The axial map of a town is one of the key components of the space syntax method – a tool for analysing urban layout. It is derived by placing the longest and fewest lines, called axial lines, to cross the adjacencies between convex polygons in a convex map of a town. Previous research has shown that placing axial lines to cross the adjacencies between a collection of convex polygons is NP-complete, even when the convex polygons are restricted to rectangles and the axial lines to have orthogonal orientation. In this document, we show that placing orthogonal axial lines in orthogonal rectangles where the adjacencies between the rectangles are restricted to be crossed only once (ALPSC- OLOR) is NP-complete. As a result, we infer the single adjacency crossing version of the general axial line placement problem is NP-complete. The transformation of NPcompleteness of ALP-SC-OLOR is from vertex cover for biconnected planar graphs. A heuristic is then presented that gives a reasonable approximate solution to ALP-SC-OLOR based on a greedy method.

Kwok-Shing Ho. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2002. / Includes bibliographical references (leaves 69-71). / Abstracts in English and Chinese. / List of Figures --- p.vi / List of Tables --- p.vii / Chapter Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Overview --- p.1 / Chapter 1.2 --- Thesis Objectives --- p.6 / Chapter 1.3 --- Outline of Thesis --- p.8 / Chapter Chapter 2 --- The Spare Capacity Planning Problem --- p.9 / Chapter 2.1 --- Mathematical Model of the Spare Capacity Planning Problem --- p.12 / Chapter 2.1.1 --- Variable Definitions --- p.12 / Chapter 2.1.2 --- Objective Function and Constraints --- p.15 / Chapter 2.1.3 --- Complexity --- p.17 / Chapter 2.2 --- Greedy Algorithm - Spare Capacity Allocation and Planning Estimator (SCAPE) --- p.19 / Chapter 2.2.1 --- Working Principle of SCAPE --- p.20 / Chapter 2.2.2 --- Implementation of SCAPE --- p.22 / Chapter 2.2.3 --- Improved SCAPE --- p.23 / Chapter 2.3 --- Experimental Results and Discussion --- p.27 / Chapter 2.3.1 --- Experimental Platform --- p.27 / Chapter 2.3.2 --- Experiment about Accuracy of SCAPE --- p.27 / Chapter 2.3.3 --- Experiment about Minimization of Network Spare Capacity --- p.30 / Chapter 2.3.4 --- Experiment about Minimization of Network Spare Cost --- p.35 / Chapter 2.4 --- Conclusions --- p.38 / Chapter Chapter 3 --- Survivable All-Optical Network Design Problem --- p.39 / Chapter 3.1 --- Mathematical Model of the Survivable Network Design Problem --- p.42 / Chapter 3.2 --- Optimization Algorithms for Survivable Network Design Problem --- p.44 / Chapter 3.2.1 --- Modified Drop Algorithm (MDA) --- p.45 / Chapter 3.2.1.1 --- Drop Algorithm Introduction --- p.45 / Chapter 3.2.1.2 --- Network Design with MDA --- p.45 / Chapter 3.2.2 --- Genetic Algorithm --- p.47 / Chapter 3.2.2.1 --- Genetic Algorithm Introduction --- p.47 / Chapter 3.2.2.2 --- Network Design with GA --- p.48 / Chapter 3.2.3 --- Complexity of MDA and GA --- p.51 / Chapter 3.3 --- Experimental Results and Discussion --- p.52 / Chapter 3.3.1 --- Experimental Platform --- p.52 / Chapter 3.3.2 --- Experiment about Accuracy of MDA and GA --- p.52 / Chapter 3.3.3 --- Experiment about Principle of Survivable Network Design --- p.55 / Chapter 3.3.4 --- Experiment about Performance of MDA and GA --- p.58 / Chapter 3.4 --- Conclusions --- p.62 / Chapter Chapter 4 --- Conclusions and Future Work --- p.63 / Appendix A The Interference Heuristic for the path restoration scheme --- p.66 / Bibliography --- p.69 / Publications --- p.72

