Showing papers on "Approximation algorithm published in 1984"

Distributed Asynchronous Deterministic and Stochastic Gradient Optimization Algorithms

Abstract: We present a model for asynchronous distributed computation and then proceed to analyze the convergence of natural asynchronous distributed versions of a large class of deterministic and stochastic gradient-like algorithms. We show that such algorithms retain the desirable convergence properties of their centralized counterparts, provided that the time between consecutive communications between processors plus communication delays are not too large.

Practical Multiprocessor Scheduling Algorithms for Efficient Parallel Processing

Abstract: This paper describes practical optimization/ approximation algorithms for scheduling a set of partially ordered computational tasks onto a multiprocessor system so that the schedule length will be minimized. Since this problem belongs to the class of ''strong'' NP-hard problems, we must foreclose the possibility of constructing not only pseudopolynomial time optimization algorithms but also fully polynomial time approximation schemes unless P = NP. This paper proposes a heuristic algorithm named CP/MISF (critical path/most immediate successors first) and an optimization/approximation algorithm named DF/IHS (d thfirst/implicit heuristic search). DF/IHS is an excellent scheduling method which can reduce markedly space complexity and average computation time by combining the branch-and-bound method with CP/MISF; it allows us to solve very large scale problems with a few hundred tasks. Numerical examples are included to demonstrate the effectiveness of the proposed algorithms.

Approximation Algorithms for Bin-Packing — An Updated Survey

Abstract: This paper updates a survey [53] written about 3 years ago. All of the results mentioned there are covered here as well. However, as a major justification for this second edition we shall be presenting many new results, some of which represent important advances. As a measure of the impressive amount of research in just 3 years, the present reference list more than doubles the list in [53].

An Aggregation Technique for the Transient Analysis of Stiff Markov Chains

Abstract: An approximation algorithm for systematically converting a stiff Markov chain into a non-stiff chain with a smaller state space is discussed in this paper. After classifying the set of all states into fast and slow states, the algorithm proceeds by further classifying fast states into fast recurrent subsets and a fast transient subset. A separate analysis of each of these fast subsets is done and each fast recurrent subset is replaced by a single slow state while the fast transient subset is replaced by a probabilistic switch. After this reduction, the remaining small and non-stiff Markov chain is analyzed by a conventional technique. The algorithm produces asymptotically exact results with respect to the aggregation of fast transient states, while for fast recurrent subsets the asymptotic accuracy depends on the degree of coupling between the fast subset and the remaining states. The algorithm is illustrated using two examples.

Graph algorithms and NP-completeness

Abstract: Vol. 2: Graph Algorithms and NP-Completeness.- IV. Algorithms on Graphs.- 1. Graphs and their Representation in a Computer.- 2. Topological Sorting and the Representation Problem.- 3. Transitive Closure of Acyclic Digraphs.- 4. Systematic Exploration of a Graph.- 5. A Close Look at Depth First Search.- 6. Strongly-Connected and Biconnected Components of Directed and Undirected Graphs.- 7. Least Cost Paths in Networks.- 7.1. Acyclic Networks.- 7.2. Non-negative Networks.- 7.3. General Networks.- 7.4. The All Pairs Problem.- 8. Minimum Spanning Trees.- 9. Maximum Network Flow and Applications.- 9.1 Algorithms for Maximum Network Flow.- 9.2 (0,1)-Networks, Bipartite Matching and Graph Connectivity.- 9.3 Weighted Network Flow and Weighted Bipartite Matching.- 10. Planar Graphs.- 11. Exercises.- 12. Bibliographic Notes.- V. Path Problems in Graphs and Matrix Multiplication.- 1. General Path Problems.- 2. Two Special Cases: Least Cost Paths and Transitive Closure.- 3. General Path Problems and Matrix Multiplication.- 4. Matrix Multiplication in a Ring.- 5. Boolean Matrix Multiplication and Transitive Closure.- 6. (Min,+)-Product of Matrices and Least Cost Paths.- 7. A Lower Bound on the Monotone Complexity of Matrix Multiplication.- 8. Exercises.- 9. Bibliographic Notes.- VI. NP-Completeness.- 1. Turing Machines and Random Access Machines.- 2. Problems, Languages and Optimization Problems.- 3. Reductions and NP-complete Problems.- 4. The Satisfiability Problem is NP-complete.- 5. More NP-complete Problems.- 6. Solving NP-complete Problems.- 6.1 Dynamic Programming.- 6.2 Branch and Bound.- 7. Approximation Algorithms.- 7.1 Approximation Algorithms for the Travelling Salesman Problem.- 7.2 Approximation Schemes.- 7.3 Full Approximation Schemes.- 8. The Landscape of Complexity Classes.- 9. Exercises.- 10. Bibliographic Notes.- IX. Algorithmic Paradigms.

