Showing papers on "Approximation algorithm published in 1990"

Approximation algorithms for scheduling unrelated parallel machines

Abstract: We consider the following scheduling problem. There arem parallel machines andn independent jobs. Each job is to be assigned to one of the machines. The processing of jobj on machinei requires timep ij . The objective is to find a schedule that minimizes the makespan. Our main result is a polynomial algorithm which constructs a schedule that is guaranteed to be no longer than twice the optimum. We also present a polynomial approximation scheme for the case that the number of machines is fixed. Both approximation results are corollaries of a theorem about the relationship of a class of integer programming problems and their linear programming relaxations. In particular, we give a polynomial method to round the fractional extreme points of the linear program to integral points that nearly satisfy the constraints. In contrast to our main result, we prove that no polynomial algorithm can achieve a worst-case ratio less than 3/2 unlessP = NP. We finally obtain a complexity classification for all special cases with a fixed number of processing times.

Towards an architecture-independent analysis of parallel algorithms

Abstract: A simple and efficient method for evaluating the performance of an algorithm, rendered as a directed acyclic graph, on any parallel computer is presented. The crucial ingredient is an efficient approximation algorithm for a particular scheduling problem. The only parameter of the parallel computer needed by our method is the message-to-instruction ratio $\tau$. Although the method used in this paper does not take into account the number of processors available, its application to several common algorithms shows that it is surprisingly accurate.

Algorithms for the maximum satisfiability problem

Abstract: Old and new algorithms for the Maximum Satisfiability problem are studied We first summarize the different heuristics previously proposed, ie, the approximation algorithms of Johnson and of Lieberherr for the general Maximum Satisfiability problem, and the heuristics of Lieberherr and Specker, Poljak and Turzik for the Maximum 2-Satisfiability problem We then consider two recent local search algorithmic schemes, the Simulated Annealing method of Kirkpatrick, Gelatt and Vecchi and the Steepest Ascent Mildest Descent method, and adapt them to the Maximum Satisfiability problem The resulting algorithms, which avoid being blocked as soon as a local optimum has been found, are shown empirically to be more efficient than the heuristics previously proposed in the literature

A Multimode Multiproduct Network Assignment Model for Strategic Planning of Freight Flows

Abstract: We present in this paper a normative model for simulating freight flows of multiple products on a multimodal network. The multimodal aspects of the transportation system considered are accounted for in the network representation chosen. The multiproduct aspects of the model are exploited in the solution procedure, which is a Gauss–Seidel-Linear Approximation Algorithm. An important component of the solution algorithm is the computation of shortest paths with intermodal transfer costs. Computational results obtained with this algorithm on a network that corresponds to the Brazil transportation network are presented. Several applications of this model are reported as well.

Exact and approximate reasoning about temporal relations

Abstract: Allen gives an algebra for representing qualitative temporal information about the relationships between pairs of intervals. In this paper, we address a fundamental reasoning task that arises in applications of the algebra: Given (possibly indefinite) knowledge about the relationships between intervals, find all feasible relationships between two intervals. We call this the minimal labels problem. Finding the minimal labels can be viewed as computing the deductive consequences of our knowledge. Determining exact solutions to this problem has been shown to be (almost assuredly) intractable. Allen gives an approximation algorithm based on constraint propagation. We present new approximation algorithms, determine analytically under what conditions the algorithms are exact, and examine, through some computational experiments, the quality of the approximate solutions produced by the algorithms. We also give a simple test for predicting when the approximation algorithms will and will not produce good quality approximations. Finally, we survey three example applications of the interval algebra chosen from the literature to show where the results of this paper could be useful.

Two point exponential approximation method for structural optimization

Abstract: This paper examines various first order approximation methods commonly used in structural optimization. It considers several attempts at improving the approximation by using previous analytical results and introduces an adaptation of a first order approximation method using an exponent adjusted to better fit the constraints and reduce the overall number of iterations needed to attain the optimum.

Multiprocessor scheduling with communication delays

Abstract: This paper adresses certain types of scheduling problems that arise when a parallel computation is to be executed on a multiprocessor. We define a model that allows for communication delays between precedence-related tasks, and propose a classification of various submodels. We also review complexity results and optimization and approximation algorithms that have been presented in the literature.

