Browsing by Author "Chen, Jianer"
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Item A Topological Theory of Weaving and Its Applications in Computer Graphics(2013-07-24) Hu, ShiyuRecent advances in the computer graphics of woven images on surfaces in 3-space motivate the development of weavings for arbitrary genus surfaces. We present herein a general framework for weaving structures on general surfaces in 3-space, and through it, we demonstrate how weavings on such surfaces are inducible from connected graph imbeddings on the same surfaces. The necessary and sufficient conditions to identify the inducible weavings in our framework are also given. For low genus surfaces, like plane and torus, we extend our framework to the weavings which are inducible from disconnected imbedded graphs. In particular, we show all weavings on a plane are inducible in our framework, including most Celtic Knots. Moreover, we study different weaving structures on general surfaces in 3-space based on our framework. We show that any weaving inducible in our framework can be converted into an alternating weaving by appropriately changing the strand orders at some crossings. By applying a topological surgery operation, called doubling operation, we can refine a weaving or convert certain non-twillable weavings into twillable weavings on the same surfaces. Interestingly, two important subdivision algorithms on graphs imbeddings, the Catmull-Clark and Doo-Sabin algorithms, correspond nicely to our doubling operation on induced weavings. Another technique we used in studying weaving structures is repetitive patterns. A weaving that can be converted into a twillable weaving by our doubling operation has a highly-symmetric structure, which consists of only two repetitive patterns. An extension of the symmetric structure leads to Quad-Pattern Coverable meshes, which can be seamlessly covered with only one periodic pattern. Both of these two topological structures can be represented with simple Permutation Voltage graphs. A considerable advantage of our model is that it is topological. This permits the graphic designer to superimpose strand colors and geometric attributes ? distances, angles, and curvatures ? that conform to manufacturing or artistic criteria. We also give a software example for plane weaving construction. A benefit of the software is that it supports plane weaving reconstructions from an image of a plane weaving, which could be useful for recording and modifying existing weavings in real life.Item Clustering and Inconsistent Information: A Kernelization Approach(2012-07-16) Cao, YixinClustering is the unsupervised classification of patterns into groups, which is easy provided the data of patterns are consistent. However, real data are almost always tempered with inconsistencies, which make it a hard problem, and actually, the most widely studied formulations, correlation clustering and hierarchical clustering, are both NP-hard. In the graph representation of data, inconsistencies also frequently present themselves as cycles, also called deadlocks, and to break cycles by removing vertices is the objective of the classical feedback vertex set (FVS) problem. This dissertation studies the three problems, correlation clustering, hierarchical clustering, and disjoint-FVS (a variation of FVS), from a kernelization approach. A kernelization algorithm in polynomial time reduces a problem instance provably to speed up the further processing with other approaches. For each of the problems studied, an efficient kernelization algorithm of linear or sub-quadratic running time is presented. All the kernels obtained in this dissertation have linear size with very small constants. Better parameterized algorithms are also designed based on the kernels for the last two problems. Finally, some concluding remarks on possible directions for future research are briefly mentioned.Item Critical Cliques and Their Application to Influence Maximization in Online Social Networks(2012-07-16) Pandey, NikhilGraph decompositions have useful applications in optimization problems that are categorized as NP-Hard. Modular Decomposition of a graph is a technique to decompose the graph into non-overlapping modules. A module M of an undirected graph G = (V, E) is commonly defined as a set of vertices such that any vertex outside of M is either adjacent or non-adjacent to all vertices in M . By the theory of modular decomposition, the modules can be categorized as parallel, series or prime modules. Series modules which are maximal and are also cliques are termed as simple series modules or critical cliques. There are modular decomposition algorithms that can be used to decompose the graph into modules and obtain critical cliques. In this current research, we present a new algorithm to decompose the graph into critical cliques without applying the process of modular decomposition. Given a simple, undirected graph G = (V, E), the runtime complexity of our proposed algorithm is O(|V| + |E|) under certain input constraints. Thus, one of our main contributions is to propose a novel algorithm for decomposing a simple, undirected graph directly into critical cliques. We apply the idea of critical cliques to propose a new way for solving the influence maximization problem in online social networks. Influence maximization in online social networks is the problem of identifying a small, initial set of influential individuals which can influence the maximum number of individuals in the network. In this