A Note on Approximation Schemes for Multidimensional Knapsack Problems

Abstract: Polynomial and fully polynomial approximation algorithms for single-dimensional knapsack problems have been extensively studied and a number of such algorithms constructed. This note shows that the problem of finding a fully polynomial approximation algorithm for multidimensional knapsack problems is NP-hard.

Approximation procedures for the optimal control of bilinear and nonlinear systems

Abstract: Optimal control problems for bilinear systems are studied and solved with a view to approximating analogous problems for general nonlinear systems. For a given bilinear optimal control problem, a sequence of linear problems is constructed, and their solutions are shown to converge to the desired solution. Also, the direct solution to the Hamilton-Jacobi equation is analyzed. A power-series approach is presented which requires offline calculations as in the linear case (Riccati equation). The methods are compared and illustrated. Relations to classical linear systems theory are discussed.

Outer approximation algorithm for nondifferentiable optimization problems

Abstract: It is known that the problem of minimizing a convex functionf(x) over a compact subsetX of ℝ n can be expressed as minimizing max{g(x, y)|y ∈X}, whereg is a support function forf[f(x) ≥g(x, y), for ally ∈X andf(x)=g(x, x)]. Standard outer-approximation theory can then be employed to obtain outer-approximation algorithms with procedures for dropping previous cuts. It is shown here how this methodology can be extended to nonconvex nondifferentiable functions.

Sparse null bases and marriage theorems

Abstract: This dissertation considers the problem of constructing the sparsest basis for the null space of a constraint matrix. This problem arises in the design of practical algorithms for large-scale numerical optimization problems. Suprisingly, this problem can be formulated as a combinatorial optimization problem under a non-degeneracy assumption on the constraint matrix. The theory of matchings in bipartite graphs--marriage theorems--can then be used to obtain the nonzero positions in a null basis. Numerically stable matrix factorizations are used in the next stage to compute the null basis. We use conformal decompositions to characterize the columns of a sparsest null basis. Matroid theory is used to prove that a greedy algorithm constructs a sparsest null basis. We prove that finding a sparsest null basis is NP-hard by showing that associated matroidal and graph-theoretic problems are NP-complete. We propose two approximation algorithms to construct sparse null bases. Both of them make use of the Dulmage-Mendelsohn decomposition of rectangular matrices. One algorithm is a sparsity exploiting variant of the variable-reduction technique. The second is a locally greedy algorithm that constructs a null basis with an upper triangular submatrix. These results are extended to computing sparse orthogonal null bases. We show that the sparsest null basis for an n-vector computed as a product of Givens rotations has n log(,2) n nonzeros. A generalization for dense t x n matrices constructs an orthogonal null basis with nt log(,2)n/t nonzeros. We also classify all known methods for constructing null bases, and show some unexpected equivalences between some of them.

When are NP-hard location problems easy?

Abstract: We discuss in this paper several location problems for which it is an NP-hard problem to find an approximate solution. Given certain assumptions on the input distributions, we present polynomial algorithms that deliver a solution asymptotically close to the optimum with probability that is asymptotically one (the exact nature of this asymptotic convergence is described in the paper). In that sense the subproblems defined on the specified family of inputs are in fact easy.