Approximation through multicommodity flow

Abstract: The first approximate max-flow-min-cut theorem for general multicommodity flow is proved. It is used to obtain approximation algorithms for minimum deletion of clauses of a 2-CNF identical to formula, via minimization problems, and other problems. Also presented are approximation algorithms for chordalization of a graph and for register sufficiency that are based on undirected and directed node separators. >

On Approximate Solutions for Combinatorial Optimization Problems

Abstract: The usefulness of a special kind of approximability-preserving transformations (called continuous reductions) among combinatorial optimization problems is demonstrated. One common measure for the approximability of an optimization problem is its best performance ratio. This parameter attains the same value for two problems (up to a bounded factor) whenever they are mutually related by continuous reductions. Therefore, lower and upper bounds or gap-theorems valid for a particular problem are transferred along reduction chains. In this paper, continuous reductions are used for the analysis of several basic combinatorial problems including graph coloring, consistent deterministic finite automaton, covering by cliques, covering by complete bipartite subgraphs, independent set, set packing, and others. The results obtained and the methods involved are a contribution towards a systematic classification of NP-complete problems with regard to their approximability.

Solving findpath by combination of goal-directed and randomized search

Abstract: An application of the findpath problem is briefly introduced, and method solving findpath by a combination of goal-directed straight-and-slide search and randomized generation of subgoals is outlined. The major parts and details of the implemented plane-findpath algorithm are discussed, and experimental results are given. The future extensions of the algorithm to a three-dimensional environment and a six-degree-of-freedom manipulator are considered. >

A class of bounded approximation algorithms for graph partitioning

Abstract: This paper considers the problem of partitioning the nodes of a weighted graph into k disjoint subsets of bounded size, such that the sum of the weights of the edges whose end vertices belong to the same subset is maximized. A class of approximation algorithms based on matching is presented. These algorithms are shown to yield practical worst-case bounds when k is large. Extensive empirical experimentation indicates that the methods produce consistently good solutions to an important VLSI design problem in a fraction of the time required by competing methods.

Layer assignment for multichip modules

Abstract: The layer assignment problem that arises in the design of a multichip module, a high-performance compact package for the interconnection of several hundred chips, is studied. The aim is to place each net in a x-y pair of layers, so as to minimize the number of such pairs. An approximation algorithm, running in O(nd) time is presented for minimizing the number of layers, where n is the number of nets and d is the (two-dimensional) density of the problem. >

Timing constraints for correct performance

Abstract: Novel methodology and algorithms for the derivation of timing constraints on all the interconnects were developed and applied to solving layout related timing problems. This methodology is based on detailed information on timing characteristics of cells and nets. A minimax approach for identifying maximal delay bounds for nets which do not violate the timing constraints on any of the logical paths in the design was proposed. An approximation algorithm with proven polynomial time behavior was described. The recursive application of this algorithm results in the distribution of the whole remaining path slacks between the comprising nets, and as a result, zero slack is achieved. The obtained timing bounds were applied to produce layouts free from timing problems. >

e-optimality for bicriteria programs and its application to minimum cost flows

Abstract: A subsetS⊂X of feasible solutions of a multicriteria optimization problem is called e-optimal w.r.t. a vector-valued functionf:X→Y $$ \subseteq $$ ℝ K if for allx∈X there is a solutionz x∈S so thatf k(z x)≤(1+e)f k (x) for allk=1,...,K. For a given accuracy e>0, a pseudopolynomial approximation algorithm for bicriteria linear programming using the lower and upper approximation of the optimal value function is given. Numerical results for the bicriteria minimum cost flow problem on NETGEN-generated examples are presented.

Quantifiers and approximation

Abstract: We investigate the relationship between logical expressibility of NP optimization problems and their approximation properties. First such attempt was made by Papadimitrou and Yannakakis (1988), who defined the class of NPO problems MAX NP. We show that many important optimization problems do not belong to MAX NP and that, in fact, there are problems in P which are not in MAX NP. The problems that we consider fit naturally in a new complexity class that we call MAX Π1. We prove that several natural optimization problems are complete for MAX Π1 under approximation-preserving reductions. All these complete problems are not approximable unless P = NP. This motivates the definition of subclasses of MAX Π1 that only contain problems which are presumably eaiser with respect to approximation. In particular, the class that we call RMAX(2) contains approximable problems and problems like MAX CLIQUE that are not known to be nonapproximable. We prove the MAX CLIQUE and several other optimization problems are complete for RMAX(2). All the complete problems in RMAX(2) share the interesting property that they either are nonapproximable or are approximable to any degree of accuracy.

