Fixed-Parameter Algorithms for Rectilinear Steiner tree and Rectilinear Traveling Salesman Problem in the plane

Given a set $P$ of $n$ points with their pairwise distances, the traveling salesman problem (TSP) asks for a shortest tour that visits each point exactly once. A TSP instance is rectilinear when the points lie in the plane and the distance considered between two points is the $l_1$ distance. In this paper, a fixed-parameter algorithm for the Rectilinear TSP is presented and relies on techniques for solving TSP on bounded-treewidth graphs. It proves that the problem can be solved in $O\left(nh7^h\right)$ where $h \leq n$ denotes the number of horizontal lines containing the points of $P$. The same technique can be directly applied to the problem of finding a shortest rectilinear Steiner tree that interconnects the points of $P$ providing a $O\left(nh5^h\right)$ time complexity. Both bounds improve over the best time bounds known for these problems.Comment: 24 pages, 13 figures, 6 table

Minimum Bend Shortest Rectilinear Route Discovery for a Moving Sink in a Grid Based Wireless Sensor Network

In a rectilinear route, a moving sink is restricted to travel either horizontally or vertically along the connecting edges. We present a new algorithm that finds the shortest round trip rectilinear route covering the specified nodes in a grid based Wireless Sensor Network.  The proposed algorithm determines the shortest round trip travelling salesman path in a two-dimensional grid graph. A special additional feature of the new path discovery technique is that it selects that path which has the least number of corners (bends) when more than one equal length shortest round trip paths are available. This feature makes the path more suitable for moving objects like Robots, drones and other types of vehicles which carry the moving sink. In the prosed scheme, the grid points are the vertices of the graph and the lines joining the grid points are the edges of the graph. The optimal edge set that forms the target path is determined using the binary integer programming

Minimum Bend Shortest Rectilinear Route Discovery for a Moving Sink in a Grid Based Wireless Sensor Network

In a rectilinear route, a moving sink is restricted to travel either horizontally or vertically along the connecting edges. We present a new algorithm that finds the shortest round trip rectilinear route covering the specified nodes in a grid based Wireless Sensor Network.  The proposed algorithm determines the shortest round trip travelling salesman path in a two-dimensional grid graph. A special additional feature of the new path discovery technique is that it selects that path which has the least number of corners (bends) when more than one equal length shortest round trip paths are available. This feature makes the path more suitable for moving objects like Robots, drones and other types of vehicles which carry the moving sink. In the prosed scheme, the grid points are the vertices of the graph and the lines joining the grid points are the edges of the graph. The optimal edge set that forms the target path is determined using the binary integer programming

Hardness and Approximation of Octilinear Steiner Trees

Given a point set K of terminals in the plane, the octilinear Steiner tree problem is to find a shortest tree that interconnects all terminals and edges run either in horizontal, vertical, or 45° diagonal direction. This problem is fundamental for the novel octilinear routing paradigm in VLSI design, the so-called X-architecture. As the related rectilinear and the Euclidian Steiner tree problem are well-known to be NP-hard, the same was widely believed for the octilinear Steiner tree problem but left open for quite some time. In this paper, we prove the NP-completeness of the decision version of the octilinear Steiner tree problem. We also show how to reduce the octilinear Steiner tree problem to the Steiner tree problem in graphs of polynomial size with the following approximation guarantee. We construct a graph of size O(n^2/epsilon^2) which contains a (1+epsilon)-approximation of a minimum octilinear Steiner tree for every epsilon > 0 and n = |K|. Hence, we can apply any k-approximation algorithm for the Steiner tree problem in graphs (the currently best known bound is k=1.55) and achieve an (k+epsilon)-approximation bound for the octilinear Steiner tree problem. This approximation guarantee also holds for the more difficult case where the Steiner tree has to avoid blockages (obstacles bounded by octilinear polygons)

Project for the analysis of technology transfer Quarterly report, 1 Jul. - 30 Sep. 1969

