Showing papers on "Approximation algorithm published in 1976"

P-Complete Approximation Problems

Abstract: For P-complete problems such as traveling salesperson, cycle covers, 0-1 integer programming, multicommodity network flows, quadratic assignment, etc., it is shown that the approximation problem is also P-complete. In contrast with these results, a linear time approximation algorithm for the clustering problem is presented.

Exact and Approximate Algorithms for Scheduling Nonidentical Processors

Abstract: Exact and approximate algorithms are presented for scheduling independent tasks in a multiprocessor environment in which the processors have different speeds. Dynamic programming type algorithms are presented which minimize finish time and weighted mean flow time on two processors. The generalization to m processors is direct. These algorithms have a worst-case complexity which is exponential in the number of tasks. Therefore approximation algorithms of low polynomial complexity are also obtained for the above problems. These algorithms are guaranteed to obtain solutions that are close to the optimal. For the case of minimizing mean flow time on m-processors an algorithm is given whose complexity is O(n log mn).

Approximation algorithms for some routing problems

Abstract: Several polynomial time approximation algorithms for some NP-complete routing problems are presented, and the worst-case ratios of the cost of the obtained route to that of an optimal are determined. A mixed-strategy heuristic with a bound of 9/5 is presented for the Stacker-Crane problem (a modified Traveling Salesman problem). A tour-splitting heuristic is given for k-person variants of the Traveling Salesman problem, the Chinese Postman problem, and the Stacker-Crane problem, for which a minimax solution is sought. This heuristic has a bound of e + 1 - 1/k, where e is the bound for the corresponding 1-person algorithm.

The use of second-order information in the approximation of discreate-time linear systems

Abstract: It is common practice to partially characterize a filter with a finite portion of its impulse response, with the objective of generating a recursive approximation. This paper discusses the use of mixed first and second information, in the form of a finite portion of the impulse response and autocorrelation sequences. The discussion encompasses a number of techniques and algorithms for this purpose. Two approximation problems are studied: an interpolation problem and a least squares problem. These are shown to be closely related. The linear systems which form the solutions to these problems are shown to be stable. An efficient algorithm for obtaining solutions is given and shown to be closely related to a well-known algorithm of Levinson and the Jury stability test. The close connection between these algorithms suggests that they are fundamental in the numerical analysis of stable discrete-time linear systems.

Optimal Algorithms for Self-Reducible Problems.

Abstract: We show that obtaining the lexicographically first four coloring of a planar graph is NP–hard. This shows that planar graph four-coloring is not self-reducible, assuming P 6= NP . One consequence of our result is that the schema of [JVV 86] cannot be used for approximately counting the number of four colorings of a planar graph. These results extend to planar graph k-coloring, for k ≥ 4. Research done while this author was supported by an IBM Graduate Fellowship at Cornell University. Work done while at Cornell University. Supported by NSF grant DCR 85-52938 and PYI matching funds from AT&T Bell Labs and Sun Microsystems, Inc.

Approximation algorithm for some routing problems

Abstract: Several polynomial time approximation algorithms for some $NP$-complete routing problems are presented, and the worst-case ratios of the cost of the obtained route to that of an optimal are determined. A mixed-strategy heuristic with a bound of 9/5 is presented for the stacker-crane problem (a modified traveling salesman problem). A tour-splitting heuristic is given for k-person variants of the traveling salesman problem, the Chinese postman problem, and the stacker-crane problem, for which a minimax solution is sought. This heuristic has a bound of $e + 1 - 1/k$, where e is the bound for the corresponding 1-person algorithm.

Optimization of System Reliability by Branch-and-Bound

Abstract: This paper compares two approximation procedures with the implicit enumeration technique for solving reliability optimization problems. The integer nature of the problem is stressed and suggestions are made for evaluating future efforts to solve the problem.

An Efficient Algorithm for Constructing Hierarchical Graphs

Abstract: An algorithm to delete redundant edges from a precedence graph is presented and proved correct. The algorithm is much more efficient than previous algorithms to perform the same task.

