Showing papers on "Approximation algorithm published in 1982"

Combinatorial optimization: algorithms and complexity

Abstract: This clearly written , mathematically rigorous text includes a novel algorithmic exposition of the simplex method and also discusses the Soviet ellipsoid algorithm for linear programming; efficient algorithms for network flow, matching, spanning trees, and matroids; the theory of NP-complete problems; approximation algorithms, local search heuristics for NPcomplete problems, more All chapters are supplemented by thoughtprovoking problems A useful work for graduate-level students with backgrounds in computer science, operations research, and electrical engineering Mathematicians wishing a self-contained introduction need look no further—American Mathematical Monthly 1982 ed

A Linear-Time Heuristic for Improving Network Partitions

Abstract: An iterative mincut heuristic for partitioning networks is presented whose worst case computation time, per pass, grows linearly with the size of the network. In practice, only a very small number of passes are typically needed, leading to a fast approximation algorithm for mincut partitioning. To deal with cells of various sizes, the algorithm progresses by moving one cell at a time between the blocks of the partition while maintaining a desired balance based on the size of the blocks rather than the number of cells per block. Efficient data structures are used to avoid unnecessary searching for the best cell to move and to minimize unnecessary updating of cells affected by each move.

An efficient approximation scheme for the one-dimensional bin-packing problem

Abstract: We present several polynomial-time approximation algorithms for the one-dimensional bin-packing problem. using a subroutine to solve a certain linear programming relaxation of the problem. Our main results are as follows: There is a polynomial-time algorithm A such that A(I) ≤ OPT(I) + O(log2 OPT(I)). There is a polynomial-time algorithm A such that, if m(I) denotes the number of distinct sizes of pieces occurring in instance I, then A(I) ≤ OPT(I) + O(log2 m(I)). There is an approximation scheme which accepts as input an instance I and a positive real number e, and produces as output a packing using as most (1 + e) OPT(I) + O(e-2) bins. Its execution time is O(e-c n log n), where c is a constant. These are the best asymptotic performance bounds that have been achieved to date for polynomial-time bin-packing. Each of our algorithms makes at most O(log n) calls on the LP relaxation subroutine and takes at most O(n log n) time for other operations. The LP relaxation of bin packing was solved efficiently in practice by Gilmore and Gomory. We prove its membership in P, despite the fact that it has an astronomically large number of variables.

Approximation Algorithms for the Set Covering and Vertex Cover Problems

Abstract: We propose a heuristic that delivers in $O(n^3 )$ steps a solution for the set covering problem the value of which does not exceed the maximum number of sets covering an element times the optimal value.

Load-Flow Solutions for Ill-Conditioned Power Systems by a Newton-Like Method

Abstract: In this paper mathematician K.M. Brown's method is used to solve load-flow problems. The method is Particularly effective for solving of ill-conditioned non- linear algebraic equations. It is a variation of Newton's method incorporating Gaussian elimination in such a way that the most recent infonnation is always used at each step of the algorithm; similar to what is done in the Gauss-Seidel process. The iteration converges locally and the convergence is quadratic in nature. A general discussion of ill-conditioning of a system of algebraic equations is given, and it is also show by the fixed-point formulation that the proposed method falls in the general category of sucessive approximation methods. Digital computer solutions by the proposed method are given for cases for which the standard load-flow methods failed to converge, namely 11-, 13- and 43-bus ill-conditioned test systems. A comparison of this method with the standard load-flow methods is also presented for the well-conditioned AEP 30-, and 57-bus systems.

Packing Rectangular Pieces—A Heuristic Approach

Abstract: The following two-dimensional bin packing problems are considered. (1) Find a way to pack an arbitrary collection of rectangular pieces into an open-ended, rectangular bin so as to minimize the height to which the pieces fill the bin. (2) Given rectangular sheets, i.e. bins of fixed width and height, allocate the rectangular pieces, the object being to obtain an arrangement that minimizes the number of sheets needed. Approximation algorithms are given where the solutions are achieved with heuristic search methods. Test results are presented to support the feasibility of the methods.

