is equal to a fixed value x. Then the following questions arise quite naturally: What does a typical object from this subclass look like? How many objects are there in this subclass? And how do the answers to the first two questions change when x changes? This thesis investigates the phase transitions in the evolution of partially ordered sets and determines the number of partially ordered sets with a given number of comparable pairs. Denote by Pn,d the class of all n-point posets with dn2 comparable pairs. In 1978, Dhar determined |Pn,d| in the range 1/8 < d < 3/16 and showed that here a typical poset consists of three layers. In 1979, Kleitman and Rothschild determined |Pn,d| in the range 0 < d < 1/8 and showed that here a typical poset consists of two layers, i.e. it is bipartite. The main result of this thesis is to complete the picture by describing the whole evolution process of Pn,d in the range 0 < d < 1/2. We determine |Pn,d| for any d and show that there exist an infinite number of phase transitions. Finally we describe how the structure of a typical partially ordered set changes during these phases.

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This dissertation explores an analogue of the prime number theorem for polynomials over finite fields as well as its connection to the necklace factorization algorithm T-transform and the string complexity measure T-complexity. Specifically, a precise asymptotic expansion for the prime polynomial counting function is derived. The approximation given is more accurate than previous results in the literature while requiring very little computational effort. In this context asymptotic series expansions for Lerch transcendent, Eulerian polynomials, truncated polylogarithm, and polylogarithms of negative integer order are also provided. The expansion formulas developed are general and have applications in numerous areas other than the enumeration of prime polynomials. A bijection between the equivalence classes of aperiodic necklaces and monic prime polynomials is utilized to derive an asymptotic bound on the maximal T-complexity value of a string. Furthermore, the statistical behaviour of uniform random sequences that are factored via the T-transform are investigated, and an accurate probabilistic model for short necklace factors is presented. Finally, a T-complexity based conditional string complexity measure is proposed and used to define the normalized T-complexity distance that measures similarity between strings. The T-complexity distance is proven to not be a metric. However, the measure can be computed in linear time and space making it a suitable choice for large data sets. / Graduate / 0544 0984 0405 / nrebenich@gmail.com