research, we propose a new model of online social networks based on the notion of critical cliques. We utilize the properties of critical cliques to assign parameters for our proposed model and select an initial set of activation nodes. We then simulate the influence propagation process in the online social network using our proposed model and experimentally compare our approach to the greedy algorithm proposed by Kempe, Kleinberg and Tardos. Our main contribution in the influence maximization research is to propose a new model of online social network taking into account the structural properties of the social network graph and a new, faster algorithm for determining the initial set of influential individuals in the online social network.Item Cuts and Partitions in Graphs/Trees with Applications(2013-07-23) Fan, Jia-HaoBoth the maximum agreement forest problem and the multicut on trees problem are NP-hard, thus cannot be solved efficiently if P /=NP. The maximum agreement forest problem was motivated in the study of evolution trees in bioinformatics, in which we are given two leaf-labeled trees and are asked to find a maximum forest that is a subgraph of both trees. The multicuton trees problem has applications in networks, in which we are given a forest and a set of pairs of termianls and are asked to find a cut that separates all pairs of terminals. We develop combinatorial and algorithmic techniques that lead to improved parameterized algorithms, approximation algorithms, and kernelization algorithms for these problems. For the maximum agreement forest problem, we proceed from the bottommost level of trees and extend solutions to whole trees. With this technique, we show that the maxi- mum agreement forest problem is fixed-parameterized tractable in general trees, resolving an open problem in this area. We also provide the first constant ratio approximation algorithm for the problem in general trees. For the multicut on trees problem, we take a new look at the problem through the eyes of vertex cover problem. This connection allows us to develop an kernelization algorithm for the problem, which gives an upper bound of O(k3) on the kernel size, significantly improving the previous best upper bound O(k6). We further exploit this connection to give a parameterized algorithm for the problem that runs in time O? (1.62k), thus improving the previous best algorithm of running time O? (2k). In the protein complex prediction problem, which comes directly from the study of bioinformatics, we are given a protein-protein interaction network, and are asked to find dense regions in this graph. We formulate this problem as a graph clustering problem and develop an algorithm to refine the results for identifying protein complexes. We test our algorithm on yeast protein- protein interaction networks, and we show that our algorithm is able to identify complexes more accurately than other existing algorithms.Item Design and Implementation of High Performance Algorithms for the (n,k)-Universal Set Problem(2010-01-14) Luo, PingThe k-path problem is to find a simple path of length k. This problem is NP-complete and has applications in bioinformatics for detecting signaling pathways in protein interaction networks and for biological subnetwork matching. There are algorithms implemented to solve the problem for k up to 13. The fastest implementation has running time O^*(4.32^k), which is slower than the best known algorithm of running time O^*(4^k). To implement the best known algorithm for the k-path problem, we need to construct (n,k)-universal set. In this thesis, we study the practical algorithms for constructing the (n,k)-universal set problem. We propose six algorithm variants to handle the increasing computational time and memory space needed for k=3, 4, ..., 8. We propose two major empirical techniques that cut the time and space tremendously, yet generate good results. For the case k=7, the size of the universal set found by our algorithm is 1576, and is 4611 for the case k=8. We implement the proposed algorithms with the OpenMP parallel interface and construct universal sets for k=3, 4, ..., 8. Our experiments show that our algorithms for the (n,k)-universal set problem exhibit very good parallelism and hence shed light on its MPI implementation. Ours is the first implementation effort for the (n,k)-universal set problem. We share the effort by proposing an extensible universal set construction and retrieval system. This system integrates universal set construction algorithms and the universal sets constructed. The sets are stored in a centralized database and an interface is provided to access the database easily. The (n,k)-universal set have been applied to many other NP-complete problems such as the set splitting problems and the matching and packing problems. The small (n,k)-universal set constructed by us will reduce significantly the time to solve those problems.Item Duotone Surfaces: Division of a Closed Surface into Exactly Two Regions(2013-04-22) Garigipati, PradeepIn this thesis work, our main motivation is to create computer aided art work which can eventually transform into a sculpting tool. The work was inspired after Taubin?s work on constructing Hamiltonian triangle strips on quadrilateral meshes. We present an algorithm that can divide a closed 2-manifold surface into exactly two regions, bounded from each other by a single continuous curve. We show that this kind of surface division is possible only if the mesh approximation of a given object