by Lui Oi Lun Irene. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2001. / Includes bibliographical references (leaves 128-132). / Abstracts in English and Chinese. / List of Figures --- p.vii / List of Tables --- p.viii / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Satisfiability Problem --- p.1 / Chapter 1.2 --- Motivation of the Research --- p.1 / Chapter 1.3 --- Overview of the Thesis --- p.2 / Chapter 2 --- Constraint Satisfaction Problem and Satisfiability Problem --- p.4 / Chapter 2.1 --- Constraint Programming --- p.4 / Chapter 2.2 --- Satisfiability Problem --- p.6 / Chapter 2.3 --- Methods in Solving SAT problem --- p.7 / Chapter 2.3.1 --- Davis-Putnam-Loveland Procedure --- p.7 / Chapter 2.3.2 --- SATZ by Chu-Min Li --- p.8 / Chapter 2.3.3 --- Local Search for SAT --- p.11 / Chapter 2.3.4 --- Integer Linear Programming Method for SAT --- p.12 / Chapter 2.3.5 --- Semidefinite Programming Method --- p.13 / Chapter 2.4 --- Softwares for SAT --- p.15 / Chapter 2.4.1 --- SAT01 --- p.15 / Chapter 2.4.2 --- "SATZ and SATZ213, contributed by Chu-Min Li" --- p.15 / Chapter 2.4.3 --- Others --- p.15 / Chapter 3 --- Integer Programming --- p.17 / Chapter 3.1 --- Introduction --- p.17 / Chapter 3.1.1 --- Formulation of IPs and BIPs --- p.18 / Chapter 3.1.2 --- Binary Search Tree --- p.19 / Chapter 3.2 --- Methods in Solving IP problem --- p.19 / Chapter 3.2.1 --- Branch-and-Bound Method --- p.20 / Chapter 3.2.2 --- Cutting-Plane Method --- p.23 / Chapter 3.2.3 --- Duality in Integer Programming --- p.26 / Chapter 3.2.4 --- Heuristic Algorithm --- p.28 / Chapter 3.3 --- Zero-one Optimization and Continuous Relaxation --- p.29 / Chapter 3.3.1 --- Introduction --- p.29 / Chapter 3.3.2 --- The Roof Dual expressed in terms of Lagrangian Relaxation --- p.30 / Chapter 3.3.3 --- Determining the Existence of a Duality Gap --- p.31 / Chapter 3.4 --- Software for solving Integer Programs --- p.33 / Chapter 4 --- Integer Programming Formulation for SAT Problem --- p.35 / Chapter 4.1 --- From 3-CNF SAT Clauses to Zero-One IP Constraints --- p.35 / Chapter 4.2 --- From m-Constrained IP Problem to Singly-Constrained IP Problem --- p.38 / Chapter 4.2.1 --- Example --- p.39 / Chapter 5 --- A Basic Branch-and-Bound Algorithm for the Zero-One Polynomial Maximization Problem --- p.42 / Chapter 5.1 --- Reason for choosing Branch-and-Bound Method --- p.42 / Chapter 5.2 --- Searching Algorithm --- p.43 / Chapter 5.2.1 --- Branch Rule --- p.44 / Chapter 5.2.2 --- Bounding Rule --- p.46 / Chapter 5.2.3 --- Fathoming Test --- p.46 / Chapter 5.2.4 --- Example --- p.47 / Chapter 6 --- Revised Bound Rule for Branch-and-Bound Algorithm --- p.55 / Chapter 6.1 --- Revised Bound Rule --- p.55 / Chapter 6.1.1 --- CPLEX --- p.57 / Chapter 6.2 --- Example --- p.57 / Chapter 6.3 --- Conclusion --- p.65 / Chapter 7 --- Revised Branch Rule for Branch-and-Bound Algorithm --- p.67 / Chapter 7.1 --- Revised Branch Rule --- p.67 / Chapter 7.2 --- Comparison between Branch Rule and Revised Branch Rule --- p.69 / Chapter 7.3 --- Example --- p.72 / Chapter 7.4 --- Conclusion --- p.73 / Chapter 8 --- Experimental Results and Analysis --- p.80 / Chapter 8.1 --- Experimental Results --- p.80 / Chapter 8.2 --- Statistical Analysis --- p.33 / Chapter 8.2.1 --- Analysis of Search Techniques --- p.83 / Chapter 8.2.2 --- Discussion of the Performance of SATZ --- p.85 / Chapter 9 --- Concluding Remarks --- p.87 / Chapter 9.1 --- Conclusion --- p.87 / Chapter 9.2 --- Suggestions for Future Research --- p.88 / Chapter A --- Searching Procedures for Solving Constraint Satisfaction Problem (CSP) --- p.91 / Chapter A.1 --- Notation --- p.91 / Chapter A.2 --- Procedures for Solving CSP --- p.92 / Chapter A.2.1 --- Generate and Test --- p.92 / Chapter A.2.2 --- Standard Backtracking --- p.93 / Chapter A.2.3 --- Forward Checking --- p.94 / Chapter A.2.4 --- Looking Ahead --- p.95 / Chapter B --- Complete Results for Experiments --- p.96 / Chapter B.1 --- Complete Result for SATZ --- p.96 / Chapter B.1.1 --- n =5 --- p.95 / Chapter B.1.2 --- n = 10 --- p.98 / Chapter B.1.3 --- n = 30 --- p.99 / Chapter B.2 --- Complete Result for Basic Branch-and-Bound Algorithm --- p.101 / Chapter B.2.1 --- n二5 --- p.101 / Chapter B.2.2 --- n = 10 --- p.104 / Chapter B.2.3 --- n = 30 --- p.107 / Chapter B.3 --- Complete Result for Revised Bound Rule --- p.109 / Chapter B.3.1 --- n = 5 --- p.109 / Chapter B.3.2 --- n = 10 --- p.112 / Chapter B.3.3 --- n = 30 --- p.115 / Chapter B.4 --- Complete Result for Revised Branch-and-Bound Algorithm --- p.118 / Chapter B.4.1 --- n = 5 --- p.118 / Chapter B.4.2 --- n = 10 --- p.121 / Chapter B.4.3 --- n = 30 --- p.124 / Bibliography --- p.128