A successive approximation algorithm for an undiscounted Markov decision process

Abstract: In this paper we consider a completely ergodic Markov decision process with finite state and decision spaces using the average return per unit time criterion. An algorithm is derived which approximates the optimal solution. It will be shown that this algorithm is finite and supplies upper and lower bounds for the maximal average return and a nearly optimal policy with average return between these bounds.

An algorithm for minimal degree linear Chebyshev approximation on a discrete set

Abstract: An algorithm for computing a linear Chebyshev approximation to a function defined on a finite set of points is presented. The method requires the accuracy of the approximation to be specified, and determines the least degree approximation which achieves this accuracy. The algorithm is based upon the simplex method of linear programming. A FORTRAN program is supplied in the Appendix.

An algorithm for minimizing a differentiable function subject to box constraints

Abstract: A quasi-Newton algorithm for minimizing a function subject to box constraints is described. Since our original motivation came from the optimization of electrical circuits, this algorithm is designed to tolerate errors in the function and gradient evaluations. Theorems are given to support the use of the symmetric rank one update.

General convergence results for constrained and unconstrained stochastic approximations

Abstract: The paper treats general convergence conditions for a class of algorithms for finding the minima of a function f(x) when f(x) is of unknown (or partly unknown) form - and when only noise corrupted observations can be taken. Such problems occur frequently in adaptive processes, and in many applications to statistics and estimation. The algorithms are of the stochastic approximation type. Several forms are dealt with - for estimation in either discrete or continuous time, with and without side constraints. The algorithms can be considered as sequential Monte Carlo methods for systems optimization. The main purpose of the paper is to show how the theory of weak convergence of a sequence of probability measures can be used to reduce the convergence problem to "basic" elements, and to yield, quickly and efficiently, a convergence theorem under rather general conditions on the noise. Many extensions are possible, and only a few applications are considered here. The ideas are useful for establishing deeper (than the usual) results on rates of convergence [17]. Extensions to problems where the parameters are abstract valued is also possible. We mention also that a study of the numerical properties of several algorithms for constrained SA (stochastic approximation) appear in [18].

On optimal approximation

Abstract: This note presents a new instance of spline approximation in which the observation of a function is its value on an interior contour or hypersurface and the coobservation is its gradient. There follow three comments relevant to the application of the theory of optimal approximation.

Use of symbolic and numeric methods in an algorithm for the approximation of multivariate functions

Abstract: In this paper, we discuss aspects of a particular algorithm, namely that of finding an approximation to a real function of n variables, that uses both symbolic and numeric manipulations. Development of this algorithm was motivated in part by an actual application; it is being used in a study of the optimal design of a geothermal energy extraction plant [4]. Our experience here with the extensible language Madcap [6], in which the approximation algorithm was developed, certainly corroborates the importance of very high level, general-purpose languages for the design of involved algorithms.

Non-linear subsidiary problems in mathematical programming☆

Abstract: IT IS proposed to use the conjugate gradient method in linear and non-linear programming problems and to estimate the Lagrange multipliers from the analytic solution of subsidiary problems. Recently iterative methods of solving non-linear programming problems using various auxiliary problems [1–5] to construct the vector step, have become more and more popular. As a rule these problems are obtained by linear [1] or non-linear [2,3] approximation of the target function and by linear approximation of the constraints. These problems are usually solved by applying the corresponding algorithms, which considerably reduces their domain of rational application [2]. In this paper we present a number of auxiliary problems with non-linear approximation of the constraints, whose solution is obtained analytically.

Dynamic Effects of Signal Normalizing on Adaptive Algorithms

Abstract: A method is shown for suppressing the adaptive switching transient that arises in complex adaptive (LMS and stochastic approximation) algorithms due to signal normalization with controlled switched gains. The method involves rescaling of the adaptive weights with the normalizing gains just prior to the first adaptation with the new gains. In addition, an effective method of rescaling the adaptive gain factor based on the maximum eigenvalues of the respective covariance matrices is shown.