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We say a graph is locally P if the induced graph on the neighbourhood of every vertex has the property P. Specically, a graph is locally traceable (LT) or locally hamiltonian (LH) if the induced graph on the neighbourhood of every vertex is traceable or hamiltonian, respectively. A locally locally hamiltonian (L2H) graph is a graph in which the graph induced by the neighbourhood of each vertex is an LH graph. This concept is generalized to an arbitrary degree of nesting, to make it possible to work with LkH graphs. This thesis focuses on the global cycle properties of LT, LH and LkH graphs. Methods are developed to construct and combine such graphs to create others with desired properties. It is shown that with the exception of three graphs, LT graphs with maximum degree no greater than 5 are fully cycle extendable (and hence hamiltonian), but the Hamilton cycle problem for LT graphs with maximum degree 6 is NP-complete. Furthermore, the smallest nontraceable LT graph has order 10, and the smallest value of the maximum degree for which LT graphs can be nontraceable is 6. It is also shown that LH graphs with maximum degree 6 are fully cycle extendable, and that there exist nonhamiltonian LH graphs with maximum degree 9 or less for all orders greater than 10. The Hamilton cycle problem is shown to be NP-complete for LH graphs with maximum degree 9. The construction of r-regular nonhamiltonian graphs is demonstrated, and it is shown that the number of vertices in a longest path in an LH graph can contain a vanishing fraction of the vertices of the graph. NP-completeness of the Hamilton cycle problem for LkH graphs for higher values of k is also investigated. / Mathematical Sciences / D. Phil. (Mathematics)

Artificial neural networks (ANNs) were originally inspired by networks of biological neurons and the interactions present in networks of these neurons. The recent revival of interest in ANNs has again focused attention on the apparent ability of ANNs to solve difficult problems, such as machine vision, in novel ways. There are many types of ANNs which differ in architecture and learning algorithms, and the list grows annually. This study was restricted to feed-forward architectures and Backpropagation- like (BP-like) learning algorithms. However, it is well known that the learning problem for such networks is NP-complete. Thus generative and incremental learning algorithms, which have various advantages and to which the NP-completeness analysis used for BP-like networks may not apply, were also studied. Various algorithms were investigated and the performance compared. Finally, the better algorithms were applied to a number of problems including music composition, image binarization and navigation and goal satisfaction in an artificial environment. These tasks were chosen to investigate different aspects of ANN behaviour. The results, where appropriate, were compared to those resulting from non-ANN methods, and varied from poor to very encouraging. / Thesis (M.Sc.)-University of Natal, Durban, 1992.

In this thesis, we investigate the complexity of the membership problem for pattern languages. A pattern is a string over the union of the alphabets A and X, where X := {x_1, x_2, x_3, ...} is a countable set of variables and A is a finite alphabet containing terminals (e.g., A := {a, b, c, d}). Every pattern, e.g., p := x_1 x_2 a b x_2 b x_1 c x_2, describes a pattern language, i.e., the set of all words that can be obtained by uniformly substituting the variables in the pattern by arbitrary strings over A. Hence, u := cacaaabaabcaccaa is a word of the pattern language of p, since substituting cac for x_1 and aa for x_2 yields u. On the other hand, there is no way to obtain the word u' := bbbababbacaaba by substituting the occurrences of x_1 and x_2 in p by words over A. The problem to decide for a given pattern q and a given word w whether or not w is in the pattern language of q is called the membership problem for pattern languages. Consequently, (p, u) is a positive instance and (p, u') is a negative instance of the membership problem for pattern languages. For the unrestricted case, i.e., for arbitrary patterns and words, the membership problem is NP-complete. In this thesis, we identify classes of patterns for which the membership problem can be solved efficiently. Our first main result in this regard is that the variable distance, i.e., the maximum number of different variables that separate two consecutive occurrences of the same variable, substantially contributes to the complexity of the membership problem for pattern languages. More precisely, for every class of patterns with a bounded variable distance the membership problem can be solved efficiently. The second main result is that the same holds for every class of patterns with a bounded scope coincidence degree, where the scope coincidence degree is the maximum number of intervals that cover a common position in the pattern, where each interval is given by the leftmost and rightmost occurrence of a variable in the pattern. The proof of our first main result is based on automata theory. More precisely, we introduce a new automata model that is used as an algorithmic framework in order to show that the membership problem for pattern languages can be solved in time that is exponential only in the variable distance of the corresponding pattern. We then take a closer look at this automata model and subject it to a sound theoretical analysis. The second main result is obtained in a completely different way. We encode patterns and words as relational structures and we then reduce the membership problem for pattern languages to the homomorphism problem of relational structures, which allows us to exploit the concept of the treewidth. This approach turns out be successful, and we show that it has potential to identify further classes of patterns with a polynomial time membership problem. Furthermore, we take a closer look at two aspects of pattern languages that are indirectly related to the membership problem. Firstly, we investigate the phenomenon that patterns can describe regular or context-free languages in an unexpected way, which implies that their membership problem can be solved efficiently. In this regard, we present several sufficient conditions and necessary conditions for the regularity and context-freeness of pattern languages. Secondly, we compare pattern languages with languages given by so-called extended regular expressions with backreferences (REGEX). The membership problem for REGEX languages is very important in practice and since REGEX are similar to pattern languages, it might be possible to improve algorithms for the membership problem for REGEX languages by investigating their relationship to patterns. In this regard, we investigate how patterns can be extended in order to describe large classes of REGEX languages.

