Francis Suraweera

Encoding Graphs for Genetic Algorithms: An Investigation Using the Minimum Spanning Tree Problem

We present a comparison of modified and unmodified GAs for the MST problem. A GA assembles successful gene substrings into improved solutions, but it does not have a mechanism to enforce global constraints. Graph theory contains many problems that are suitable for GAs because polynomial-time algorithms do not exist, but they often have global constraints. Special encodings and modifications of GA operators have been developed to deal with this difficulty. We use the MST as an example problem because it is representative of the encoding difficulty, while the existence of polynomial-time algorithms makes the evaluation of performance relatively simple. We modify the GA crossover operator to preserve the property that an MST has 1 fewer edges than the number of vertices. Although this restricts the search space substantially, our results show that the expected benefits are not obtained. The GA demonstrates its power by successfully restricting the search without help.

NP-completeness and FPT Results for Rectilinear Covering Problems

This paper discusses three rectilinear (that is, axis-parallel) covering prob- lems in d dimensions and their variants. The first problem is the Rectilinear Line Cover where the inputs are n points in R d and a positive integer k ,a nd we are asked to answer if we can cover these n points with at most k lines where these lines are restricted to be axis parallel. We show that this problem has efficient fixed-parameter tractable (FPT) algorithms. The second problem is the Rectilinear k-Links Span- ning Path Problem where the inputs are also n points in R d and a positive integer k but here we are asked to answer if there is a piecewise linear path through these n points having at most k line-segments (links) where these line-segments are axis- parallel. We prove that this second problem is FPT under the assumption that no two line-segments share the same line. The third problem is the Rectilinear Hyper- plane Cover problem and we are asked to cover a set of n points in d dimensions with k axis-parallel hyperplanes of d − 1 dimensions. We also demonstrate this has an FPT-algorithm. Previous to the results above, only conjectures were enunciated over several years on the NP-completeness of the Rectilinear Minimum Link Traveling Salesman Problem ,t heMinimum Link Spanning Path Problem and the Recti- linear Hyperplane Cover. We provide the proof that the Rectilinear Minimum Link Traveling Salesman Problem and the Rectilinear Minimum Link Span- ning Path Problem are NP-complete by a reduction from the One-In-Three 3-SAT problem. The NP-completeness of the Rectilinear Hyperplane Cover problem is proved by a reduction from 3-SAT. This suggests dealing with the intractability just discovered with fixed-parameter tractability. Moreover, if we extend our problems to a finite set of orientations, our approach proves these problems remain FPT.

A fast algorithm for the minimum spanning tree

Abstract This paper presents a fast algorithm for the construction of minimum-weight spanning trees in connected graphs or networks. The algorithm incorporates an old idea to sort the weights or costs associated with the edges in linear time. Also the algorithm exploits the fact that a connected graph with n nodes and having exactly ( n − 1 ) edges is a tree. The overall complexity of the algorithm is bounded by O( m log log n ) where m is the number of edges and n is the number of nodes of the network.

NC algorithms for the single most vital edge problem with respect to shortest paths

Abstract For a weighted, undirected graph G = (V,E), the single most vital edge in a network with respect to shortest paths is the edge that, when removed, results in the greatest increase in the shortest distance between two nodes s and t. We give a sequential algorithm for the Single Most Vital Edge problem on weighted and undirected graphs. Our algorithm has a time complexity O(mα(m,n)), where n = ¦ V ¦, m = ¦ E ¦ , and α(m,n) is a functional inverse of Ackermanns function. This algorithm eliminates the inherent sequentiality of the algorithm due to Malik et al. We also obtain a set of parallel algorithms running in O(log n) time using m processors and O(m) space on the CRCW PRAM, in O(log n) time using mn log n CREW processors and O(m + n log m) space, and in O(log n) time using mn log n EREW processors and O(mn) respectively. These are the first NC algorithms for solving this problem on the PRAM.

