Showing papers on "Approximation algorithm published in 1975"

Fast Approximation Algorithms for the Knapsack and Sum of Subset Problems

Abstract: Given a positive integer M and n pairs of positive integers (p~, cD, , (p. , c.), maximize the s u m ~ ~p~ subject to the cons t ramts~ ~c, < M and ~, = 0 or 1 This is the well-known 0/1 knapsack problem An algorithm is presented which finds for any 0 < e < 1 an approximate solution P satisfying (P* P)/P* < ~, where P* is the desired optimal sum Moreover, for any fixed e, the algorithm has time complexity 0(n log n) and space complexity O(n) Modification of the algorithm for the unbounded knapsack problem where the ~,'s can be any nonnegative integer results in a O(n) computing time A hnear-time algorithm is also obtained for a special class of 0/1 knapsack problems having the property that p,/c, is the same for all 1 < z < n

Using duality to solve discrete optimization problems: theory and computational experience*

Abstract: Meaningful dual problems have recently been identified for the integer programming problem, the resource constrained network scheduling problem and the traveling salesman problem. In this paper a general class of discrete optimization problem is given for which dual problems of this type may be derived. We discuss the use of dual problems for obtaining strong bounds, feasible solutions, and for guiding the search in enumeration schemes for this class of problems. Properties of dual problems and three algorithms are discussed, a primal-dual ascent algorithm, a simplicial approximation algorithm and an algorithm based on the relaxation method for solving systems of inequalities. Finally, computational experience is given for integer programming and resource constrained network scheduling dual problems.

Exact, Approximate, and Guaranteed Accuracy Algorithms for the Flow-Shop Problem n / 2 / F / F¯

Abstract: Improved exact and approximate algorithms for the n-job two-machine mean finishing time flow-shop problem, n/2JF/P, are presented While other researchers have used a variety of approximate methods to generate suboptimal solutions and branch-and-bound algorithms to gen- erate exact solutmns to sequencing problems, thin work demonstrates the computatmnal effectiveness of couphng the two methods to generate solutmns with a guaranteed accuracy. The computational reqmrements of exact, approximate, and guaranteed accuracy algorithms are compared expem- mentally on a set of test problems ranging in size from 10 to 50 jobs The approach is readily apphca- ble to other sequencing problems

Optimal Piecewise Polynomial L 2 Approximation of Functions of One and Two Variables

Abstract: The problem of piecewise polynomial L2 approximation with variable boundaries is considered. Necdssary and sufficient conditions for local optima are derived. These suggest simple functional iteration, algorithms for locating the boundaries.

Approximation of digital filters in one and two dimensions

Abstract: An algorithm for the time domain approximation of discrete systems with a recursion is described. The algorithm iterates towards a solution minimizing the sum of squared differences between the desired and the actual output. Convergence is guaranteed. The scheme is applied to the design of low-pass filters by time domain approximation. The results compare well with other design strategies. We have extended the algorithm to two dimensions. This algorithm is essentially the same iterative scheme used for the one-dimensional case. The two-dimensional iteration achieves better results than the currently used two-dimensional filter synthesis techniques since the starting point of the iteration is the solution of the latter approach. Usually, a few iterations suffice to improve the solution in a satisfactory amount. Convergence is also guaranteed. An example of two-dimensional impulse response approximation is also given as an illustration.

Floating-point computation of functions with maximum accuracy

Abstract: Algorithms are given which compute multiple sums and products and arbitray roots of floating-point numbers with maximum accuracy. The summation algorithm can be applied to compute scalar products, matrix products, etc. For all these functions, simple error formulas and the smallest floating-point intervals containing the exact result can be obtained.

A mean cost approximation for transportation problems with stochastic demand

Abstract: Among the many tools of the operations researcher is the transportation algorithm which has been used to solve a variety of problems ranging from shipping plans to plant location. An important variation of the basic transportation problem is the transportation problem with stochastic demand or stochastic supply. This paper presents a simple approximation technique which may be used as a starting solution for algorithms that determine exact solutions. The paper indicates that the approximation technique offered here is superior to a starting solution obtained by substituting expected demand for the random variables.

Synthesis of spectrum shaping digital filters of recursive design

Abstract: Several techniques are presented, both iterative and noniterative, for synthesizing a' recursive digital filter whose amplitude spectrum approximates a desired shape. The synthesis is carried out by implementing known algorithms for rational function approximation of a continuous function on a finite interval. In particular, techniques for synthesizing the best rational function approximation in the Chebyshev sense and in the least squares sense are treated. Examples are given which illustrate the quality of approximation achievable through each technique.

On Two—Processor Scheduling of One— or Two—Unit Time Tasks with Precedence Constraints

Abstract: This paper is concerned with the problem of scheduling on two processors tasks with 1- or 2-unit execution time and having arbitrary precedence constraints. An analysis is made of the algorithm (called f LPTS-schedule) which chooses as the next task to be scheduled, that task for which the sum of the execution times of all its successors is maximal. It is shown that LPTS produces a schedule with an error where f * and f are the finish times of an optimal and LPTS schedules, respectively. An example is given which shows that the error bound is tight, even for the case when all tasks have 1-unit execution time. A comparison of the LPTS algorithm with other scheduling algorithms is also presented.

Parallel computations in graph theory

Abstract: In parallel computation two approaches are common; namely unbounded parallelism and bounded parallelism. In this paper both approaches will be considered. The problem of unbounded parallelism is studied in section II and some lower and upper bounds on different connectivity problems for directed and undirected graphs are presented. In section III we mention bounded parallelism and three different k-parallel graph search techniques, namely k-depth search, breadth depth search, and breadth-first search. Each algorithm is analyzed with respect to the optimal serial algorithm. It is shown that for sufficiently dense graphs the parallel breadth first search technique is very close to the optimal bound. Techniques for searching sparse graphs are also discussed.

Comments on "Parameter identification and control of linear discrete-time systems"

Abstract: A new proof of convergence of the stochastic approximation algorithm for parameter identification of closed-loop linear discrete-time control systems is proposed. This algorithm relates very effectively in terms of a sufficient condition the stability properties of the closed-loop system with the convergence of the identification algorithms, which were previously treated independently.