In the 1970's computer scientists developed the theory of computational complexity. Some problems seemed hard-to-compute, while others were easy. It turned out that many of the hard problems were equally hard in a way that could be precisely specified. They became known as the NP-complete problems. The SATISFIABILITY problem (SAT) was the first problem to be proved NP-complete in 1971. Since then numerous other hard-to-solve problems have been proved to be in NP-complete. In this paper we will examine the problem of how to find a maximum independent set of vertices for a graph. This problem is known as Maximum Independent Set (MIS) for a graph. The corresponding decision problem for MIS is the question, given an integer K, is there a independent set with at least K vertices? This decision problem is INDEPENDENT SET (IS). The intention of this paper is to show through polynomial transformation that IS is in the set of NP-complete Problems. We intend to show that 3SAT is NP-complete and then using this fact, that IS is NP-complete.

IEEE 802.16e ¡]also known as Mobile WiMAX¡^ is currently the international MAC ¡]medium access control¡^ standard for wireless metropolitan area networks. To enhance the network throughput and extend the coverage of base station, IEEE then defined the 802.16j standard. Clearly, one of the popular applications for WiMAX is the multicast service. On the other hand, the design of power saving technologies is important since mobile stations are often powered by batteries. In this thesis, we study the maximum energy-efficient multicast scheduling ¡]MEMS¡^ problem for an IEEE 802.16j network with transparent mode. Specifically, the base station should determine how to schedule the multicast data in a multicast superframe such that the multicast energy efficiency of network is maximal. We first prove that the MEMS problem is NP-complete. Then on the basis of SMBC-AMC, we propose its variant, called SMBC-relay, to solve this problem. However, in SMBC-relay, the base station may send the same multicast data several times, wasting the scarce bandwidth. Hence we we propose a genetic algorithm-based multicast scheduling algorithm, called GAMS. One of the key features of GAMS is that the base station can control when to terminate the algorithm by stopping the evolution at any time. Simulation results show that GAMS significantly outperforms SMBC-relay in terms of multicast energy efficiency.

Bin packing is a problem with many applications in various industries. This thesis addresses a specific instance of the this problem, known as the Container Packing problem. Special attention is paid to the Pallet Loading problem which is a restricted sub-problem of the general Container Packing problem. Since the Bin Packing problem is NP-complete, it is customary to apply a heuristic measure in order to approximate solutions in a reasonable amount of computation time rather than to attempt to produce optimal results by applying some exact algorithm. Several heuristics are examined for the problems under consideration, and the results produced by each are shown and compared where relevant. / Thesis (M.Sc.)-University of Natal, Durban, 1991.

Thesis (M.S.)--State University of New York at Binghamton, Thomas J. Watson School of Engineering and Applied Science, Department of Computer Science, 2009. / Includes bibliographical references.

Title: The knapsack and its applications Author: Romana Linkeová Department: Department of Algebra Supervisor: doc. Mgr. Pavel Příhoda, Ph.D., Department of Algebra Abstract: This thesis is focused on various aspects of cryptosystems based on NP (non-deterministic polynomial) complete knapsack problem. From the theory of complexity point of view, the less known parts of the proof of knapsack problem NP completeness are shown in detail. From the cryptographical point of view, a demonstration of breaking of the Merkle-Hellman cryptosystem (the basic de- sign of knapsack-type cryptosystems) is provided, showing that poor parameters choice can lead to easy obtaining of the whole private key. Another contribution of this thesis consists in a presented proposal of a new cryptosystem concept based on the matrix 0-1 knapsack problem. This concept was developed in order to prevent known attacks, however, in the thesis we provide a proof analogous to J. C. Lagarias and A. M. Odlyzko, 1985, which shows that an attack based on the LLL algorithm will be successful on the majority of the matrix 0-1 kna- psack problem cryptosystems. Finally, a list of modern cryptosystems based on the knapsack problem is provided and a cryptanalysis thereof is given. Keywords: knapsack problem, NP complete problems, LLL algorithm 1