Tracking bees - a 3D, outdoor small object environment

The automatic tracking of bees while pollinating macadamia trees is important for the understanding of reproductive biology and fruit yield. We present techniques for tracking these small targets in an uncontrolled illumination environment and where the background is not fixed. Our results indicate we can track, based on color, bees moving in 3D in paths over 3s long (72 frames).

Reduction rules deliver efficient FPT-algorithms for covering points with lines

We present efficient algorithms to solve the Line Cover Problem exactly. In this NP-complete problem, the inputs are n points in the plane and a positive integer k, and we are asked to answer if we can cover these n points with at most k lines. Our approach is based on fixed-parameter tractability and, in particular, kernelization. We propose several reduction rules to transform instances of Line Cover into equivalent smaller instances. Once instances are no longer susceptible to these reduction rules, we obtain a problem kernel whose size is bounded by a polynomial function of the parameter k and does not depend on the size n of the input. Our algorithms provide exact solutions and are easy to implement. We also describe the design of algorithms to solve the corresponding optimization problem exactly. We experimentally evaluated ten variants of the algorithms to determine the impact and trade-offs of several reduction rules. We show that our approach provides tractability for a larger range of values of the parameter and larger inputs, improving the execution time by several orders of magnitude with respect to earlier algorithms that use less rules.

FPT-ALGORITHMS FOR MINIMUM-BENDS TOURS

This paper discusses the κ-BENDS TRAVELING SALESMAN PROBLEM. In this NP-complete problem, the inputs are n points in the plane and a positive integer κ, and we are asked whether we can travel in straight lines through these n points with at most κ bends. There are a number of applications where minimizing the number of bends in the tour is desirable because bends are considered very costly. We prove that this problem is fixed-parameter tractable (FPT). The proof is based on the kernelization approach. We also consider the RECTILINEAR κ-BENDS TRAVELING SALESMAN PROBLEM, which requires that the line-segments be axis-parallel.1 Note that a rectilinear tour with κ bends is a cover with κ-line segments, and therefore a cover by lines. We introduce two types of constraints derived from the distinction between line-segments and lines. We derive FPT-algorithms with different techniques and improved time complexity for these cases.

NC algorithms for the single most vital edge problem with respect to all pairs shortest paths

For a weighted, undirected graph G = (V, E) where |V| = n and |E| = m, we examine the single most vital edge with respect to two measurements related to all-pairs shortest paths (APSP). The first measurement considers only the impact of the removal of a single edge from the APSP on the shortest distance between each vertex pair. The second considers the total weight of all the edges which make up the APSP, that is, calculate the sum of the distance between each vertex pair after the deletion of a tree edge. We give a sequential algorithm for this problem, and show how to obtain an NC algorithm running in O(log n) time using mn2 processors and O(mn2) space on the MINIMUM CROW PRAM. Given the shortest distance between each pair of vertices u and v, the diameter of the graph is defined as the longest of these distances. The Most vital edge with respect to the diameter is the edge lying on such a u - v shortest path which when removed causes the greatest increase in the diameter. We show how to modify the above algorithm to solve this problem using the same time and number of processors. Both algorithms compare favourably with the straightforward solution which simply recalculates the all pairs shortest path information.

The rectilinear k-bends TSP

We study a hard geometric problem. Given n points in the plane and a positive integer k, the RECTILINEAR k-BENDS TRAVELLING Salesman Problem asks if there is a piecewise linear tour through the n points with at most k bends where every line-segment in the path is either horizontal or vertical. The problem has applications in VLSI design. We prove that this problem belongs to the class FPT (fixedparameter tractable). We give an algorithm that runs in O(kn2 + k4kn) time by kernelization. We present two variations on the main result. These variations are derived from the distinction between line-segments and lines. Note that a rectilinear tour with k bends is a cover with k line-segments, and therefore a cover by lines. We derive FPT-algorithms using bounded-search

A parallel algorithm for the minimum spanning tree on an SIMD machine

A parallel algorithm for constructing a minimum spanning tree of a connected, weighted, undirected graph in O(log <italic>m</italic>) time using O(<italic>m + </italic>) processors is presented for an SIMD machine where <italic>m</italic> and <italic>n</italic> denote the number of edges and vertices respectively.