Domination in graphs with application to network reliability

In this thesis we investigate different domination-related graph polynomials, like the connected domination polynomial, the independent domination polynomial, and the total domination polynomial. We prove some basic properties of these polynomials and obtain formulas for the calculation in special graph classes. Furthermore, we also prove results about the calculation of the different graph polynomials in product graphs and different representations of the graph polynomials. One focus of this thesis lays on the generalization of domination-related polynomials. In this context the trivariate domination polynomial is defined and some results about the bipartition polynomial, which is also a generalization of the domination polynomial, is presented. These two polynomials have many useful properties and interesting connections to other graph polynomials. Furthermore, some more general domination-related polynomials are defined in this thesis, which shows some possible directions for further research.In dieser Dissertation werden verschiedene, zum Dominationspolynom verwandte, Graphenpolynome, wie das zusammenhängende Dominationspolynom, das unabhängige Dominationspolynom und das totale Dominationspolynom, untersucht. Es werden grundlegende Eigenschaften erforscht und Sätze für die Berechnung dieser Polynome in speziellen Graphenklassen bewiesen. Weiterhin werden Ergebnisse für die Berechnung in Produktgraphen und verschiedene Repräsentationen für diese Graphenpolynome gezeigt. Ein Fokus der Dissertation liegt auf der Verallgemeinerung der verschiedenen Dominationspolynome. In diesem Zusammenhang wird das trivariate Dominationspolynom definiert. Außerdem werden Ergebnisse für das Bipartitionspolynom bewiesen. Diese beiden Polynome haben viele interessante Eigenschaften und Beziehungen zu anderen Graphenpolynomen. Darüber hinaus werden weitere multivariate Graphenpolynome definiert, die eine mögliche Richtung für weitere Forschung auf diesem Gebiet aufzeigen

Independent sets, matchings, and occupancy fractions

We prove tight upper bounds on the logarithmic derivative of the independence and matching polynomials of d-regular graphs. For independent sets, this theorem is a strengthening of Kahn's result that a disjoint union of copies of Kd;d maximizes the number of independent sets of a bipartite d-regular graph, Galvin and Tetali's result that the independence polynomial is maximized by the same, and Zhao's extension of both results to all d-regular graphs. For matchings, this shows that the matching polynomial and the total number of matchings of a d-regular graph are maximized by a union of copies of Kd;d. Using this we prove the asymptotic upper matching conjecture of Friedland, Krop, Lundow, and Markstrom. In probabilistic language, our main theorems state that for all d-regular graphs and all �, the occupancy fraction of the hard-core model and the edge occupancy fraction of the monomer-dimer model with fugacity � are maximized by Kd;d. Our method involves constrained optimization problems over distributions of random variables and applies to all d-regular graphs directly, without a reduction to the bipartite case. Using a variant of the method we prove a lower bound on the occupancy fraction of the hard-core model on any d-regular, vertex-transitive, bipartite graph: the occupancy fraction of such a graph is strictly greater than the occupancy fraction of the unique translationinvariant hard-core measure on the infinite d-regular tre

Separate, measure and conquer: faster polynomial-space algorithms for Max 2-CSP and counting dominating sets

We show a method resulting in the improvement of several polynomial-space, exponential-time algorithms. The method capitalizes on the existence of small balanced separators for sparse graphs, which can be exploited for branching to disconnect an instance into independent components. For this algorithm design paradigm, the challenge to date has been to obtain improvements in worst-case analyses of algorithms, compared with algorithms that are analyzed with advanced methods, notably Measure and Conquer. Our contribution is the design of a general method to integrate the advantage from the separator-branching into Measure and Conquer, for a more precise and improved running time analysi

Parallel black-box complexity with tail bounds

We propose a new black-box complexity model for search algorithms evaluating λ search points in parallel. The parallel unary unbiased black-box complexity gives lower bounds on the number of function evaluations every parallel unary unbiased black-box algorithm needs to optimise a given problem. It captures the inertia caused by offspring populations in evolutionary algorithms and the total computational effort in parallel metaheuristics. We present complexity results for LeadingOnes and OneMax. Our main result is a general performance limit: we prove that on every function every λ-parallel unary unbiased algorithm needs at least a certain number of evaluations (a function of problem size and λ) to find any desired target set of up to exponential size, with an overwhelming probability. This yields lower bounds for the typical optimisation time on unimodal and multimodal problems, for the time to find any local optimum, and for the time to even get close to any optimum. The power and versatility of this approach is shown for a wide range of illustrative problems from combinatorial optimisation. Our performance limits can guide parameter choice and algorithm design; we demonstrate the latter by presenting an optimal λ-parallel algorithm for OneMax that uses parallelism most effectively

Delocalisation and continuity in 2D: loop O(2), six-vertex, and random-cluster models

We prove the existence of macroscopic loops in the loop O(2) model with $\frac12\leq x^2\leq 1$ or, equivalently, delocalisation of the associated integer-valued Lipschitz function on the triangular lattice. This settles one side of the conjecture of Fan, Domany, and Nienhuis (1970s-80s) that $x^2 = \frac12$ is the critical point. We also prove delocalisation in the six-vertex model with $0<a,\,b\leq c\leq a+b$. This yields a new proof of continuity of the phase transition in the random-cluster and Potts models in two dimensions for $1\leq q\leq 4$ relying neither on integrability tools (parafermionic observables, Bethe Ansatz), nor on the Russo-Seymour-Welsh theory. Our approach goes through a novel FKG property required for the non-coexistence theorem of Zhang and Sheffield, which is used to prove delocalisation all the way up to the critical point. We also use the $\mathbb T$-circuit argument in the case of the six-vertex model. Finally, we extend an existing renormalisation inequality in order to quantify the delocalisation as being logarithmic, in the regimes $\frac12\leq x^2\leq 1$ and $a=b\leq c\leq a+b$. This is consistent with the conjecture that the scaling limit is the Gaussian free field.Comment: 50 pages, 10 figure