Showing papers on "Approximation algorithm published in 1989"

New Approximation Method for Stress Constraints in Structural Synthesis

Abstract: A new approximation method for dealing with stress constraints in structural synthesis is presented. The finite element nodal forces are approximated and these are used to create an explicit, but often nonlinear, approximation to the original problem. The principal motivation is to create the best approximation possible, in order to reduce the number of detailed finite element analyses needed to reach the optimum. Examples are offered and compared with published results, to demonstrate the efficiency and reliability of the proposed method.

Beyond Steiner's Problem: A VLSI Oriented Generalization

Abstract: We consider a generalized version of Steiner's problem in graphs, motivated by the wire routing phase in physical VLSI design: given a connected, undirected distance graph with groups of required vertices and Steiner vertices, find a shortest connected subgraph containing at least one required vertex of each group We propose two efficient approximation algorithms computing different approximate solutions in time O(|E| + |V|log|V|) and in time O(g · (|E| + |V|log|V|)), respectively, where |E| is the number of edges in the given graph, |V| is the number of vertices, and g is the number of groups The latter algorithm propagates a set of wavefronts with different distances simultaneously through the graph; it is interesting in its own right

The strongly connecting problem on multihop packet radio networks

Abstract: The problem of strongly connecting a multihop packet radio network by using a minimal total amount of transmission power is investigated. This problem is shown to be NP-complete. An approximation algorithm with the same computational complexity as that of finding a minimum spanning tree is given. It is also shown that the approximation algorithm can find a solution no greater than twice that of the optimal solution. Experimental results show that the approximation solution may be close to the optimal solution. >

Efficient NC algorithms for set cover with applications to learning and geometry

Abstract: NC approximation algorithms are given for the unweighted and weighted set cover problems. The algorithms use a linear number of processors and give a cover that has at most log n times the optimal size/weight, thus matching the performance of the best sequential algorithms. The set cover algorithm is applied to learning theory, providing an NC algorithm for learning the concept class obtained by taking the closure under finite union or finite intersection of any concept class of finite VC dimension which has an NC hypothesis finder. In addition, a linear-processor NC algorithm is given for a variant of the set cover problem and used to obtain NC algorithms for several problems in computational geometry. >

Approximation algorithms for the shortest common superstring problem

Abstract: The object of the shortest common superstring problem (SCS) is to find the shortest possible string that contains every string in a given set as substrings. As the problem is NP-complete, approximation algorithms are of interest. The value of an aproximate solution to SCS is normally taken to be its length, and we seek algorithms that make the length as small as possible. A different measure is given by the sum of the overlaps between consecutive strings in a candidate solution. When considering this measure, the object is to find solutions that make it as large as possible. These two measures offer different ways of viewing the problem. While the two viewpoints are equivalent with respect to optimal solutions, they differ with respect to approximate solutions. We describe several approximation algorithms that produce solutions that are always within a factor of two of optimum with respect to the overlap measure. We also describe an efficient implementation of one of these, using McCreight's compact suffix tree construction algorithm. The worstcase running time is O ( m log n ) for small alphabets, where m is the sum of the lengths of all the strings in the set and n is the number of strings. For large alphabets, the algorithm can be implemented in O ( m log m ) time by using Sleator and Tarjan's lexicographic splay tree data structure.

Approximation algorithms for temporal reasoning

Abstract: We consider a representation for temporal relations between intervals introduced by James Allen, and its associated computational or reasoning problem: given possibly indefinite knowledge of the relations between some intervals, how do we compute the strongest possible assertions about the relations between some or all intervals. Determining exact solutions to this problem has been shown to be (almost assuredly) intractable. Allen gives an approximation algorithm based on constraint propagation. We giv e new approximation algorithms, examine their effectiveness, and determine under what conditions the algorithms are exact.

Approximation schemes for constrained scheduling problems

Abstract: Several constrained scheduling problems are considered. The first polynomial approximation schemes for the problem of minimizing maximum completion time in a two-machine flow shop with release dates and for the problem of minimizing maximum lateness for the single and parallel-machine problem with release dates are described. All of these algorithms are based upon the notion of an outline, a set of information with which it is possible to compute, with relatively simple procedures and in polynomial time, an optimal or near-optimal solution to the problem instance under consideration. Two related precedence-constrained scheduling problems are discussed, and new approximation results are presented. >

Approximation algorithms for geometric embeddings in the plane with applications to parallel processing problems

