Hybrid Rounding Techniques for Knapsack Problems

We address the classical knapsack problem and a variant in which an upper bound is imposed on the number of items that can be selected. We show that appropriate combinations of rounding techniques yield novel and powerful ways of rounding. As an application of these techniques, we present a linear-storage Polynomial Time Approximation Scheme (PTAS) and a Fully Polynomial Time Approximation Scheme (FPTAS) that compute an approximate solution, of any fixed accuracy, in linear time. This linear complexity bound gives a substantial improvement of the best previously known polynomial bounds.Comment: 19 LaTeX page

A multiscale approximation algorithm for the cardinality constrained knapsack problem

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Civil and Environmental Engineering, 2006.Includes bibliographical references (leaves 83-86).I develop a multiscale approximation algorithm for the cardinality constrained knapsack problem. The algorithm consists of three steps: a rounding and reduction step where a hierarchical representation of the problem data ranging from coarse to fine is generated, a solution step where a coarse solution is computed, and a refinement step where the accuracy of the solution is improved by refining the problem representation. I demonstrate that the algorithm is fully polynomial with a runtime complexity that improves upon the previous best known fully polynomial approximation scheme. Through an extensive computational study, I show that the running times of the algorithm is less than or equal to that of a commercial integer programming package with little loss in solution accuracy.by Bharath Kumar Krishnan.Ph.D

Efficient Semidefinite Branch-and-Cut for MAP-MRF Inference

We propose a Branch-and-Cut (B&C) method for solving general MAP-MRF inference problems. The core of our method is a very efficient bounding procedure, which combines scalable semidefinite programming (SDP) and a cutting-plane method for seeking violated constraints. In order to further speed up the computation, several strategies have been exploited, including model reduction, warm start and removal of inactive constraints. We analyze the performance of the proposed method under different settings, and demonstrate that our method either outperforms or performs on par with state-of-the-art approaches. Especially when the connectivities are dense or when the relative magnitudes of the unary costs are low, we achieve the best reported results. Experiments show that the proposed algorithm achieves better approximation than the state-of-the-art methods within a variety of time budgets on challenging non-submodular MAP-MRF inference problems.Comment: 21 page

The Interpolating Random Spline Cryptosystem and the Chaotic-Map Public-Key Cryptosystem

The feasibility of implementing the interpolating cubic spline function as encryption and decryption transformations is presented. The encryption method can be viewed as computing a transposed polynomial. The main characteristic of the spline cryptosystem is that the domain and range of encryption are defined over real numbers, instead of the traditional integer numbers. Moreover, the spline cryptosystem can be implemented in terms of inexpensive multiplications and additions. Using spline functions, a series of discontiguous spline segments can execute the modular arithmetic of the RSA system. The similarity of the RSA and spline functions within the integer domain is demonstrated. Furthermore, we observe that such a reformulation of RSA cryptosystem can be characterized as polynomials with random offsets between ciphertext values and plaintext values. This contrasts with the spline cryptosystems, so that a random spline system has been developed. The random spline cryptosystem is an advanced structure of spline cryptosystem. Its mathematical indeterminacy on computing keys with interpolants no more than 4 and numerical sensitivity to the random offset t( increases its utility. This article also presents a chaotic public-key cryptosystem employing a one-dimensional difference equation as well as a quadratic difference equation. This system makes use of the El Gamal’s scheme to accomplish the encryption process. We note that breaking this system requires the identical work factor that is needed in solving discrete logarithm with the same size of moduli