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

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