is a two colorable quadrilateral mesh. For such a quadrilateral mesh, appropriate texturing of the faces of the mesh using a pair of Truchet tiles will give us a Duotone Surface. Catmull-Clark subdivision can convert any given mesh with arbitrary sided polygons into a two colorable quadrilateral mesh. Using such vertex insertion schemes, we modify the mesh and classify the vertices of the new mesh into two sets. By appropriately texturing each face of the mesh such that the color of the vertices of the face match with the colored regions of the corresponding Truchet tile, we can get a continuous curve that splits the surface of the mesh into two regions. Now, coloring the thus obtained two regions with two different colors gives us a Duotone Surface.Item Effective algorithms and protocols for wireless networking: a topological approach(Texas A&M University, 2008-10-10) Zhang, FenghuiMuch research has been done on wireless sensor networks. However, most protocols and algorithms for such networks are based on the ideal model Unit Disk Graph (UDG) model or do not assume any model. Furthermore, many results assume the knowledge of location information of the network. In practice, sensor networks often deviate from the UDG model significantly. It is not uncommon to observe stable long links that are more than five times longer than unstable short links in real wireless networks. A more general network model, the quasi unit-disk graph (quasi-UDG) model, captures much better the characteristics of wireless networks. However, the understanding of the properties of general quasi-UDGs has been very limited, which is impeding the design of key network protocols and algorithms. In this dissertation we study the properties for general wireless sensor networks and develop new topological/geometrical techniques for wireless sensor networking. We assume neither the ideal UDG model nor the location information of the nodes. Instead we work on the more general quasi-UDG model and focus on figuring out the relationship between the geometrical properties and the topological properties of wireless sensor networks. Based on such relationships we develop algorithms that can compute useful substructures (planar subnetworks, boundaries, etc.). We also present direct applications of the properties and substructures we constructed including routing, data storage, topology discovery, etc. We prove that wireless networks based on quasi-UDG model exhibit nice properties like separabilities, existences of constant stretch backbones, etc. We develop efficient algorithms that can obtain relatively dense planar subnetworks for wireless sensor networks. We also present efficient routing protocols and balanced data storage scheme that supports ranged queries. We present algorithmic results that can also be applied to other fields (e.g., information management). Based on divide and conquer and improved color coding technique, we develop algorithms for path, matching and packing problem that significantly improve previous best algorithms. We prove that it is unlikely for certain problems in operation science and information management to have any relatively effective algorithm or approximation algorithm for them.Item Kernelization and Enumeration: New Approaches to Solving Hard Problems(2011-08-08) Meng, JieNP-Hardness is a well-known theory to identify the hardness of computational problems. It is believed that NP-Hard problems are unlikely to admit polynomial-time algorithms. However since many NP-Hard problems are of practical significance, different approaches are proposed to solve them: Approximation algorithms, randomized algorithms and heuristic algorithms. None of the approaches meet the practical needs. Recently parameterized computation and complexity has attracted a lot of attention and been a fruitful branch of the study of efficient algorithms. By taking advantage of the moderate value of parameters in many practical instances, we can design efficient algorithms for the NP-Hard problems in practice. In this dissertation, we discuss a new approach to design efficient parameterized algorithms, kernelization. The motivation is that instances of small size are easier to solve. Roughly speaking, kernelization is a preprocess on the input instances and is able to significantly reduce their sizes. We present a 2k kernel for the cluster editing problem, which improves the previous best kernel of size 4k; We also present a linear kernel of size 7k 2d for the d-cluster editing problem, which is the first linear kernel for the problem. The kernelization algorithm is simple and easy to implement. We propose a quadratic kernel for the pseudo-achromatic number problem. This implies that the problem is tractable in term of parameterized complexity. We also study the general problem, the vertex grouping problem and prove it is intractable in term of parameterized complexity. In practice, many problems seek a set of good solutions instead of a good solution. Motivated by this, we present the framework to study enumerability in term of parameterized complexity. We study three popular techniques for the design of parameterized algorithms, and show that combining with effective enumeration techniques, they could be transferred to design efficient enumeration algorithms.Item Measure-Driven Algorithm Design and Analysis: A New Approach for Solving NP-hard Problems(2010-10-12) Liu, YangNP-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.Item On Covering Points with Conics and Strips in the