Path planning in the presence of vertical obstacles

Abstract: Consideration is given to the problem of finding a shortest path between two points in 3-D space with a restricted class of polyhedral obstacles (vertical buildings with a fixed number k of distinct heights). For the case when all the obstacles have equal heights, a shortest-path algorithm is presented with complexity O(n/sup 2/), i.e. the same complexity as for the 2-D case (n is the total number of corners in all the obstacles). For the general case (k distinct heights), an algorithm is presented for finding a shortest path in time O(n/sup 6k-1/). Also presented is an O(n/sup 2/) approximation algorithm that finds paths that are, at most, 8% longer than the shortest path for the case of k distinct heights when certain minimum separation requirements are satisfied, and a description is given of how the approximation algorithm can be extended to the general case (arbitrary separations). >

A Newton-Raphson method for moving-average spectral factorization using the Euclid algorithm

Abstract: An implementation of the Newton-Raphson approach to compute the minimum phase moving-average spectral factor of a finite positive definite correlation sequence is presented. Each step in the successive approximation method involves a system of linear equations that is solved using either the Levinson algorithm backwards (the Jury stability test), or a symmetrized version of the Euclid algorithm. Various properties of the Newton-Raphson map are studied. The algorithm is generalized to other symmetries (other than with respect to the unit circle). The special case of the symmetry with respect to the imaginary axis is presented and related to the Routh-Hurwitz stability test for continuous time transfer function. >

Time-optimal trajectories for a robot manipulator: a provably good approximation algorithm

Abstract: An algorithm is presented for generating near-time-optimal trajectories for an open-kinematic-chain manipulator moving in a cluttered workspace. This is the first algorithm to guarantee bounds on the closeness of an approximation to a time-optimal trajectory. The running time and space required are polynomial in the desired accuracy of the approximation. The user may also specify tolerances by which the trajectories must clear obstacles in the workspace to allow modeling of control errors. >

Non-asymptotic confidence bounds for stochastic approximation algorithms with constant step size

Abstract: We consider the Robbins—Monro process with constant step-size in ℝ k and design confidence regions (via a comparison result) and a stopping time which is based on the inner products of consecutive gradients. We show that this stopping time leads to confidence regions which are uniform in the starting value.

Bounds on the quality of approximate solutions to the Group Steiner Problem

Abstract: The Group Steiner Problem (GSP) is a generalized version of the well known Steiner Problem. For an undirected, connected distance graph with groups of required vertices and Steiner vertices, GSP asks for a shortest connected subgraph, containing at least one vertex of each group. As the Steiner Problem is NP-hard, GSP is too, and we are interested in approximation algorithms. Efficient approximation algorithms have already been proposed, but nothing about the quality of any approximate solution is known so far. Especially for the VLSI design application of the problem, bounds on the quality of approximate solutions are of great importance. We present a simple polynomial time approximation algorithm that computes a tree with no more than $g-1$ times the length of a minimal tree, where $g$ is the number of required groups. In addition, we propose an extended version of this algorithm, trading quality of the solution for computation time. Here, one extreme is just our proposed approximation, and the other is an optimal solution. Moreover, we will prove the quality bound $g-1$ for a modification of an efficient approximation algorithm proposed in the literature.

Leighton-Rao might be practical: faster approximation algorithms for concurrent flow with uniform capacities

Abstract: In this paper, we describe new algorithms for approximately solving the concurrent multicommodity flow problem with uniform capacities. Our algorithms are much faster than previously known algorithms. Besides being an important problem in its own right, the concurrent flow problem has many interesting applications. Leighton and Rao used concurrent flow to find an approximately "sparsest cut" in a graph, and thereby approximately solve a wide variety of graph problems, including minimum feedback arc set, minimum cut linear arrangement, and minimum area layout. We show that their method might be practical by giving an O(m~logm) expected-time randomized algorithm for their concurrent flow problem on an m-edge graph. l~aghavan and Thompson used concurrent flow to approximately solve a channel width minimization problem in VLSI. We give an O(k3/2(m+n log n)) expectedtime randomized algorithm and an O(k min{n, k}(m + n log n) log k) deterministic algorithm for this problem when the channel width is O(logn), where k denotes the number of wires to be routed in an n-node, m-edge network, *Research partially suppor ted by ONR grant N00014-88-K0243 and DARPA grant N00039-88-C0113 at Harvard University tSuppor t provided by Air Force Contract AFOSI:t-86-0078, and by an NSF PYI awarded to David Shmoya, with matching funds from IBM, Sun Microsysterns, and UPS. t Research partially supported by NSF grant DMS87-06133. 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. 1 I n t r o d u c t i o n The multicommodity flow problem involves shipping several different commodities from their respective sources to their sinks in a single network with the total amount of flow going through an edge limited by its capacity. Here we consider the concurrent flow problem with uniform capacities, which is the problem of finding a multicommodity flow that minimizes the maximum total flow on any edge (the congestion). An important special case of the problem, called the unit-demand unit-capacity concurrent flow problem, arises when the amount of flow that needs to be shipped is the same for each commodity. For both versions of the problem we give algorithms that, for any positive e, find a solution whose congestion is no more than (1 + e) times the minimum congestion. Our algorithms significantly improve the time required for finding such approximately optimal solutions. We shall use n, m and k to denote the number of nodes, edges and commodities. Leighton and Rao [7] showed how to use the solution to a unit-capacity unit-demand concurrent fiow problem to find an approximate "sparsest cut" of a graph. As a consequence, they gave the first polylog-timesoptimal approximation algorithms for a wide variety of graph problems. The computational bottleneck of their method is solving a unit-capacity unit-demand concurrent flow problem with O(n) commodities. They appealed to linear programming techniques to show that the problem can be solved in polynomial time. The new approximation algorithm greatly improves the resulting running time. T h e o r e m 1.1 For any fixed 0 < e < 1, a (1 + e)-factor approximation to the unit-capacity unit-demand concurrent flow problem can be found by a randomized algorithm in O((k + m)mlogm) time, where the constant depends

