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Given an undirected edge-weighted graph G and a positive integer m, the Constrained Forest Problem (CFP) seeks a lowest-cost (or minimum-weight) forest F which spans G while satisfying the requirement that each tree in F contain at least m vertices. This problem has been shown to be NP-hard for values of m greater than three, giving rise to a number of approximation strategies for finding reasonable m-forest solutions. This research presents a new genetic algorithm (GA) which can consistently find equal-or-better solutions to the problem when compared to non-genetic alternatives. This GA is unique in that it uses chromosomes which are actual candidate solutions (m-forests) and performs genetic operations (random creation, selection, recombination, and mutation) on these candidate solutions. Experiments were run using 180 different GA configurations on 50 benchmark graphs to determine which operators and techniques would be most successful in solving the m-forest problem. The "heaviest edge first" or HEF algorithm run against the minimum spanning tree (MST) of a graph was used as a performance metric. Previously, the HEF(MST) algorithm had been shown to produce the best results on m-forest problems. When the GA was able to find better results than HEF(MST) on the same problem instance, this was considered a GA success. Since the GA's initial population included heuristic candidate solutions such as HEF(MST), the GA never did worse than the best of these. GA solution quality did vary, however, often finding several different better-than-HEF(MST) solutions, illustrating the stochastic nature of the process. Based on data collected from the 9000 initial problem instances, several factors were shown to significantly improve the quality of the GA solution. Problem instances which did not include mutation had a much lower success rate than those which did. Adding calculated heuristic solutions such as HEF(MST) to the initial population allowed the GA to converge more quickly and improved its likelihood of finding better-than-HEF(MST) solutions. Building an initial population using randomly-generated candidate solutions whose edges were restricted to the problem graph's MST proved equally successful. GA configuration options were analyzed using all 9000 test cases and again using only those 403 cases in which the GA was able to find the very best solution for each graph. These analyses were consistent, and resulted in the identification of a single "best" GA configuration which combined the best overall initial population strategy, random seeding algorithms, mutation and crossover strategy. The selected configuration was then further tested using various values of m to ensure that the resulting GA could in fact find better-than-HEF(MST) solutions for the majority of problem instances.

NP-hard problems have numerous applications in various fields such as networks, computer systems, circuit design, etc. However, no efficient algorithms have been found for NP-hard problems. It has been commonly believed that no efficient algorithms for NP-hard problems exist, i.e., that P6=NP. Recently, it has been observed that there are parameters much smaller than input sizes in many instances of NP-hard problems in the real world. In the last twenty years, researchers have been interested in developing efficient algorithms, i.e., fixed-parameter tractable algorithms, for those instances with small parameters. Fixed-parameter tractable algorithms can practically find exact solutions to problem instances with small parameters, though those problems are considered intractable in traditional computational theory. In this dissertation, we propose a new approach of algorithm design and analysis: discovering better measures for problems. In particular we use two measures instead of the traditional single measure?input size to design algorithms and analyze their time complexity. For several classical NP-hard problems, we present improved algorithms designed and analyzed with this new approach, First we show that the new approach is extremely powerful for designing fixedparameter tractable algorithms by presenting improved fixed-parameter tractable algorithms for the 3D-matching and 3D-packing problems, the multiway cut problem, the feedback vertex set problems on both directed and undirected graph and the max-leaf problems on both directed and undirected graphs. Most of our algorithms are practical for problem instances with small parameters. Moreover, we show that this new approach is also good for designing exact algorithms (with no parameters) for NP-hard problems by presenting an improved exact algorithm for the well-known satisfiability problem. Our results demonstrate the power of this new approach to algorithm design and analysis for NP-hard problems. In the end, we discuss possible future directions on this new approach and other approaches to algorithm design and analysis.

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A flexible workforce or a versatile machine is employed to perform various types of operations. Often these resources are associated with setups. Whenever a worker or machine switches from processing one type of operation to another a setup time may be required although several operations of a same type can be processed in succession after a single setup. The presence of setups gives rise to the problem of choosing batch sizes that are neither too large nor too small. In the last one and a half decade, many researchers have addressed the problem of scheduling with batching. A majority of articles assumes that there is only one type of scarce resource, which is typically machine. Often there can be two scarce resources such as a worker and a machine or a machine and a tool. We propose a resource constrained scheduling model with a single operator and two or more machines. Whenever the operator changes machine, a setup time is required that may be sequence dependent or sequence independent. We consider the two cases of an open shop and a flow shop. In the open shop case, the order in which a job visits the machines is unrestricted. In the flow shop case, every job must visit the machines in the same order. We consider various scheduling objectives. For variable number of machines, many cases are intractable. We discuss some dominance properties that narrow down the search for an optimal schedule. We present a dynamic programming approach which solves a large number of cases. The running time of the dynamic program is polynomial for a fixed number of machines. For the case of two machines, we show that the dominance properties have a nice interpretation. We develop some algorithms and justify their use by establishing running times, comparing the running times with those of the existing algorithms, and testing the performance of the algorithms.