Metric spaces, which generalise the properties of commonly-encountered physical and abstract spaces into a mathematical framework, frequently occur in computer science applications. Three major kinds of questions about metric spaces are considered here: the intrinsic dimensionality of a distribution, the maximum number of distance permutations, and the difficulty of reverse similarity search. Intrinsic dimensionality measures the tendency for points to be equidistant, which is diagnostic of high-dimensional spaces. Distance permutations describe the order in which a set of fixed sites appears while moving away from a chosen point; the number of distinct permutations determines the amount of storage space required by some kinds of indexing data structure. Reverse similarity search problems are constraint satisfaction problems derived from distance-based index structures. Their difficulty reveals details of the structure of the space. Theoretical and experimental results are given for these three questions in a wide range of metric spaces, with commentary on the consequences for computer science applications and additional related results where appropriate.

Metric spaces, which generalise the properties of commonly-encountered physical and abstract spaces into a mathematical framework, frequently occur in computer science applications. Three major kinds of questions about metric spaces are considered here: the intrinsic dimensionality of a distribution, the maximum number of distance permutations, and the difficulty of reverse similarity search. Intrinsic dimensionality measures the tendency for points to be equidistant, which is diagnostic of high-dimensional spaces. Distance permutations describe the order in which a set of fixed sites appears while moving away from a chosen point; the number of distinct permutations determines the amount of storage space required by some kinds of indexing data structure. Reverse similarity search problems are constraint satisfaction problems derived from distance-based index structures. Their difficulty reveals details of the structure of the space. Theoretical and experimental results are given for these three questions in a wide range of metric spaces, with commentary on the consequences for computer science applications and additional related results where appropriate.

In this thesis, we proposed a simple yet novel multicast scheduling scheme for IEEE 802.16e wireless metropolitan area networks. Specifically, we want to solve the problem that how the base station schedules data messages in a multicast superframe such that mobile stations can receive their required multicast data and the total awake time of mobile stations is minimal. We first prove that this problem is NP-complete, and then propose a greedy k-approximation algorithm, named G-EEMS, whose running time is , where n is the total number of multicast data messages and k is the size of MBS (multicast and broadcast service) zone in a frame. Simulation results show that, in terms of energy throughput, G-EEMS significantly outperforms the existing scheme, called SMBC-D.

published_or_final_version / abstract / toc / Mathematics / Master / Master of Philosophy

Given graphs G,H, is it possible to find a matching which, when deleted from G, destroys all copies of H? The answer is obvious for some inputs—notably, when G is a large complete graph the answer is “no”—but in general this can be a very difficult question. In this thesis, we study this decision problem when H is a fixed tree or cycle; our aim is to identify those H for which it can be solved efficiently. The H-transverse matching problem, TM(H) for short, asks whether an input graph admits a matching M such that no subgraph of G − M is isomorphic to H. The main goal of this thesis is the following dichotomy. When H is a triangle or one of a few small-diameter trees, there is a polynomial-time algorithm to find an H-transverse matching if one exists. However, TM(H) is NP-complete when H is any longer cycle or a tree of diameter ≥ 4. In addition, we study the restriction of these problems to structured graph classes. / Graduate

Constraint programming is an invaluable tool for solving many of the complex NP-complete problems that we need solutions to. These problems can be easily described as Constraint Satisfaction Problems (CSPs) and then passed to constraint solvers: complex pieces of software written to solve general CSPs efficiently. Many of the problems we need solutions to are real world problems: planning (e.g. vehicle routing), scheduling (e.g. job shop schedules) and timetabling problems (e.g. staff rotas) to name but a few. In the real world, we place structure on objects to make them easier to deal with. This manifests itself as symmetry. The symmetry in these real world problems make them easier to deal with for humans. However, they lead to a great deal of redundancy when using computational methods of problem solving. Thus, this thesis examines some of the many aspects of utilising the symmetry of CSPs to reduce the amount of computation needed by constraint solvers. In this thesis we look at the ease of use of previous symmetry breaking methods. We introduce a new and novel method of describing the symmetries of CSPs. We look at previous methods of symmetry breaking and show how we can drastically reduce their computation while still breaking all symmetry. We give the first detailed investigation into the behaviour of breaking only subsets of all symmetry. We look at how this affects the performance of constraint solvers before discovering the properties of a good symmetry. We then present an original method for choosing the best symmetries to use. Finally, we look at areas of redundant computation in constraint solvers that no other research has examined. New ways of dealing with this redundancy are proposed with results of an example implementation which improves efficiency by several orders of magnitude.

Machine Learning Techniques have been used on a wide array of input types: images, sound waves, text, and so forth. In articulating these input types to the almighty machine, there have been all sorts of amazing problems that have been solved for many practical purposes. Nevertheless, there are some input types which don’t lend themselves nicely to the standard set of machine learning tools we have. Moreover, there are some provably difficult problems which are abysmally hard to solve within a reasonable time frame. This thesis addresses several of these difficult problems. It frames these problems such that we can then attempt to marry the allegedly powerful utility of existing machine learning techniques to the practical solvability of said problems.