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

Recognizing Visibility Graphs of Polygons with Holes and Internal-External Visibility Graphs of Polygons

Visibility graph of a polygon corresponds to its internal diagonals and boundary edges. For each vertex on the boundary of the polygon, we have a vertex in this graph and if two vertices of the polygon see each other there is an edge between their corresponding vertices in the graph. Two vertices of a polygon see each other if and only if their connecting line segment completely lies inside the polygon, and they are externally visible if and only if this line segment completely lies outside the polygon. Recognizing visibility graphs is the problem of deciding whether there is a simple polygon whose visibility graph is isomorphic to a given input graph. This problem is well-known and well-studied, but yet widely open in geometric graphs and computational geometry. Existential Theory of the Reals is the complexity class of problems that can be reduced to the problem of deciding whether there exists a solution to a quantifier-free formula F(X1,X2,...,Xn), involving equalities and inequalities of real polynomials with real variables. The complete problems for this complexity class are called Existential Theory of the Reals Complete. In this paper we show that recognizing visibility graphs of polygons with holes is Existential Theory of the Reals Complete. Moreover, we show that recognizing visibility graphs of simple polygons when we have the internal and external visibility graphs, is also Existential Theory of the Reals Complete.Comment: Sumbitted to COCOON2018 Conferenc

Homomorphism Polynomials Complete for VP

The VP versus VNP question, introduced by Valiant, is probably the most important open question in algebraic complexity theory. Thanks to completeness results, a variant of this question, VBP versus VNP, can be succinctly restated as asking whether the permanent of a generic matrix can be written as a determinant of a matrix of polynomially bounded size. Strikingly, this restatement does not mention any notion of computational model. To get a similar restatement for the original and more fundamental question, and also to better understand the class itself, we need a complete polynomial for VP. Ad hoc constructions yielding complete polynomials were known, but not natural examples in the vein of the determinant. We give here several variants of natural complete polynomials for VP, based on the notion of graph homomorphism polynomials

Combinatorial Nullstellensatz modulo prime powers and the Parity Argument

We present new generalizations of Olson's theorem and of a consequence of Alon's Combinatorial Nullstellensatz. These enable us to extend some of their combinatorial applications with conditions modulo primes to conditions modulo prime powers. We analyze computational search problems corresponding to these kinds of combinatorial questions and we prove that the problem of finding degree-constrained subgraphs modulo $2^d$ such as $2^d$-divisible subgraphs and the search problem corresponding to the Combinatorial Nullstellensatz over $\mathbb{F}_2$ belong to the complexity class Polynomial Parity Argument (PPA)

Generalized characteristic polynomials of graph bundles

In this paper, we find computational formulae for generalized characteristic polynomials of graph bundles. We show that the number of spanning trees in a graph is the partial derivative (at (0,1)) of the generalized characteristic polynomial of the graph. Since the reciprocal of the Bartholdi zeta function of a graph can be derived from the generalized characteristic polynomial of a graph, consequently, the Bartholdi zeta function of a graph bundle can be computed by using our computational formulae