A Quasi-Polynomial Time Partition Oracle for Graphs with an Excluded Minor

Motivated by the problem of testing planarity and related properties, we study the problem of designing efficient {\em partition oracles}. A {\em partition oracle} is a procedure that, given access to the incidence lists representation of a bounded-degree graph $G= (V,E)$ and a parameter \eps, when queried on a vertex $v\in V$, returns the part (subset of vertices) which $v$ belongs to in a partition of all graph vertices. The partition should be such that all parts are small, each part is connected, and if the graph has certain properties, the total number of edges between parts is at most \eps |V|. In this work we give a partition oracle for graphs with excluded minors whose query complexity is quasi-polynomial in 1/\eps, thus improving on the result of Hassidim et al. ({\em Proceedings of FOCS 2009}) who gave a partition oracle with query complexity exponential in 1/\eps. This improvement implies corresponding improvements in the complexity of testing planarity and other properties that are characterized by excluded minors as well as sublinear-time approximation algorithms that work under the promise that the graph has an excluded minor.Comment: 13 pages, 1 figur

Planar diagrams from optimization

We propose a new toy model of a heteropolymer chain capable of forming planar secondary structures typical for RNA molecules. In this model the sequential intervals between neighboring monomers along a chain are considered as quenched random variables. Using the optimization procedure for a special class of concave--type potentials, borrowed from optimal transport analysis, we derive the local difference equation for the ground state free energy of the chain with the planar (RNA--like) architecture of paired links. We consider various distribution functions of intervals between neighboring monomers (truncated Gaussian and scale--free) and demonstrate the existence of a topological crossover from sequential to essentially embedded (nested) configurations of paired links.Comment: 10 pages, 10 figures, the proof is added. arXiv admin note: text overlap with arXiv:1102.155

Book Embeddings of Nonplanar Graphs with Small Faces in Few Pages

An embedding of a graph in a book, called book embedding, consists of a linear ordering of its vertices along the spine of the book and an assignment of its edges to the pages of the book, so that no two edges on the same page cross. The book thickness of a graph is the minimum number of pages over all its book embeddings. For planar graphs, a fundamental result is due to Yannakakis, who proposed an algorithm to compute embeddings of planar graphs in books with four pages. Our main contribution is a technique that generalizes this result to a much wider family of nonplanar graphs, which is characterized by a biconnected skeleton of crossing-free edges whose faces have bounded degree. Notably, this family includes all 1-planar and all optimal 2-planar graphs as subgraphs. We prove that this family of graphs has bounded book thickness, and as a corollary, we obtain the first constant upper bound for the book thickness of optimal 2-planar graphs

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

The Structure of Minimum Vertex Cuts

In this paper we continue a long line of work on representing the cut structure of graphs. We classify the types of minimum vertex cuts, and the possible relationships between multiple minimum vertex cuts. As a consequence of these investigations, we exhibit a simple O(? n)-space data structure that can quickly answer pairwise (?+1)-connectivity queries in a ?-connected graph. We also show how to compute the "closest" ?-cut to every vertex in near linear O?(m+poly(?)n) time

A (1.5+ϵ)(1.5+\epsilon)-Approximation Algorithm for Weighted Connectivity Augmentation

Connectivity augmentation problems are among the most elementary questions in Network Design. Many of these problems admit natural $2$-approximation algorithms, often through various classic techniques, whereas it remains open whether approximation factors below $2$ can be achieved. One of the most basic examples thereof is the Weighted Connectivity Augmentation Problem (WCAP). In WCAP, one is given an undirected graph together with a set of additional weighted candidate edges, and the task is to find a cheapest set of candidate edges whose addition to the graph increases its edge-connectivity. We present a $(1.5+\varepsilon)$-approximation algorithm for WCAP, showing for the first time that factors below $2$ are achievable. On a high level, we design a well-chosen local search algorithm, inspired by recent advances for Weighted Tree Augmentation. To measure progress, we consider a directed weakening of WCAP and show that it has highly structured planar solutions. Interpreting a solution of the original problem as one of this directed weakening allows us to describe local exchange steps in a clean and algorithmically amenable way. Leveraging these insights, we show that we can efficiently search for good exchange steps within a component class for link sets that is closely related to bounded treewidth subgraphs of circle graphs. Moreover, we prove that an optimum solution can be decomposed into smaller components, at least one of which leads to a good local search step as long as we did not yet achieve the claimed approximation guarantee

Semiconductor Wastewater Treatment With Natural Starches As Coagulants Using Response Surface Methodology [TD746. F252 2007 f rb].

Air sisa semikonduktor yang digunakan dalam kajian ini diambil dari sebuah syarikat semikonduktor antarabangsa di Pulau Pinang. The semiconductor wastewater used in this study was obtained from a multinational semiconductor company located in Penang

Benznidazole

The conformation of the title compound [systematic name: N-benzyl-2-(2-nitro­imidazol-1-yl)acetamide], C12H12N4O3, can be described in terms of the relative orientation of three planar fragments, the imidazol group (A), benzyl group (B), and the acetamide fragment (C), with corresponding dihedral angles: A/C = 88.17 (4), B/C = 67.12 (5) and A/B = 21.11 (4)°. The crystal packing is enhanced by a network of strong inter­molecular N—H⋯O hydrogen bonds