A formula (in conjunctive normal form) is said to be minimal unsatisfiable if it is unsatisfiable and deleting any clause makes it satisfiable. The deficiency of a formula is the difference of the number of clauses and the number of variables. It is known that every minimal unsatisfiable formula has positive deficiency. Until recently, polynomial–time algorithms were known to recognize minimal unsatisfiable formulas with deficiency 1 and 2. We state an algorithm which recognizes minimal unsatisfiable formulas with any fixed deficiency in polynomial time.

Abstract. A digraph is upward planar if it has a planar drawing such that all the edges are monotone with respect to the vertical direction. Testing upward planarity and constructing upward planar drawings is important for displaying hierarchical network structures, which frequently arise in software engineering, project management, and visual languages. In this paper we investigate upward planarity testing of single-source digraphs; we provide a new combinatorial characterization of upward planarity and give an optimal algorithm for upward planarity testing. Our algorithm tests whether a single-source digraph with n vertices is upward planar in O(n) sequential time, and in O(log n) time on a CRCW PRAM with n log log n / log n processors, using O(n) space. The algorithm also constructs an upward planar drawing if the test is successful. The previously known best result is an O(n2)-time algorithm by Hutton and Lubiw [Proc. 2nd ACM–SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, 1991, pp. 203–211]. No efficient parallel algorithms for upward planarity testing were previously known.

ix Chapter 1 Introduction 1 1.1 Algorithmic Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Technical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.2 Parallel Computational Complexity . . . . . . . . . . . . . . . . . . . . . 7 1.2.3 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.4 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.2.5 Random Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.2.6 Group Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.3 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Chapter 2 Parallel Uniform Generation of Unlabelled Graphs 25 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.2 Sampling O...

Valuations are measure-like functions mapping the open sets of a topological space into positive real numbers. They can be classified according to some additional properties. Some topological spaces are defined whose elements are valuations from various classes. The relationships among these spaces are studied, and universal properties are shown for some of them. 1 Introduction For a topological space X , a valuation on X is a function which maps the open sets of X to real numbers in the range from zero to infinity (inclusively) with the following properties: (1) The empty set is mapped to zero: ; = 0 (strictness). (2) The values assigned to binary union and intersection are related by the following equation: (U [ V ) + (U " V ) = U + V for all opens U and V (modularity). (3) Bigger sets are mapped to bigger numbers: if U ` V , then U V (monotonicity). Most often, we consider Scott continuous valuations which enjoy the additional property ( S i2I V i ) = t i2I V i for every dir...

We present new graph-theoretical conditions for polyhedra of inscribable type and Delaunay triangulations. We establish several sufficient conditions of the following general form: if a polyhedron has a sufficiently rich collection of Hamiltonian subgraphs, then it is of inscribable type. These results have several consequences: ffl All 4-connected polyhedra are of inscribable type. ffl All simplicial polyhedra in which all vertex degrees are between 4 and 6, inclusive, are of inscribable type. ffl All triangulations without chords or nonfacial triangles are realizable as combinatorially equivalent Delaunay triangulations. We also strengthen some earlier results about matchings in polyhedra of inscribable type. Specifically, we show that any nonbipartite polyhedron of inscribable type has a perfect matching containing any specified edge, and that any bipartite polyhedron of inscribable type has a perfect matching containing any two specified disjoint edges. We give examples showing t...

Abstract. We outline a general theory of graph polynomials which covers all the examples we found in the vast literature, in particular, the chromatic polynomial, various generalizations of the Tutte polynomial, matching polynomials, interlace polynomials, and the cover polynomial of digraphs. We introduce two classes of (hyper)graph polynomials definable in second order logic, and outline a research program for their classification in terms of definability and complexity considerations, and various notions of reducibilities. 1

. Two lower bag domain constructions are introduced: the initial construction which gives free lower monoids, and the final construction which is defined explicitly in terms of second order functions. The latter is analyzed closely. For sober dcpo's, the elements of the final lower bag domains can be described concretely as bags. For continuous domains, initial and final lower bag domains coincide. They are continuous again and can be described via a basis which is constructed from a basis of the argument domain. The lower bag domain construction preserves algebraicity and the properties I and M, but does not preserve bounded completeness, property L, or bifiniteness. 1 Introduction Power domain constructions [13, 15, 16] were introduced to describe the denotational semantics of non-deterministic programming languages. A power domain construction is a domain constructor P , which maps domains to domains, together with some families of continuous operations. If X is the semantic domain ...

Abstract. In the network querying problem, one is given a protein complex or pathway of species A and a protein–protein interaction network of species B; the goal is to identify subnetworks of B that are similar to the query. Existing approaches mostly depend on knowledge of the interaction topology of the query in the network of species A; however, in practice, this topology is often not known. To combat this problem, we develop a topology-free querying algorithm, which we call Torque. Given a query, represented as a set of proteins, Torque seeks a matching set of proteins that are sequence-similar to the query proteins and span a connected region of the network, while allowing both insertions and deletions. The algorithm uses alternatively dynamic programming and integer linear programming for the search task. We test Torque with queries from yeast, fly, and human, where we compare it to the QNet topology-based approach, and with queries from less studied species, where only topology-free algorithms apply. Torque detects many more matches than QNet, while in both cases giving results that are highly functionally coherent. 1

It has been known for a long time now that the problem of counting the number of perfect matchings in a planar graph is in NC. This result is based on the notion of a pfaffian orientation of a graph. (Recently, Galluccio and Loebl [7] gave a P-time algorithm for the case of graphs of small genus.) However, it is not known if the corresponding search problem, that of finding one perfect matching in a planar graph, is in NC. This situation is intriguing as it seems to contradict our intuition that search should be easier than counting. For the case of planar bipartite graphs, Miller and Naor [22] showed that a perfect matching can indeed be found using an NC algorithm. We present a very different NC-algorithm for this problem. Unlike the Miller...

We study various versions of the univariate and multivariate matching and rook polynomials. We show that there is most general multivariate matching polynomial, which is, up the some simple substitutions and multiplication with a prefactor, the original multivariate matching polynomial introduced by C. Heilmann and E. Lieb. We follow here a line of investigation which was very successfully pursued over the years by, among others, W. Tutte, B. Bollobas and O. Riordan, and A. Sokal in studying the chromatic and the Tutte polynomial. We show here that evaluating these polynomials over the reals is ♯P-hard for all points in R k but possibly for an exception set which is semi-algebraic and of dimension strictly less than k. This result is analoguous to the characterization due to F. Jaeger, D. Vertigan and D. Welsh (1990) of the points where the Tutte polynomial is hard to evaluate. Our proof, however, builds mainly on the work by M. Dyer and C. Greenhill (2000). 1