Abstract: Given an undirected graph G with N vertices and a set P of N points in the plane, the geometric embedding problem consists of finding a bijection from the vertices of G to the points in the plane which minimize the sum total of edge lengths of the embedded graph. Fast approximation algorithms are given for embedding d-dimensional grids in the plane within a factor of O(log N) times optimal cost for d>2 and O(log/sup 2/N) for d=2. It is shown that any embedding of a hypercube, butterfly, or shuffle-exchange graph must be within an O(log N) factor of optimal cost. When the points of P are randomly distributed or arranged in a grid, a polynomial-time algorithm that can embed arbitrary weighted graphs in these points with cost within an O(log/sup 2/N) factor of optimal is given. It is shown how the algorithms developed for geometric embeddings can be used to give O(log/sup 2/N) times optimal solutions to problems of performance optimization for parallel processors. >

A provably good approximation algorithm for optimal-time trajectory planning

Abstract: The authors describe the first implementation of a good polynomial-time approximation algorithm for kinodynamic planning. Attention is given to the following problem: given a robot system, find a minimal-time trajectory from a start position and velocity to a goal position and velocity, while avoiding obstacles and respecting dynamic constraints on velocity and acceleration. From the class of approximate minimal-time trajectories for a given problem instance that the theoretical algorithm would find, the proposed implementation will find a trajectory with minimal useless chattering. In addition, the authors present an improved analysis of the accuracy of the approximation strength of this approach. This analysis reveals that the algorithm produces approximations good to a small additive error in state space and exact in time while only sacrificing the epsilon -approximation factor in safety, where epsilon is an error term. In addition, the analysis halves the polynomial complexity of the algorithm in (1/ epsilon ), and it provides a simple characterization of when the algorithm will find a trajectory that is exact at the start and goal. >

Polynomial approximation of functions of matrices and applications

Abstract: In solving a mathematical problem numerically, we frequently need to operate on a vector υ by an operator that can be expressed asf(A), whereA is anN ×N matrix [e.g., exp(A), sin(A), A−-]. Except for very simple matrices, it is impractical to construct the matrixf (A) explicitly. Usually an approximation to it is used. This paper develops an algorithm based upon a polynomial approximation tof (A). First the problem is reduced to a problem of approximatingf (z) by a polynomial in z, where z belongs to a domainD in the complex plane that includes all the eigenvalues ofA. This approximation problem is treated by interpolatingf (z) in a certain set of points that is known to have some maximal properties. The approximation thus achieved is “almost best.” Implementing the algorithm to some practical problems is described. Since a solution to a linear systemAx=b isx=A−1b, an iterative solution algorithm can be based upon a polynomial approximation tof (A)=A−1. We give special attention to this important problem.

Completeness in Approximation Classes

Abstract: We introduce a formal framework for studying approximation properties of NP optimization (NPO) problems. The classes we consider are those appearing in the literature, namely the class of approximable problems within a constant e (APX), the class of problems having a Polynomial-time Approximation Scheme (PAS) and the class of problems having a Fully Polynomial-time Approximation Scheme (FPAS). We define natural approximation preserving reductions and obtain completeness results for these classes. A complete problem in a class can not have stronger approximation properties unless P=NP. We also show that the degree structure of NPO allows intermediate degrees, that is, if P≠NP, there are problems which are neither complete nor belong to a lower class.

New Approximation Algorithms for the Steiner Tree Problem

Abstract: In this paper we consider several approximation algorithms for the Steiner tree problem in an attempt to find one whose worst case performance is better than two times optimal. We first examine Rayward-Smith's algorithm to gain insight into why it has worst case performance no better than two. Based on these ideas we propose several new algorithms (approximation schemes). We eliminate from further consideration those which we have bene able to showed have worst case performance that is still no better than two. Then we conjecture that one of these schemes not only has improved worst case performance, but is also the basis for a polynomial scheme. That is, given an e greater than 0 we conjecture that this scheme specifies a polynomial time approximation algorithm with worst case performance within 1 + e times optimal. ... Read complete abstract on page 2.

