Evaluations of topological Tutte polynomials

We find new properties of the topological transition polynomial of embedded graphs, $Q(G)$. We use these properties to explain the striking similarities between certain evaluations of Bollob\'as and Riordan's ribbon graph polynomial, $R(G)$, and the topological Penrose polynomial, $P(G)$. The general framework provided by $Q(G)$ also leads to several other combinatorial interpretations these polynomials. In particular, we express $P(G)$, $R(G)$, and the Tutte polynomial, $T(G)$, as sums of chromatic polynomials of graphs derived from $G$; show that these polynomials count $k$-valuations of medial graphs; show that $R(G)$ counts edge 3-colourings; and reformulate the Four Colour Theorem in terms of $R(G)$. We conclude with a reduction formula for the transition polynomial of the tensor product of two embedded graphs, showing that it leads to additional relations among these polynomials and to further combinatorial interpretations of $P(G)$ and $R(G)$.Comment: V2: major revision, several new results, and improved expositio

Fast Evaluation of Interlace Polynomials on Graphs of Bounded Treewidth

We consider the multivariate interlace polynomial introduced by Courcelle (2008), which generalizes several interlace polynomials defined by Arratia, Bollobas, and Sorkin (2004) and by Aigner and van der Holst (2004). We present an algorithm to evaluate the multivariate interlace polynomial of a graph with n vertices given a tree decomposition of the graph of width k. The best previously known result (Courcelle 2008) employs a general logical framework and leads to an algorithm with running time f(k)*n, where f(k) is doubly exponential in k. Analyzing the GF(2)-rank of adjacency matrices in the context of tree decompositions, we give a faster and more direct algorithm. Our algorithm uses 2^{3k^2+O(k)}*n arithmetic operations and can be efficiently implemented in parallel.Comment: v4: Minor error in Lemma 5.5 fixed, Section 6.6 added, minor improvements. 44 pages, 14 figure

Computational complexity of graph polynomials

The thesis provides hardness and algorithmic results for graph polynomials. We observe VNP-completeness of the interlace polynomial, and we prove VNP-completeness of almost all q-restrictions of Z(G; q; x), the multivariate Tutte polynomial. Using graph transformations, we obtain point-to-point reductions for graph polynomials.We develop two general methods: Vertex/edge cloning and, more general,uniform local graph transformations. These methods unify known and new hardness-of-evaluation results for graph polynomials. We apply both methods to several examples. We show that, almost everywhere, it is #P-hard to evaluate the two-variable interlace polynomial and the (normal as well as extended) bivariate chromatic polynomial. Almost everywhere" means that the dimension of the set of exceptional points is strictly less than the dimension of the domain of the graph polynomial. We also give an inapproximability result for evaluation of the independent set polynomial. Providing a new family of reductions for the interlace polynomial that increases the instance size only polylogarithmically, we obtain an exp(Ω (n= log3 n)) time lower bound for evaluation of the independent set polynomial under a counting version of the exponential time hypothesis. We observe that the extended bivariate chromatic polynomial can be computed in vertex-exponential time. We devise a means to compute the interlace polynomial using tree decompositions. This enables a parameterized algorithm to evaluate the interlace polynomial in time linear in the size of the graph and single-exponential in the treewidth. We give several versions of the algorithm, including a parallel one and a faster way to compute the interlace polynomial of any graph. Finally, we propose two faster algorithms to compute/evaluate the interlace polynomial in special cases.Diese Arbeit beinhaltet Härteresultate und Algorithmen für Graphpolynome. Wir stellen zunächst fest, dass das Interlacepolynom VNP-vollständig ist, und wir zeigen die VNP-Vollständigkeit fast aller q-Restriktionen des multivariaten Tutte-Polynoms Z(G; q; x). Unter Verwendung von Graphtransformationen erhalten wir Punkt-zu-Punkt-Reduktionen für Graphpolynome. Dabei entwickeln wir auch zwei allgemeine Methoden: Das Klonen von Knoten bzw. Kanten und, allgemeiner, uniforme lokale Graphtransformationen. Beide Methoden vereinheitlichen bekannte und neue Härteresultate für das Auswerten von Graphpolynomen. Wir wenden beide Methoden auf verschiedene Beispiele an. Wir zeigen, dass es fast überall #P-schwer ist, das Interlacepolynom in zwei Variablen bzw. das (normale oder erweiterte) bivariatechromatische Polynom auszuwerten. Fast überall heißt hier: Überall, außerauf einer Ausnahmemenge, deren Dimension um mindestens eins kleiner ist als der Definitionsbereich des Graphpolynoms. Wir zeigen auch, dass näherungsweises Auswerten des Independent-Set-Polynoms schwer ist. Wir entwickeln eine neue Familie von Reduktionen für das Interlacepolynom, die die Instanz nur polylogarithmisch vergrößert. Damit zeigen wir, unter Annahme einer Variante der Exponentialzeit-Hypothese, dass das Auswerten des Independent-Set-Polynoms fast überall Zeit exp(Ω(n= log3 n)) benötigt. Wir stellen fest, dass das erweiterte bivariate chromatische Polynom in Zeit exponentiell in der Knotenzahl berechnet werden kann. Wir entwickeln ein Mittel, um das Interlacepolynom mit Hilfe von Baumzerlegungen zu berechnen. Das führt zu einem parametrisierten Algorithmus zum Auswerten des Interlacepolynoms mit Laufzeit linear in der Anzahl der Knoten und einfach exponentiell in der Weite der gegebenen Baumzerlegung. Wir diskutieren verschiedene Varianten dieses Algorithmus, einschließlich Parallelisierung und einer Möglichkeit, das Interlacepolynom jedes Graphen asymptotisch schneller zu berechnen. Schließlich geben wir zwei schnellere Algorithmen an, die das Interlacepolynomin speziellen Situationen berechnen