Research activities in technology transfer progra

On Covering Points with Conics and Strips in the Plane

Geometric covering problems have always been of focus in computer scientific research. The generic geometric covering problem asks to cover a set S of n objects with another set of objects whose cardinality is minimum, in a geometric setting. Many versions of geometric cover have been studied in detail, one of which is line cover: Given a set of points in the plane, find the minimum number of lines to cover them. In Euclidean space Rm, this problem is known as Hyperplane Cover, where lines are replaced by affine hyperplanes bounded by dimension d. Line cover is NP-hard, so is its hyperplane analogue. Our thesis focuses on few extensions of hyperplane cover and line cover. One of the techniques used to study NP-hard problems is Fixed Parameter Tractability (FPT), where, in addition to input size, a parameter k is provided for input instance. We ask to solve the problem with respect to k, such that the running time is a function in both n and k, strictly polynomial in n, while the exponential component is limited to k. In this thesis, we study FPT and parameterized complexity theory, the theory of classifying hard problems involving a parameter k. We focus on two new geometric covering problems: covering a set of points in the plane with conics (conic cover) and covering a set of points with strips or fat lines of given width in the plane (fat line cover). A conic is a non-degenerate curve of degree two in the plane. A fat line is defined as a strip of finite width w. In this dissertation, we focus on the parameterized versions of these two problems, where, we are asked to cover the set of points with k conics or k fat lines. We use the existing techniques of FPT algorithms, kernelization and approximation algorithms to study these problems. We do a comprehensive study of these problems, starting with NP-hardness results to studying their parameterized hardness in terms of parameter k. We show that conic cover is fixed parameter tractable, and give an algorithm of running time O∗ ((k/1.38)^4k), where, O∗ implies that the running time is some polynomial in input size. Utilizing special properties of a parabola, we are able to achieve a faster algorithm and show a running time of O∗ ((k/1.15)^3k). For fat line cover, first we establish its NP-hardness, then we explore algorithmic possibilities with respect to parameterized complexity theory. We show W [1]-hardness of fat line cover with respect to the number of fat lines, by showing a parameterized reduction from the problem of stabbing axis-parallel squares in the plane. A parameterized reduction is an algorithm which transforms an instance of one parameterized problem into an instance of another parameterized problem using a FPT-algorithm. In addition, we show that some restricted versions of fat line cover are also W [1]-hard. Further, in this thesis, we explore a restricted version of fat line cover, where the set of points are integer coordinates and allow only axis-parallel lines to cover them. We show that the problem is still NP-hard. We also show that this version is fixed parameter tractable having a kernel size of O (k^2) and give a FPT-algorithm with a running time of O∗ (3^k). Finally, we conclude our study on this problem by giving an approximation algorithm for this version having a constant approximation ratio 2

Ant colony optimization based simulation of 3d automatic hose/pipe routing

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.This thesis focuses on applying one of the rapidly growing non-deterministic optimization algorithms, the ant colony algorithm, for simulating automatic hose/pipe routing with several conflicting objectives. Within the thesis, methods have been developed and applied to single objective hose routing, multi-objective hose routing and multi-hose routing. The use of simulation and optimization in engineering design has been widely applied in all fields of engineering as the computational capabilities of computers has increased and improved. As a result of this, the application of non-deterministic optimization techniques such as genetic algorithms, simulated annealing algorithms, ant colony algorithms, etc. has increased dramatically resulting in vast improvements in the design process. Initially, two versions of ant colony algorithms have been developed based on, respectively, a random network and a grid network for a single objective (minimizing the length of the hoses) and avoiding obstacles in the CAD model. While applying ant colony algorithms for the simulation of hose routing, two modifications have been proposed for reducing the size of the search space and avoiding the stagnation problem. Hose routing problems often consist of several conflicting or trade-off objectives. In classical approaches, in many cases, multiple objectives are aggregated into one single objective function and optimization is then treated as a single-objective optimization problem. In this thesis two versions of ant colony algorithms are presented for multihose routing with two conflicting objectives: minimizing the total length of the hoses and maximizing the total shared length (bundle length). In this case the two objectives are aggregated into a single objective. The current state-of-the-art approach for handling multi-objective design problems is to employ the concept of Pareto optimality. Within this thesis a new Pareto-based general purpose ant colony algorithm (PSACO) is proposed and applied to a multi-objective hose routing problem that consists of the following objectives: total length of the hoses between the start and the end locations, number of bends, and angles of bends. The proposed method is capable of handling any number of objectives and uses a single pheromone matrix for all the objectives. The domination concept is used for updating the pheromone matrix. Among the currently available multi-objective ant colony optimization (MOACO) algorithms, P-ACO generates very good solutions in the central part of the Pareto front and hence the proposed algorithm is compared with P-ACO. A new term is added to the random proportional rule of both of the algorithms (PSACO and P-ACO) to attract ants towards edges that make angles close to the pre-specified angles of bends. A refinement algorithm is also suggested for searching an acceptable solution after the completion of searching the entire search space. For all of the simulations, the STL format (tessellated format) for the obstacles is used in the algorithm instead of the original shapes of the obstacles. This STL format is passed to the C++ library RAPID for collision detection. As a result of using this format, the algorithms can handle freeform obstacles and the algorithms are not restricted to a particular software package