Finite Element Approximation of Parabolic Time Optimal Control Problems

Abstract: The numerical approximation of a parabolic time optimal control problem via piecewise linear splines, is considered. At each stage of the approximation a bang-bang approximate control is selected by solving for its switching times as the solution of a constrained nonlinear least squares optimization problem. The well-posedness of the approximation scheme is shown and the rate of convergence to the exact solution investigated. Numerical results for some one- and two-dimensional parabolic control problems are given.

Lower bounds for on-line two-dimensional packing algorithms

Abstract: Many problems, such as cutting stock problems and the scheduling of tasks with a shared resource, can be viewed as two-dimensional bin packing problems. Using the two-dimensional packing model of Baker, Coffman, and Rivest, a finite list L of rectangles is to be packed into a rectangular bin of finite width but infinite height, so as to minimize the total height used. An algorithm which packs the list in the order given without looking ahead or moving pieces already packed is called an on-line algorithm. Since the problem of finding an optimal packing is NP-hard, previous work has been directed at finding approximation algorithms. Most of the approximation algorithms which have been studied are on-line except that they require the list to have been previously sorted by height or width. This paper examines lower bounds for the worst-case performance of on-line algorithms for both non-preordered lists and for lists preordered by increasing or decreasing height or width.

On approximating a vertex cover for planar graphs

Abstract: The approximation problem for vertex cover of n-vertex planar graphs is treated. Two results are presented: (1) A linear time approximation algorithm for which the (error) performance bound is 2/3. (2) An O(n log n) time approximation scheme.

On the computational complexity of clustering and related problems

Abstract: The problem of clustering a set of n points into k groups under various objective functions is studied. It is shown that under some objective functions clustering problems are NP-hard even when the points to be grouped are restricted to lie in the two dimensional euclidean space. Our results can be extended to show that their corresponding approximation problems are also NP-hard. It is shown that some restricted graph partition problems are also NP-hard.

Probabilistic analysis of some bin-packing problems

Abstract: We analyze the average-case behaviour of the Next-Fit algorithm for bin-packing, and obtain closed-form expressions for distributions of interest. Our analysis is based on a novel technique of partitioning the interval (0, 1) suitably and then formulating the problem as a matrix-differential equation. We compare our analytic results with previously known simulation results and show that there is an excellent agreement between the two. We also explain a certain empirically observed anomaly in the behaviour of the algorithm. Finally we establish that asymptotically perfect packing is possible when input items are drawn from a monotonically decreasing density function.

The computational complexity of simultaneous Diophantine approximation problems

Abstract: Simultaneous Diophantine approximation in d dimensions deals with the approximation of a vector α = (α1, ..., αd) of d real numbers by vectors of rational numbers all having the same denominator. This paper considers the computational complexity of algorithms to find good simultaneous approximations to a given vector α of d rational numbers. We measure the goodness of an approximation using the sup norm. We show that a result of H. W. Lenstra, Jr. produces polynomial-time algorithms to find sup norm best approximations to a given vector α when the dimension d is fixed. We show that a recent algorithm of A. K. Lenstra, H. W. Lenstra, Jr., and L. Lovasz to find short vectors in an integral lattice can be used to find a good approximation to a given vector α in d dimensions with a denominator Q satisfying 1 ≤ Q ≤ 2d/2 N which is within a factor √5d 2d+1/2 of the best approximation with denominator Q* with 1 ≤ Q* ≤ N. This algorithm runs in time polynomial in the input size, independent of the dimension d. We prove results complementing these, showing certain natural simultaneous Diophantine approximation problems are NP-hard. We show that the problem of deciding whether a given vector α of rational numbers has a simultaneous approximation of specified accuracy with respect to the sup norm with denominator Q in a given interval 1 ≤ Q ≤ N is NP-complete. (Here the dimension d is allowed to vary.) We prove two other complexity results, which suggest that the problem of locating best (sup norm) simultaneous approximations is harder than this NP-complete problem.

