A Linear-Time Algorithm for Finding Induced Planar Subgraphs

In this paper we study the problem of efficiently and effectively extracting induced planar subgraphs. Edwards and Farr proposed an algorithm with O(mn) time complexity to find an induced planar subgraph of at least 3n/(d+1) vertices in a graph of maximum degree d. They also proposed an alternative algorithm with O(mn) time complexity to find an induced planar subgraph graph of at least 3n/(bar{d}+1) vertices, where bar{d} is the average degree of the graph. These two methods appear to be best known when d and bar{d} are small. Unfortunately, they sacrifice accuracy for lower time complexity by using indirect indicators of planarity. A limitation of those approaches is that the algorithms do not implicitly test for planarity, and the additional costs of this test can be significant in large graphs. In contrast, we propose a linear-time algorithm that finds an induced planar subgraph of n-nu vertices in a graph of n vertices, where nu denotes the total number of vertices shared by the detected Kuratowski subdivisions. An added benefit of our approach is that we are able to detect when a graph is planar, and terminate the reduction. The resulting planar subgraphs also do not have any rigid constraints on the maximum degree of the induced subgraph. The experiment results show that our method achieves better performance than current methods on graphs with small skewness

On positive realness of descriptor systems

In this brief, the positive realness of descriptor systems is studied. For the continuous-time case, two positive real lemmas are given, based on a generalized algebraic Riccati equation and inequality respectively. For the discrete-time case, the positive real lemma is given in terms of a generalized algebraic Riccati inequality.published_or_final_versio

Graph Planarization Problem Optimization Based on Triple-Valued Gravitational Search Algorithm

This article presents a triple-valued gravitational search algorithm (TGSA) to tackle the graph planarization problem (GPP). GPP is one of the most important tasks in graph theory, and has proved to be an NP-hard problem. To solve it, TGSA uses a triple-valued encoding scheme and models the search space into a triangular hypercube quantitatively based on the well-known single-row routing representation method. The agents in TGSA, whose interactions are driven by the gravity law, move toward the global optimal position gradually. The position updating rule for each agent is based on two indices: one is a velocity index which is a function of the current velocity of the agent, and the other is a population index based on the cumulative information in the whole population. To verify the performance of the algorithm, 21 benchmark instances are tested. Experimental results indicate that TGSA can solve the GPP by finding its maximum planar subgraph and embedding the resulting edges into a plane simultaneously. Compared with traditional algorithms, a novelty of TGSA is that it can find multiple optimal solutions for the GPP. Comparative results also demonstrate that TGSA outperforms the traditional meta-heuristics in terms of the solution qualities within reasonable computational times. © 2013 Institute of Electrical Engineers of Japan

An evolutionary algorithm for graph planarisation by vertex deletion

A non-planar graph can only be planarised if it is structurally modified. This work presents a new heuristic algorithm that uses vertices deletion to modify a non-planar graph in order to obtain a planar subgraph. The proposed algorithm aims to delete a minimum number of vertices to achieve its goal. The vertex deletion number of a graph G = (V,E) is the smallest integer k ? 0 such that there is an induced planar subgraph of G obtained by the removal of k vertices of G. Considering that the corresponding decision problem is NPcomplete and an approximation algorithm for graph planarisation by vertices deletion does not exist, this work proposes an evolutionary algorithm that uses a constructive heuristic algorithm to planarise a graph. This constructive heuristic has time complexity of O(n+m), where m = |V| and n = |E|, and it is based on the PQ-trees data structure and on the vertex deletion operation. The algorithm performance is verified by means of case studies

A nonmonotone GRASP

A greedy randomized adaptive search procedure (GRASP) is an itera- tive multistart metaheuristic for difficult combinatorial optimization problems. Each GRASP iteration consists of two phases: a construction phase, in which a feasible solution is produced, and a local search phase, in which a local optimum in the neighborhood of the constructed solution is sought. Repeated applications of the con- struction procedure yields different starting solutions for the local search and the best overall solution is kept as the result. The GRASP local search applies iterative improvement until a locally optimal solution is found. During this phase, starting from the current solution an improving neighbor solution is accepted and considered as the new current solution. In this paper, we propose a variant of the GRASP framework that uses a new “nonmonotone” strategy to explore the neighborhood of the current solu- tion. We formally state the convergence of the nonmonotone local search to a locally optimal solution and illustrate the effectiveness of the resulting Nonmonotone GRASP on three classical hard combinatorial optimization problems: the maximum cut prob- lem (MAX-CUT), the weighted maximum satisfiability problem (MAX-SAT), and the quadratic assignment problem (QAP)

