Charakterisierung Pfaff’scher Graphen mittels verbotener Teilgraphen

Pfaff’sche Graphen sind genau jene, auf die man Kasteleyns Methode zum Abzählen perfekter Matchings anwenden kann, womit dieses Problem in polynomieller Zeit lösbar ist. Diese Arbeit soll darlegen, für welche Klassen von Graphen eine einfache und schöne Charakterisierung Pfaff’scher Graphen existiert. Das Problem ergibt sich aus der unhandlichen und nur umständlich zu überprüfenden Definition. Dabei wird insbesondere die Charakterisierung mittels verbotener Teilgraphen im Mittelpunkt stehen. Die Idee ist, eine Liste von (möglichst wenigen) Graphen anzugeben, deren “nicht-enthalten-Sein” als sogenannter „Matching Minor” eine notwendige und hinreichende Bedingung dafür darstellt, dass es sich um einen Pfaff’schen Graphen handelt.Pfaffian graphs are exactly those, on which one can apply Kasteleyns method of counting perfect matchings, which implies the polynomial-time-solvability of our problem. This paper shall demonstrate for which classes of graphs there exists a facile characterisation of Pfaffian graphs. The difficulties here originate in the unhandy definition, whose requirements are hard to check. Thereby, the characterisation in terms of forbidden subgraphs will be the central issue of our studies. The idea is to state a list of (preferably few) graphs, so that for a given graph, containing one of the graphs in the list as a so-called matching minor is equivalent with beeing non-Pfaffian

Combinatorial Optimization

Combinatorial Optimization is a very active field that benefits from bringing together ideas from different areas, e.g., graph theory and combinatorics, matroids and submodularity, connectivity and network flows, approximation algorithms and mathematical programming, discrete and computational geometry, discrete and continuous problems, algebraic and geometric methods, and applications. We continued the long tradition of triannual Oberwolfach workshops, bringing together the best researchers from the above areas, discovering new connections, and establishing new and deepening existing international collaborations

Exactly soluble models in many-body physics

Almost all phenomena in the universe are described, at the fundamental level, by quantum manybody models. In general, however, a complete understanding of large systems with many degrees of freedom is impossible. While in general many-body quantum systems are intractable, there are special cases for which there are techniques that allow for an exact solution. Exactly soluble models are interesting because they are soluble; beyond this, they can be used to gain intuition for further reaching many-body systems, including when they can be leveraged to help with numerical approximations for general models. The work presented in this thesis considers exactly soluble models of quantum many-body systems. The first part of this thesis extends the family of many-body spin models for which we can find a freefermion solution. A solution method that was developed for a specific free-fermion model is generalized in such a way that allows application to a broader class of many-body spin system than was previously known to be free. Models which admit a solution via this method are characterized by a graph theory invariants: in brief it is shown that a quantum spin system has an exact description via non-interacting fermions if its frustration graph is claw-free and contains a simplicial clique. The second part of this thesis gives an explicit example of how the usefulness of exactly soluble models can extend beyond the solution itself. This chapter pertains to the calculation of the topological entanglement entropy in topologically ordered loop-gas states. Topological entanglement entropy gives an understanding of how correlations may extend throughout a system. In this chapter the topological entanglement entropy of two- and three-dimensional loop-gas states is calculated in the bulk and at the boundary. We obtain a closed form expression for the topological entanglement in terms of the anyonic theory that the models support

Matchings, matroids and submodular functions

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2008.Includes bibliographical references (p. 111-118).This thesis focuses on three fundamental problems in combinatorial optimization: non-bipartite matching, matroid intersection, and submodular function minimization. We develop simple, efficient, randomized algorithms for the first two problems, and prove new lower bounds for the last two problems. For the matching problem, we give an algorithm for constructing perfect or maximum cardinality matchings in non-bipartite graphs. Our algorithm requires O(n") time in graphs with n vertices, where w < 2.38 is the matrix multiplication exponent. This algorithm achieves the best-known running time for dense graphs, and it resolves an open question of Mucha and Sankowski (2004). For the matroid intersection problem, we give an algorithm for constructing a common base or maximum cardinality independent set for two so-called "linear" matroids. Our algorithm has running time O(nrw-1) for matroids with n elements and rank r. This is the best-known running time of any linear matroid intersection algorithm. We also consider lower bounds on the efficiency of matroid intersection algorithms, a question raised by Welsh (1976). Given two matroids of rank r on n elements, it is known that O(nr1.5) oracle queries suffice to solve matroid intersection. However, no non-trivial lower bounds are known. We make the first progress on this question. We describe a family of instances for which (log2 3)n - o(n) queries are necessary to solve these instances. This gives a constant factor improvement over the trivial lower bound for a certain range of parameters. Finally, we consider submodular functions, a generalization of matroids. We give three different proofs that [omega](n) queries are needed to find a minimizer of a submodular function, and prove that [omega](n2/ log n) queries are needed to find all minimizers.by Nicholas James Alexander Harvey.Ph.D