String compression algorithms

Abstract: This thesis deals with three related problems involving strings, specifically, finding superstrings, the string sequencing problem, and macro compression schemes. In each section, the author shows several natural variants of the problem at hand to be computationally intractable (i.e., NP-complete). He then suggests a number of approximation algorithms and gives bounds on their performance. This work is well motivated, competently executed and constitutes an acceptable Ph.D. thesis. Sections of the work have already been accepted for journal publication.

Simplicial approximation of unemployment equilibria

Abstract: In this paper we consider an exchange economy with fixed prices and quantity rationing. An unemployment equilibrium is an equilibrium with quantity constraints on the supplies only. A simplicial approximation algorithm is used to find an unemployment equilibrium such that at least one commodity is not rationed. Moreover, some insight about the structure of the set of unemployment equilibria is obtained from the sequence of simplices generated by the algorithm.

A New Approach for Network Reliability Analysis

Abstract: This paper presents a novel approach to network reliability analysis based on a network function. Once this function has been calculated for a given system, the network is no longer needed in the analysis. The approach is far more efficient than Monte Carlo simulation and much more flexible than cut-set techniques.

Digital Simulation Algorithms Using Parallel Processing

Abstract: Algorithms are presented which reduce the computation time per integration step compared to the more common methods. This is accomplished by using only algorithms which permit the simultaneous calculation of x and f(x) where x = f(x). These results are important not only in the field of digital simulation, but also in realizing dynamic controllers/estimators digitally, e.g., realizing a nonlinear observer.

The ellipsoid algorithm for linear inequalities in exact arithmetic

Abstract: A modification of the ellipsoid algorithm is shown to be capable of testing for satisfiability a system of linear equations and inequalities with integer coefficients of the form Ax = b, x ≥ 0. All the rational arithmetic is performed exactly, without losing polynomiality of the computing time. In case of satisfiability, the approach always provides a rational feasible point. The bulk of the computations consists of a sequence of linear least squares problems, each a rank one modification of the preceding one. The continued fractions jump is used to compute some of the coordinates of a feasible point.

An Introduction to Proof Techniques for Bin-Packing Approximation Algorithms

Abstract: Introductory Remarks — Upon introducing a seemingly small change or generalization into either the model or an approximation algorithm for a previously studied bin packing problem, it has been common to find that very little of the analysis of the original problem can be exploited in analyzing the new problem. Such experiences may suggest that the mathematics of bin packing does not contain a central, well-structured theory that provides powerful, broadly applicable techniques for the analysis of approximation algorithms. It would be difficult to repudiate completely such an observation, but we hope to show that the extent to which it is true is largely inevitable. Specifically, our aim in this brief tutorial extension of the survey in [GJ] will be to explain somewhat informally certain techniques that do enjoy a moderately broad applicability, while making it clear where the novelty and perhaps ingenuity of the approach to an individual problem may be required.

Least-Squares Sequential Parameter and State Estimation for Large Space Structures

Abstract: This paper presents the formulation of simultaneous state and parameter estimation problems for flexible structures in terms of least-squares minimization problems. The approach combines an on-line order determination algorithm, with least-squares algorithms for finding estimates of modal approximation functions, modal amplitudes, and modal parameters. The approach combines previous results on separable nonlinear least squares estimation with a regression analysis formulation of the state estimation problem. The technique makes use of sequential Householder transformations. This allows for sequential accumulation of matrices required during the identification process. The technique is used to identify the modal parameters of a flexible beam.

An improved finite-difference approximation to the diffusion equation

Abstract: A higher order approximation to field quantities than the piecewise constant approximation commonly used and implicit in the second-order central difference approximation to the Helmholtz equation is described. The continuous finite-difference (CFD) approximation results in reduced error without an increase in complexity of the associated computational algorithm.