Permuting elements within columns of a matrix in order to minimize maximum row sum

Abstract: We study the problem of permuting the elements within columns of a given m × n matrix A so as to minimize its maximum row sum (sum of the elements in a row). We introduce and analyze an approximation algorithm of greedy type for this NP-complete problem. We prove that our algorithm produces matrices with maximum row sums no more than 3/2 − 1/2m times greater than those found by an optimization rule. Moreover, examples are presented which achieve this relative performance. Thus, our algorithm represents a substantial improvement in that all earlier algorithms have a worst-case performance that is asymptotically twice that of an optimization rule. We verify that our algorithm requires at most O(m2n) time, which is a modest increase over the earlier algorithms requiring Θ(mn log n) time in the worst-case.

Approximation Algorithms for the Assembly Line Crew Scheduling Problem

Abstract: Given an m × n nonnegative matrix, we study the problem of independently permuting the elements in each column so as to minimize the maximum row sum. This problem is NP-hard even when we restrict it to the m × 3 case. A special case of our problem is the multiprocessor scheduling problem without precedence constraints. In this paper we give a (2 − 1/m)-algorithm for the general m × n case. We also design a 3/2-algorithm for the m × 3 case, which applies a restricted longest-processing-time-first heuristic. Finally, using an edge matching algorithm, we produce a more elaborate algorithm that guarantees a bound of 4/3 for the m × 3 case.

An approximation algorithm for Manhattan routing

Abstract: @n) tracks. In practice, it appears that the flux never exceeds a small constant. In this case the algorithm performs asymptotically better than the best-known knock-knee algorithm [21], and almost as well as the best-known three-layer algorithm [19], without requiring the use of either knock-knees or three layers of interconnect. In addition, the three-parameter model, which is closer to the design rules of current fabrication technologies, is presented. Under this model, it is shown that every channel can be routed using 2d+O(1) tracks.

On the algorithmic complexity of coloring simple hypergraphs and steiner triple systems

Abstract: In this paper we establish that decidingt-colorability for a simplek-graph whent≧3,k≧3 is NP-complete. Next, we establish that if there is a polynomial time algorithm for finding the chromatic number of a Steiner Triple system then there exists a polynomial time “approximation” algorithm for the chromatic number of simple 3-graphs. Finally, we show that the existence of such an approximation algorithm would imply that P=NP.

New Algorithms for Classic Economic Load Dispatch

Abstract: This paper presents efficient and reliable algorithms to solve the classic economic load dispatch problem. In the conventional equal incremental method, an ambiguity exists for selecting a value of a relaxation coefficient. Since the value of the relaxation coefficient can be determined only from experience, there exist some examples whose convergence is very slow or convergence is not accomplishable. In order to overcome this defect, three algorithms are proposed, i.e., the parametric quadratic programming, the modified parametric quadratic programming and the recursive quadratic programming algorithm. A number of numerical tests for real system have been carried out to demonstrate the effectiveness of the proposed algorithms. The numerical results show that the proposed algorithms are practical for real-time applications.

A computationally efficient approximation to the nearest neighbor interchange metric

Abstract: The nearest neighbor interchange (nni) metric is a distance measure providing a quantitative measure of dissimilarity between two unrooted binary trees with labeled leaves. The metric has a transparent definition in terms of a simple transformation of binary trees, but its use in nontrivial problems is usually prevented by the absence of a computationally efficient algorithm. Since recent attempts to discover such an algorithm continue to be unsuccessful, we address the complementary problem of designing an approximation to the nni metric. Such an approximation should be well-defined, efficient to compute, comprehensible to users, relevant to applications, and a close fit to the nni metric; the challenge, of course, is to compromise these objectives in such a way that the final design is acceptable to users with practical and theoretical orientations. We describe an approximation algorithm that appears to satisfy adequately these objectives. The algorithm requires O(n) space to compute dissimilarity between binary trees withn labeled leaves; it requires O(n logn) time for rooted trees and O(n 2 logn) time for unrooted trees. To help the user interpret the dissimilarity measures based on this algorithm, we describe empirical distributions of dissimilarities between pairs of randomly selected trees for both rooted and unrooted cases.

Approximation Schemes for Covering and Packing Problems in Robotics and VLSI

Abstract: The approach described in this paper, the shifting strategy, has proved useful in a large variety of contexts. Via the use of this approach we were able to derive algorithms that are the best possible in the sense that the exponential dependence on 1/ɛ cannot be removed unless NP=P. We also note that all other polynomial approximation schemes that we are familiar with rely on dynamic programming. The technique we introduced is an alternative to dynamic programming for the construction of polynomial approximation schemes for strongly NP-complete problems.