Two new approximation algorithms for the maximum planar subgraph problem

The maximum planar subgraph problem (MPS) is defined as follows: given a graph G, find a largest planar subgraph of G. The problem is NP-hard and it has applications in graph drawing and resource location optimization. Călinescu et al. [J. Alg. 27, 269-302 (1998)] presented the first approximation algorithms for MPS with nontrivial performance ratios. Two algorithms were given, a simple algorithm which runs in linear time for bounded-degree graphs with a ratio 7/18 and a more complicated algorithm with a ratio 4/9. Both algorithms produce outerplanar subgraphs. In this article we present two new versions of the simpler algorithm. The first new algorithm still runs in the same time, produces outerplanar subgraphs, has at least the same performance ratio as the original algorithm, but in practice it finds larger planar subgraphs than the original algorithm. The second new algorithm has similar properties to the first algorithm, but it produces only planar subgraphs. We conjecture that the performance ratios of our algorithms are at least 4/9 for MPS. We experimentally compare the new algorithms against the original simple algorithm. We also apply the new algorithms for approximating the thickness and outerthickness of a graph. Experiments show that the new algorithms produce clearly better approximations than the original simple algorithm by Călinescu et al

Three Dimensional Integration (3DI) of semiconductor circuit layers: new devices and fabrication process

The device density of Integrated Circuits (ICs) manufactured by current VLSI technology is reaching it\u27s theoretical limit. Nevertheless, the demand for integration of more devices per chip is growing. To accommodate this need three main possibilities can be explored: Wafer Scale Integration (WSI), Ultra Large Scale Integration (ULSI), and Three Dimensional Integration (3DI). A brief review of these techniques along with their comparative advantages and disadvantages is presented. It has been concluded that 3DI technology is superior to others. Therefore, an attempt is made to develop a viable fabrication process for this technology. This is done by first reviewing the current technologies that are utilized for fabrication of Integrated Circuits (ICs) and their compatibility with 3DI stringent requirements.;Based on this review, a set of fabrication procedure for realization of 3DI technology, are presented in chapter 3. In Chapter 1 the compatibility of the currently used devices, such as BJTs and FETS, with 3DI technology is examined. Moreover, a new active device is developed for 3DI technology to replace BJTs and FETs in circuits. This new device is more compatible to the constrains of 3DI technology. Chapter 2 is devoted to solving the overall problems of 3DI circuits. The problem of heat and power dispassion and signal coupling (Cross-Talk) between the layers are reviewed, and an inter-layer shield is proposed to overcome these problems. The effectiveness of such a thin shield is considered theoretically. In Chapter 3 a fabrication process for 3DI technology is proposed. This is done after a short analysis of previous attempts in developing 3DI technologies.;Chapter 4 focuses on analog extension of 3DI technology. Moreover, in this chapter microwave 3DI circuits or 3DI MMIC is investigated. Practical considerations in choice of material for the proposed device is the subject of study in Chapter 5. Low temperature ohmic contact and utilization of metal-silicides for the proposed device are considered in this chapter. Finally in Chapter 6 various computer verifications for this work is presented, and in Chapter 7 experimental results to support this work is included

Combinatorial optimization and metaheuristics

Today, combinatorial optimization is one of the youngest and most active areas of discrete mathematics. It is a branch of optimization in applied mathematics and computer science, related to operational research, algorithm theory and computational complexity theory. It sits at the intersection of several fields, including artificial intelligence, mathematics and software engineering. Its increasing interest arises for the fact that a large number of scientific and industrial problems can be formulated as abstract combinatorial optimization problems, through graphs and/or (integer) linear programs. Some of these problems have polynomial-time (“efficient”) algorithms, while most of them are NP-hard, i.e. it is not proved that they can be solved in polynomial-time. Mainly, it means that it is not possible to guarantee that an exact solution to the problem can be found and one has to settle for an approximate solution with known performance guarantees. Indeed, the goal of approximate methods is to find “quickly” (reasonable run-times), with “high” probability, provable “good” solutions (low error from the real optimal solution). In the last 20 years, a new kind of algorithm commonly called metaheuristics have emerged in this class, which basically try to combine heuristics in high level frameworks aimed at efficiently and effectively exploring the search space. This report briefly outlines the components, concepts, advantages and disadvantages of different metaheuristic approaches from a conceptual point of view, in order to analyze their similarities and differences. The two very significant forces of intensification and diversification, that mainly determine the behavior of a metaheuristic, will be pointed out. The report concludes by exploring the importance of hybridization and integration methods

Scalable Exact Visualization of Isocontours in Road Networks via Minimum-Link Paths

Isocontours in road networks represent the area that is reachable from a source within a given resource limit. We study the problem of computing accurate isocontours in realistic, large-scale networks. We propose isocontours represented by polygons with minimum number of segments that separate reachable and unreachable components of the network. Since the resulting problem is not known to be solvable in polynomial time, we introduce several heuristics that run in (almost) linear time and are simple enough to be implemented in practice. A key ingredient is a new practical linear-time algorithm for minimum-link paths in simple polygons. Experiments in a challenging realistic setting show excellent performance of our algorithms in practice, computing near-optimal solutions in a few milliseconds on average, even for long ranges