In this paper, we obtain convergence of solutions of stochastic differential systems with memory gap to those with full finite memory. More specifically, solutions of stochastic differential systems with memory gap are processes in which the intrinsic dependence of the state on its history goes only up to a specific time in the past. As a consequence of this convergence, we obtain a new existence proof and approximation scheme for stochastic functional differential equations (SFDEs) whose coefficients have linear growth. In mathematical finance, an option pricing formula with full finite memory is obtained through convergence of stock dynamics with memory gap to stock dynamics with full finite memory.

Piercing problems arise often in facility location, which is a well-studied area of computational geometry. The general form of the piercing problem discussed in this dissertation asks for the minimum number of facilities for a set of given rectangular demand regions such that each region has at least one facility located within it. It has been shown that even if all regions are uniform sized squares, the problem is NP-hard. Therefore we concentrate on approximation algorithms for the problem. As the known approximation ratio for arbitrarily sized rectangles is poor, we restrict our effort to designing approximation algorithms for unit-height rectangles. Our e-approximation scheme requires <I>n</I>^{<I>O</I>(1/&epsilon;??)} time. We also consider the problem with restrictions like bounding the depth of a point and the width of the rectangles. The approximation schemes for these two cases take <I>n</I>^{<I>O</I>(1/&epsilon;)} time. We also show how to maintain a factor 2 approximation of the piercing set in <I>O</I>(log <I>n</I>) amortized time in an insertion-only scenario.

In this work we formalize a new natural objective (or cost) function for bi-clustering - Monochromatic bi-clustering. Our objective function is suitable for detecting meaningful homogenous clusters based on categorical valued input matrices. Such problems have arisen recently in systems biology where researchers have inferred functional classifications of biological agents based on their pairwise interactions. We analyze the computational complexity of the resulting optimization problems. We show that finding optimal solutions is NP-hard and complement this result by introducing a polynomial time approximation algorithm for this bi-clustering task. This is the first positive approximation guarantee for bi-clustering algorithms. We also show that bi-clustering with our objective function can be viewed as a generalization of correlation clustering.

Piercing problems arise often in facility location, which is a well-studied area of computational geometry. The general form of the piercing problem discussed in this dissertation asks for the minimum number of facilities for a set of given rectangular demand regions such that each region has at least one facility located within it. It has been shown that even if all regions are uniform sized squares, the problem is NP-hard. Therefore we concentrate on approximation algorithms for the problem. As the known approximation ratio for arbitrarily sized rectangles is poor, we restrict our effort to designing approximation algorithms for unit-height rectangles. Our e-approximation scheme requires <I>n</I>^{<I>O</I>(1/&epsilon;²)} time. We also consider the problem with restrictions like bounding the depth of a point and the width of the rectangles. The approximation schemes for these two cases take <I>n</I>^{<I>O</I>(1/&epsilon;)} time. We also show how to maintain a factor 2 approximation of the piercing set in <I>O</I>(log <I>n</I>) amortized time in an insertion-only scenario.

In this work we formalize a new natural objective (or cost) function for bi-clustering - Monochromatic bi-clustering. Our objective function is suitable for detecting meaningful homogenous clusters based on categorical valued input matrices. Such problems have arisen recently in systems biology where researchers have inferred functional classifications of biological agents based on their pairwise interactions. We analyze the computational complexity of the resulting optimization problems. We show that finding optimal solutions is NP-hard and complement this result by introducing a polynomial time approximation algorithm for this bi-clustering task. This is the first positive approximation guarantee for bi-clustering algorithms. We also show that bi-clustering with our objective function can be viewed as a generalization of correlation clustering.

Given a finite set K of terminals in the plane. A rectilinear Steiner minimum tree for K (RST) is a tree which interconnects among these terminals using only horizontal and vertical lines of shortest possible length containing Steiner point. We show the complexity of RST i.e. belongs to NP-complete. Moreover we present an approximative method of determining the solution of RST problem proposed by Sanjeev Arora in 1996, Arora's Approximation Scheme. This algorithm has time complexity polynomial in the number of terminals for a fixed performance ratio 1 + Epsilon.

We propose multigrid methods for solving Hamilton-Jacobi-Bellman (HJB) and Hamilton-Jacobi-Bellman-Isaacs (HJBI) equations. The methods are based on the full approximation scheme. We propose a damped-relaxation method as smoother for multigrid. In contrast with policy iteration, the relaxation scheme is convergent for both HJB and HJBI equations. We show by local Fourier analysis that the damped-relaxation smoother effectively reduces high frequency error. For problems where the control has jumps, restriction and interpolation methods are devised to capture the jump on the coarse grid as well as during coarse grid correction. We will demonstrate the effectiveness of the proposed multigrid methods for solving HJB and HJBI equations arising from option pricing as well as problems where policy iteration does not converge or converges slowly.

We propose multigrid methods for solving Hamilton-Jacobi-Bellman (HJB) and Hamilton-Jacobi-Bellman-Isaacs (HJBI) equations. The methods are based on the full approximation scheme. We propose a damped-relaxation method as smoother for multigrid. In contrast with policy iteration, the relaxation scheme is convergent for both HJB and HJBI equations. We show by local Fourier analysis that the damped-relaxation smoother effectively reduces high frequency error. For problems where the control has jumps, restriction and interpolation methods are devised to capture the jump on the coarse grid as well as during coarse grid correction. We will demonstrate the effectiveness of the proposed multigrid methods for solving HJB and HJBI equations arising from option pricing as well as problems where policy iteration does not converge or converges slowly.

Given a finite set K of terminals in the plane. A rectilinear Steiner minimum tree for K (RST) is a tree which interconnects among these terminals using only horizontal and vertical lines of shortest possible length containing Steiner point. We show the complexity of RST i.e. belongs to NP-complete. Moreover we present an approximative method of determining the solution of RST problem proposed by Sanjeev Arora in 1996, Arora's Approximation Scheme. This algorithm has time complexity polynomial in the number of terminals for a fixed performance ratio 1 + Epsilon.