An unknown input observer provides perfect asymptotic tracking of the state of a system affected by unknown inputs. Such an observer exists (possibly requiring a delay in estimation) if and only if the system satisfies a property known as strong detectability. In this thesis, we consider the problem of selecting (at design-time) a minimum cost subset of sensors from a given set to make a given system strongly detectable. We show this problem is NP-hard even when the system is stable. Furthermore, we show it is not possible to approximate the minimum cost within a factor of log(n) in polynomial-time (unless P=NP). However, we prove if a given system (with a selected set of sensors) is already strongly detectable, finding the smallest set of additional sensors to install to obtain a zero-delay observer can be done in polynomial time. Next we consider the problem of attacking a set of deployed sensors to remove the property of strong detectability. We show finding the smallest number of sensors to remove is NP-hard. Lastly through simulations, we analyze two greedy approaches for approximating the strong detectability sensor selection problem.

No description available.

A flexible workforce or a versatile machine is employed to perform various types of operations. Often these resources are associated with setups. Whenever a worker or machine switches from processing one type of operation to another a setup time may be required although several operations of a same type can be processed in succession after a single setup. The presence of setups gives rise to the problem of choosing batch sizes that are neither too large nor too small. In the last one and a half decade, many researchers have addressed the problem of scheduling with batching. A majority of articles assumes that there is only one type of scarce resource, which is typically machine. Often there can be two scarce resources such as a worker and a machine or a machine and a tool. We propose a resource constrained scheduling model with a single operator and two or more machines. Whenever the operator changes machine, a setup time is required that may be sequence dependent or sequence independent. We consider the two cases of an open shop and a flow shop. In the open shop case, the order in which a job visits the machines is unrestricted. In the flow shop case, every job must visit the machines in the same order. We consider various scheduling objectives. For variable number of machines, many cases are intractable. We discuss some dominance properties that narrow down the search for an optimal schedule. We present a dynamic programming approach which solves a large number of cases. The running time of the dynamic program is polynomial for a fixed number of machines. For the case of two machines, we show that the dominance properties have a nice interpretation. We develop some algorithms and justify their use by establishing running times, comparing the running times with those of the existing algorithms, and testing the performance of the algorithms.

Parameterized NP-hard problems are NP-hard problems that are associated with special variables called parameters. One example of the problem is to find simple paths of length k in a graph, where the integer k is the parameter. We call this problem the p-path problem. The p-path problem is the parameterized version of the well-known NP-complete problem - the longest simple path problem. There are two main reasons why we study parameterized NP-hard problems. First, many application problems are naturally associated with certain parameters. Hence we need to solve these parameterized NP-hard problems. Second, if parameters take only small values, we can take advantage of these parameters to design very effective algorithms. If a parameterized NP-hard problem can be solved by an algorithm of running time in form of f(k)nO(1), where k is the parameter, f(k) is independent of n, and n is the input size of the problem instance, we say that this parameterized NP-hard problem is fixed parameter tractable (FPT). If a problem is FPT and the parameter takes only small values, the problem can be solved efficiently (it can be solved almost in polynomial time). In this dissertation, first, we introduce several techniques that can be used to design efficient algorithms for parameterized NP-hard problems. These techniques include branch and bound, divide and conquer, color coding and dynamic programming, iterative compression, iterative expansion and kernelization. Then we present our results about how to use these techniques to solve parameterized NP-hard problems, such as the p-path problem and the pd-feedback vertex set problem. Especially, we designed the first algorithm of running time in form of f(k)nO(1) for the pd-feedback vertex set problem. Thus solved an outstanding open problem, i.e. if the pd-feedback vertex set problem is FPT. Finally, we will introduce how to use parameterized algorithm techniques to solve the signaling pathway problem and the motif finding problem from bioinformatics.

According to the theory of NPcompleteness, many problems that have important realworld applications are NPhard. This excludes the possibility of solving them in polynomial time unless P=NP. A number of approaches have been proposed in dealing with NPhard problems, among them are approximation algorithms and parameterized algorithms. The study of approximation algorithms tries to find good enough solutions instead of optimal solutions in polynomial time, while parameterized algorithms try to give exact solutions when a natural parameter is small. In this thesis, we study the structural properties of parameterized computation and approximation algorithms for NP optimization problems. In particular, we investigate the relationship between parameterized complexity and polynomialtime approximation scheme (PTAS) for NP optimization problems. We give nice characterizations for two important subclasses in PTAS: Fully Polynomial Time Approximation Scheme (FPTAS) and Effcient Polynomial Time Approximation Scheme (EPTAS), using the theory of parameterized complexity. Our characterization of the class FPTAS has its advantages over the former characterizations, and our characterization of EPTAS is the first systematic investigation of this new but important approximation class. We develop new techniques to derive strong computational lower bounds for certain parameterized problems based on the theory of parameterized complexity. For example, we prove that unless an unlikely collapse occurs in parameterized complexity theory, the clique problem could not be solved in time O(f (k)no(k)) for any function f . This lower bound matches the upper bound of the trivial algorithm that simply enumerates and checks all subsets of k vertices in the given graph of n vertices. We then extend our techniques to derive computational lower bounds for PTAS and EPTAS algorithms of NP optimization problems. We prove that certain NP optimization problems with known PTAS algorithms have no PTAS algorithms of running time O(f (1/Epsilon)no(1/Epsilon)) for any function f . Therefore, for these NP optimization problems, although theoretically they can be approximated in polynomial time to an arbitrarily small error bound Epsilon, they have no practically effective approximation algorithms for small error bound Epsilon. To our knowledge, this is the first time such lower bound results have been derived for PTAS algorithms. This seems to open a new direction for the study of computational lower bounds on the approximability of NP optimization problems.