Plane(2012-12-10) Tiwari, Praveen 1985-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.Item Parameterized algorithms and computational lower bounds: a structural approach(Texas A&M University, 2006-10-30) Xia, GeMany problems of practical significance are known to be NP-hard, and hence, are unlikely to be solved by polynomial-time algorithms. There are several ways to cope with the NP-hardness of a certain problem. The most popular approaches include heuristic algorithms, approximation algorithms, and randomized algorithms. Recently, parameterized computation and complexity have been receiving a lot of attention. By taking advantage of small or moderate parameter values, parameterized algorithms provide new venues for practically solving problems that are theoretically intractable. In this dissertation, we design efficient parameterized algorithms for several wellknown NP-hard problems and prove strong lower bounds for some others. In doing so, we place emphasis on the development of new techniques that take advantage of the structural properties of the problems. We present a simple parameterized algorithm for Vertex Cover that uses polynomial space and runs in time O(1.2738k + kn). It improves both the previous O(1.286k + kn)-time polynomial-space algorithm by Chen, Kanj, and Jia, and the very recent O(1.2745kk4 + kn)-time exponential-space algorithm, by Chandran and Grandoni. This algorithm stands out for both its performance and its simplicity. Essential to the design of this algorithm are several new techniques that use structural information of the underlying graph to bound the search space. For Vertex Cover on graphs with degree bounded by three, we present a still better algorithm that runs in time O(1.194k + kn), based on an ??????almost-global?????? analysis of the search tree. We also show that an important structural property of the underlying graphs ?????? the graph genus ?????? largely dictates the computational complexity of some important graph problems including Vertex Cover, Independent Set and Dominating Set. We present a set of new techniques that allows us to prove almost tight computational lower bounds for some NP-hard problems, such as Clique, Dominating Set, Hitting Set, Set Cover, and Independent Set. The techniques are further extended to derive computational lower bounds on polynomial time approximation schemes for certain NP-hard problems. Our results illustrate a new approach to proving strong computational lower bounds for some NP-hard problems under reasonable conditions.Item Parameterized complexity and polynomial-time approximation schemes(Texas A&M University, 2005-02-17) Huang, XiuzhenAccording 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.Item Privacy-preserving data mining(2009-05-15) Zhang, NanIn the research of privacy-preserving data mining, we address issues related to extracting knowledge from large amounts of data without violating the privacy of the data owners. In this study, we first introduce an integrated baseline architecture, design principles, and implementation techniques for privacy-preserving data mining systems. We then discuss the key components of privacy-preserving data mining systems which include three protocols: data collection, inference control, and information sharing. We present and compare strategies for realizing these protocols. Theoretical analysis and experimental evaluation show that our protocols can generate accurate data mining models while protecting the privacy of the data being mined.Item Randomized and Deterministic Parameterized Algorithms and Their Applications in Bioinformatics(2010-07-14) Lu, SongjianParameterized 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.Item Study on Two Optimization Problems: Line Cover and Maximum Genus Embedding(2012-07-16) Cao, ChengIn this thesis, we study two optimization problems which have a lot of important applications in diverse domains: Line Cover Problem (LCP) in Computational Geometry and Maximum Genus Embedding (MGE) in Topological Graph Theory. We study LCP whose decision version is known NP-Complete from the perspective of Parameterized Complexity, as well as classical techniques in Algorithm Design. In particular, we provide an exact algorithm in time O(n^3 2n) based on Dynamic Programming and initiate a dual problem of LCP in terms of Linear Programming Duality. We study the dual problem by applying approximation and kernelization, obtaining an approximation algorithm with ratio k - 1 and a kernel of size O(k^4). Then we survey related geometric properties on LCP. Finally we propose a Parameterized Algorithm to solve LCP with running time O*(k^k/1:35^k). We explore connections between the maximum genus of a graph and its cycle space consisting of fundamental cycles only. We revisit a known incorrect approach of finding a maximum genus embedding via computing a maximum pairing of intersected fundamental cycles with respect to an arbitrary spanning tree. We investigate the reason it failed and conclude it confused the concept of deficiency. Also, we characterize the upper-embeddablity of a graph in terms of maximum pairings of intersected fundamental cycles, i.e. a graph is upper-embeddable if and only if the number of maximum pairings of intersected fundamental cycles for any spanning tree is the same. Finally, we present a lower and an upper bound of the maximum number of vertex-disjoint cycles in a general graph, beta(G) - 2gammaM(G) and beta(G) - gammaM(G), only depending on maximum genus and cycle rank.