Types of triangle in plane Hamiltonian triangulations and applications to domination and k-walks

We investigate the minimum number t(0)(G) of faces in a Hamiltonian triangulation G so that any Hamiltonian cycle C of G has at least t(0)(G) faces that do not contain an edge of C. We prove upper and lower bounds on the maximum of these numbers for all triangulations with a fixed number of facial triangles. Such triangles play an important role when Hamiltonian cycles in triangulations with 3-cuts are constructed from smaller Hamiltonian cycles of 4-connected subgraphs. We also present results linking the number of these triangles to the length of 3-walks in a class of triangulation and to the domination number

The Complexity of Routing with Few Collisions

We study the computational complexity of routing multiple objects through a network in such a way that only few collisions occur: Given a graph $G$ with two distinct terminal vertices and two positive integers $p$ and $k$, the question is whether one can connect the terminals by at least $p$ routes (e.g. paths) such that at most $k$ edges are time-wise shared among them. We study three types of routes: traverse each vertex at most once (paths), each edge at most once (trails), or no such restrictions (walks). We prove that for paths and trails the problem is NP-complete on undirected and directed graphs even if $k$ is constant or the maximum vertex degree in the input graph is constant. For walks, however, it is solvable in polynomial time on undirected graphs for arbitrary $k$ and on directed graphs if $k$ is constant. We additionally study for all route types a variant of the problem where the maximum length of a route is restricted by some given upper bound. We prove that this length-restricted variant has the same complexity classification with respect to paths and trails, but for walks it becomes NP-complete on undirected graphs

Pseudoknots in a Homopolymer

After a discussion of the definition and number of pseudoknots, we reconsider the self-attracting homopolymer paying particular attention to the scaling of the number of pseudoknots at different temperature regimes in two and three dimensions. Although the total number of pseudoknots is extensive at all temperatures, we find that the number of pseudoknots forming between the two halves of the chain diverges logarithmically at (in both dimensions) and below (in 2d only) the theta-temparature. We later introduce a simple model that is sensitive to pseudoknot formation during collapse. The resulting phase diagram involves swollen, branched and collapsed homopolymer phases with transitions between each pair.Comment: submitted to PR

Walking Through Waypoints

We initiate the study of a fundamental combinatorial problem: Given a capacitated graph $G=(V,E)$, find a shortest walk ("route") from a source $s\in V$ to a destination $t\in V$ that includes all vertices specified by a set $\mathscr{W}\subseteq V$: the \emph{waypoints}. This waypoint routing problem finds immediate applications in the context of modern networked distributed systems. Our main contribution is an exact polynomial-time algorithm for graphs of bounded treewidth. We also show that if the number of waypoints is logarithmically bounded, exact polynomial-time algorithms exist even for general graphs. Our two algorithms provide an almost complete characterization of what can be solved exactly in polynomial-time: we show that more general problems (e.g., on grid graphs of maximum degree 3, with slightly more waypoints) are computationally intractable

Stronger ILPs for the Graph Genus Problem

The minimum genus of a graph is an important question in graph theory and a key ingredient in several graph algorithms. However, its computation is NP-hard and turns out to be hard even in practice. Only recently, the first non-trivial approach - based on SAT and ILP (integer linear programming) models - has been presented, but it is unable to successfully tackle graphs of genus larger than 1 in practice. Herein, we show how to improve the ILP formulation. The crucial ingredients are two-fold. First, we show that instead of modeling rotation schemes explicitly, it suffices to optimize over partitions of the (bidirected) arc set A of the graph. Second, we exploit the cycle structure of the graph, explicitly mapping short closed walks on A to faces in the embedding. Besides the theoretical advantages of our models, we show their practical strength by a thorough experimental evaluation. Contrary to the previous approach, we are able to quickly solve many instances of genus > 1

Plane and simple : using planar subgraphs for efficient algorithms

In this thesis, we showcase how planar subgraphs with special structural properties can be used to fi nd efficient algorithms for two NP-hard problems in combinatorial optimization. In the fi rst part, we develop algorithms for the computation of Tutte paths and show how these special subgraphs can be used to efficiently compute long cycles and other relaxations of Hamiltonicity if we restrict the input to planar graphs. We give an O(n^2) time algorithm for the computation of Tutte paths in circuit graphs and generalize it to the computation of Tutte paths between any two given vertices and a prescribed intermediate edge in 2-connected planar graphs. In the second part, we study the Maximum Planar Subgraph Problem (MPS) and show how dense planar subgraphs can be used to develop new approximation algorithms for this problem. All new algorithms and arguments we present are based on a novel approach that focuses on maximizing the number of triangular faces in the computed subgraph. For this, we define a new optimization problem called Maximum Planar Triangles (MPT). We show that this problem is NP-hard and quantify how good an approximation algorithm for MPT performs as an approximation for MPS. We give a greedy 1/11-approximation algorithm for Mpt and show that the approximation ratio can be improved to 1/6 by using locally optimal triangular cactus subgraphs.In dieser Dissertation zeigen wir, wie planare Teilgraphen mit speziellen Eigenschaften verwendet werden können, um effiziente Algorithmen für zwei NP-schwere Probleme in der kombinatorischen Optimierung zu fi nden. Im ersten Teil entwickeln wir Algorithmen zur Berechnung von Tutte-Wegen und zeigen, wie diese verwendet werden können, um lange Kreise und andere Lockerungen der Hamilton-Charakteristik zu finden, wenn wir uns auf Graphen in der Ebene beschränken. Wir beschreiben zunächst einen O(n^2)-Algorithmus in Circuit-Graphen und verallgemeinern diesen anschließend für die Berechnung von Tutte-Wegen in 2-zusammenhängenden planaren Graphen. Im zweiten Teil untersuchen wir das Maximum Planar Subgraph Problem (MPS) und zeigen, wie besonders dichte planare Teilgraphen verwendet werden können, um neue Approximationsalgorithmen zu entwickeln. Unsere Ergebnisse basieren auf einem neuartigen Ansatz, bei dem die Anzahl der dreieckigen Gebiete im berechneten Teilgraphen maximiert wird. Dazu de finieren wir ein neues Optimierungsproblem namens Maximum Planar Triangles (MPT). Wir zeigen, dass dieses Problem NP-schwer ist und quantifi zieren, wie gut ein Approximationsalgorithmus für MPT als Approximation für MPS funktioniert. Wir geben einen 1/11-Approximationsalgorithmus für MPT und zeigen, wie dies durch die Verwendung von lokal optimaler Kaktus-Teilgraphen auf 1/6 verbessert werden kann