Approximate mean value analysis algorithms for queuing networks: existence, uniqueness, and convergence results

Abstract: This paper is concerned with the properties of nonlinear equations associated with the Scheweitzer-Bard (S-B) approximate mean value analysis (MVA) heuristic for closed product-form queuing networks. Three forms of nonlinear S-B approximate MVA equations in multiclass networks are distinguished: Schweitzer, minimal, and the nearly decoupled forms. The approximate MVA equations have enabled us to: (a) derive bounds on the approximate throughput; (b) prove the existence and uniqueness of the S-B throughput solution, and the convergence of the S-B approximation algorithm for a wide class of monotonic, single-class networks; (c) establish the existence of the S-B solution for multiclass, monotonic networks; and (d) prove the asymptotic (i.e., as the number of customers of each class tends to ∞) uniqueness of the S-B throughput solution, and (e) the convergence of the gradient projection and the primal-dual algorithms to solve the asymptotic versions of the minimal, the Schweitzer, and the nearly decoupled forms of MVA equations for multiclass networks with single server and infinite server nodes. The convergence is established by showing that the approximate MVA equations are the gradient vector of a convex function, and by using results from convex programming and the convex duality theory.

Approximation algorithms for VLSI partition problems

Abstract: Polynomial time approximation algorithms are described for several VLSI partitioning problems, including graph and hypergraph partitioning, and nonplanar edge deletion. The algorithms find solutions that are within a polylogarithmic factor of the optimal solution for several of the problems. All of the algorithms use the Leighton-Rao (7988) graph-separator algorithm as a subroutine. >

Provably good approximation algorithms for optimal kinodynamic planning for Cartesian robots and open chain manipulators

Abstract: We consider the following problem: given a robot system, find a minimal-time trajectory from a start state to a goal state, while avoiding obstacles by a speed-dependent safety margin and respecting dynamics bounds. In [CDRX] we developed a provably good approximation algorithm for the minimum-time trajectory problem for a robot system with decoupled dynamics bounds. This algorithm differed from previous work in three ways: it is possible (1) to bound the goodness of the approximation by an error term e (2) to polynomially bound the running time (complexity) of our algorithm; and (3) to express the complexity as a polynomial function of the error term.We extend these results to d-link, revolute-joint 3D robots will full rigid body dynamics. Specifically, we first prove a generalized trajectory-tracking lemma for robots with coupled dynamics bounds. Using this result we describe polynomial-time approximation algorithms for Cartesian robots obeying L2 dynamics bounds and open kinematic chain manipulators with revolute and prismatic joints; the latter class includes most industrial manipulators. We obtain a general O(n2 (log n)(1/e6d-1) algorithm, where n is the geometric complexity. The algorithm is simple, but the new game-theoretic proof techniques we introduce are subtle, and employ tools from disparate parts of computational geometry, robotics, and dynamical systems.

Deterministic load balancing in computer networks

Abstract: The paper presents theoretical analysis of the deterministic complexity of the load balancing problem (LBP). Because of difficulty of the general problem, research in the area mostly restricts itself to probabilistic or approximation algorithms, or to the average behavior of a network. The paper provides deterministic analysis of the problem for general networks. It focuses on the worst-case complexity analysis of the problem. It shows certain cases of the LBP to be NP-complete. The paper also discusses situations closely related to computer networks, where there is a global view of load distribution in the network; it provides a polynomial algorithm for solving the load balancing problem in this network. >

Convergence of Recursive Optimization Algorithms using IPA Derivative Estimates

Abstract: We propose two estimation and updating schemes which use infinitesimal perturbation analysis based derivative estimates for the recursive optimization of queues. With the aid of extensions of convergence theorems from stochastic approximation, we prove convergence of the two proposed algorithms, when applied to an M/G/1 queue and to a multi-queue system. We also present simulation results illustrating our theorems for the case of an M/M/1 queue, and a three-queue system.