Asymptotic behavior of stochastic approximation and large deviations

Abstract: The theory of large deviations is applied to the study of the asymptotic properties of the stochastic approximation algorithms

Powers of graphs: A powerful approximation technique for bottleneck problems

Abstract: In this paper we investigate a powerful, and yet simple, technique for devising approximation algorithms for a wide variety of NP-complete problems in routing, location, and communication network design. Each of the algorithms presented here delivers an approximate solution guaranteed to be within a constant factor of the optimal solution. In addition, for several of these problems we can show that unless P=NP, there does not exist a polynomial-time algorithm that has a better performance guarantee.

Fast expected-time and approximation algorithms for geometric minimum spanning trees

Abstract: Kenneth L. Clarkson Stanford University §1 I n t r o d u c t i o n We present algorithms for solving the geometric minimum spanning tree problem: Given a set of n points hi d-dimensional space, find a minimum spraining tree for the complete weighted graph G dcfined by these points. The graph G has n vertices, each vertex identified with a point, with the weight of an edge given by the distance between the two points deffining tha t edge. This problem has many applications, including wire routing, statistical pa t tern classification, and heuristics for the traveling salesman problem. It is in fact a ftunily of problems, depending on the dimension and on the distance measure used to determine the edge weights. Three khlds of algorithms will be described: *For the planar case (d = 2), with the Ll (Manhattan} metric used for the distance between points, an algorithm will bc described having nearly linear expected time, for independently indcntically distr ibuted random points with an (unknown) underlying distribution from an extremely broad class. The "nearly linear" time mentioned above mcans that the algorithm requires linear expected tinm for all stcps except the last. That step requlrcs that disjoint sets be maintained [Tar], with n unions ea, d m = O(n) Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission. © 1984 ACM 0-89791-133-4184/004/0342 $00.75 thxds, requiring O(n~(m, n)} time in the worst case, where a (m, n) grows extremely slowly. ,For the Li distance measure, a class of approximation algorithms will be presented having a runifing time of O (n(c~(m, n)--I-load1 (1/£) log e)), where w$ = O(n), to find a spanning tree with weight less than 1 + e of the minimum spanning tree weight. Herc 6 is the ratio of the distance between the faxthest pair of input points to the distance between the closest pair. The space required is OCn(logC1/e ) + log6)}. The constant factor in the bound depends on d. ,A class of approximation algorithms will be described for d = 3 and the L2, or Euclidean, distance. These algorithms require O(n(log n+(1/e) log 6)) time, with a space bound of O (n log 6). For the plazmx Manhat tan case, algorithms requiring O(nlogn) time in the worst case have been known for some time [SH][LWl. These Mgorithms require t h e construction of the Voronoi diagrasn for the Ll metric, arid do not generalize to fast algorithms for d > 2. Yao [Yao] has described a reduction from the geometric minimum spazming tree problem to the geographic neighbor problem, and Guibas mid Stolfi [GS] have recently found an O(nlogn) algorithm for the Li pla. nar case, using this reduction. The algorithms we will describe use neither of these approaches, nor the "spiral search" teclmique described by Bentley, Weide, and Yao [BWY] that has an O(nloglogn) expectedtime bound mtdcr much more restrictive conditions. (This bound was recently improved to the nc'axly linear time O(nlog* n), with the general mildmunt spanning tree algorithm of Fredman said Taxjan [FT], and to O(no~(rn, n)) expected time, using a bucket sorting algorithm [C2].) The complexity bounds given assume that thc floor, logarithm, and bitwisc exclusiveor fmlctions axe available at unit cost. For the general L1 case, quite recently Bentley, Gabow, said Tarjmt [BGT] have found algorithms requiring

A Piecewise Linear Approximation Based on a Statistical Model

Abstract: A statistical model is introduced and then, based on it, a piecewise linear approximation algorithm of linear computational complexity is presented. The advantages of the algorithm are proved experimentally in small sample cases and theoretically in the large sample case. The paper is closed with a discussion on some possible extensions.