A note on robust stability bounds

Abstract: The authors comment on stability robustness bounds for linear systems with multiple uncertain parameters which were obtained by means of Lyapunov functions. Results obtained with quadratic and with vector Lyapunov functions are compared. Simple examples are used to illustrate the effects of the choice of functions, corresponding majorizations, and system decompositions. >

Approximation Algorithms for the Chromatic Sum

Abstract: The chromatic sum of a graph G is the smallest total among all proper colorings of G using natural numbers. It was shown that computing the chromatic sum is NP-hard. In this article we prove that a simple greedy algorithm applied to sparse graphs gives a "good" approximation of the chromatic sum. For all graphs the existence of a polynomial time algorithm that approximates the chromatic sum with a linear function error implies P = NP.

e-regularized two-level optimization problems: approximation and existence results

Abstract: The purpose of this work is to improve some results given in [12], relating to approximate solutions for two-level optimization problems By considering an e-regularized problem, we get new properties, under convexity assumptions in the lower level problems In particular, we prove existence results for the solutions to the e-regularized problem, whereas the initial two-level optimization problem may fail to have a solution Finally, as an example, we consider an approximation method with interior penalty functions

A Strongly Polynomial Algorithm for Minimum Cost Submodular Flow Problems

Abstract: The only known strongly polynomial algorithm for solving minimum cost submodular flow problems is due to Frank and Tardos Frank, A., E. Tardos. 1985. An application of the simultaneous approximation in combinatorial optimization. Report No. 85375, Institut fur Okonometrie und Operations Research, Bonn, May. and is based on the simultaneous approximation algorithm of Lenstra, Lenstra, and Lovasz Lenstra, A. K., H. W. Lenstra, L. Lovasz. 1982. Factoring polynomials with rational coefficients. Math. Ann.261 515--534.. We propose a purely combinatorial strongly polynomial algorithm. It consists in solving a sequence of at most m + nn-1 minimum cost submodular flow problems with cost coefficients bounded by n2, where n is the number of the vertices and m is the number of the arcs in the underlying graph. The current cost coefficients are calculated by means of tree projection and scaling.

On the maximum likelihood method for estimating molecular trees: uniqueness of the likelihood point.

Abstract: Studies are carried out on the uniqueness of the stationary point on the likelihood function for estimating molecular phylogenetic trees, yielding proof that there exists at most one stationary point, i.e., the maximum point, in the parameter range for the one parameter model of nucleotide substitution. The proof is simple yet applicable to any type of tree topology with an arbitrary number of operational taxonomic units (OTUs). The proof ensures that any valid approximation algorithm be able to reach the unique maximum point under the conditions mentioned above. An algorithm developed incorporating Newton's approximation method is then compared with the conventional one by means of computers simulation. The results show that the newly developed algorithm always requires less CPU time than the conventional one, whereas both algorithms lead to identical molecular phylogenetic trees in accordance with the proof.

Efficient algorithm for graph-partitioning problem using a problem transformation method

Abstract: This paper shows that the uniform k- way partitioning problem can be transformed into the max-cut problem using a graph modification technique. An iterative algorith, based on Kernighan-Lin's (KL) method, for partitioning graphs is presented that exploits the problem equivalence property. The algorithm deals with nodes of various sizes without any performance degradation. The computing time per pass of the algorithm is O(itkN2), where N is the number of nodes in the given graph. In practice, only a small number of passes are typically needed, leading to a fast approximation algorithm for k- way partitioning. Experimental results show that the proposed algorithm outperforms the KL algorithm in the quality of solutions. The performance gap between the proposed and KL algorithms becomes much bigger as the amount of size differences between nodes increases.

An O(n0.4)-approximation algorithm for 3-coloring

Abstract: This paper presents a polynomial-time algorithm to color any 3-colorable n-node graph with O(n2/5 log8/5n) colors, improving the best previously known bound of O(√n/√logn) colors. By reducing the number of colors needed to color a 3-colorable graph, the algorithm also improves the bound for k-coloring for fixed k ≥ 3 from the previous O((n/log n)1-1/(k-1)) colors to O(n1-1/(k-4/3) log8/5n) colors. An extension of the algorithm further improves the bounds. Precise values appear in a table at the end of this paper.

Layer assignment for VLSI interconnect delay minimization

Abstract: A formulation of the layer assignment problem for VLSI circuits is presented in which the objective is to minimize the interconnect delay by taking into account the resistance and capacitance of interconnect wires and contacts. For MOS circuits with two layers of interconnections the problem is shown to be equivalent to that of minimizing a weighted resistance of the corresponding RC network. This formulation readily handles wires with preassigned layers, such as power supply lines or module terminals. With user-defined weights assigned to selected nets, this method can be used to minimize critical path delays. The problem is shown to be NP-complete. A polynomial-time approximation algorithm, based on graph partitioning technique, is presented along with some experimental results. The layer assignment algorithm presented in this paper has been implemented in LISP and tested on several design examples with complexity ranging from tens to a few hundred nets. Computational complexity of this algorithm is on the order of O(n/sup 2/) for building the required data structure, and O(n/sup 1.5/) for actual layer assignment, where n is the number of wire segments in the routing. >