Showing papers on "Approximation algorithm published in 1998"

Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications

Abstract: Preface to the Second Edition. Preface to the First Edition. 1. Introduction. 1.1 Motivation. 1.2 Methodological Background. 1.3 Basics of Probability and Statistics. 2. Markov Chains. 2.1 Markov Processes. 2.2 Performance Measures. 2.3 Generation Methods. 3. Steady-State Solutions of Markov Chains. 3.1 Solution for a Birth Death Process. 3.2 Matrix-Geometric Method: Quasi-Birth-Death Process. 3.3 Hessenberg Matrix: Non-Markovian Queues. 3.4 Numerical Solution: Direct Methods. 3.5 Numerical Solution: Iterative Methods. 3.6 Comparison of Numerical Solution Methods. 4. Steady-State Aggregation/Disaggregation Methods. 4.1 Courtois' Approximate Method. 4.2 Takahashi's Iterative Method. 5. Transient Solution of Markov Chains. 5.1 Transient Analysis Using Exact Methods. 5.2 Aggregation of Stiff Markov Chains. 6. Single Station Queueing Systems. 6.1 Notation. 6.2 Markovian Queues. 6.3 Non-Markovian Queues. 6.4 Priority Queues. 6.5 Asymmetric Queues. 6.6 Queues with Batch Service and Batch Arrivals. 6.7 Retrial Queues. 6.8 Special Classes of Point Arrival Processes. 7. Queueing Networks. 7.1 Definitions and Notation. 7.2 Performance Measures. 7.3 Product-Form Queueing Networks. 8. Algorithms for Product-Form Networks. 8.1 The Convolution Algorithm. 8.2 The Mean Value Analysis. 8.3 Flow Equivalent Server Method. 8.4 Summary. 9. Approximation Algorithms for Product-Form Networks. 9.1 Approximations Based on the MVA. 9.2 Summation Method. 9.3 Bottapprox Method. 9.4 Bounds Analysis. 9.5 Summary. 10. Algorithms for Non-Product-Form Networks. 10.1 Nonexponential Distributions. 10.2 Different Service Times at FCFS Nodes. 10.3 Priority Networks. 10.4 Simultaneous Resource Possession. 10.5 Prograrns with Internal Concurrency. 10.6 Parallel Processing. 10.7 Networks with Asymmetric Nodes. 10.8 Networks with Blocking. 10.9 Networks with Batch Service. 11. Discrete-Event Simulation. 11.1 Introduction to Simulation. 11.2 Simulative or Analytic Solution? 11.3 Classification of Simulation Models. 11.4 Classification of Tools in DES. 11.5 The Role of Probability and Statistics in Simulation. 11.6 Applications. 12. Performance Analysis Tools. 12.1 PEPSY. 12.2 SPNP. 12. 3 MOSEL-2. 12.4 SHARPE. 12.5 Characteristics of Some Tools. 13. Applications. 13.1 Case Studies of Queueing Networks. 13.2 Case Studies of Markov Chains. 13.3 Case Studies of Hierarchical Models. Glossary. Bibliography. Index.

Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems

Abstract: We present a polynomial time approximation scheme for Euclidean TSP in fixed dimensions. For every fixed c > 1 and given any n nodes in ℛ2, a randomized version of the scheme finds a (1 + 1/c)-approximation to the optimum traveling salesman tour in O(n(log n)O(c)) time. When the nodes are in ℛd, the running time increases to O(n(log n)(O(√c))d-1). For every fixed c, d the running time is n · poly(logn), that is nearly linear in n. The algorithmm can be derandomized, but this increases the running time by a factor O(nd). The previous best approximation algorithm for the problem (due to Christofides) achieves a 3/2-aproximation in polynomial time.We also give similar approximation schemes for some other NP-hard Euclidean problems: Minimum Steiner Tree, k-TSP, and k-MST. (The running times of the algorithm for k-TSP and k-MST involve an additional multiplicative factor k.) The previous best approximation algorithms for all these problems achieved a constant-factor approximation. We also give efficient approximation schemes for Euclidean Min-Cost Matching, a problem that can be solved exactly in polynomial time.All our algorithms also work, with almost no modification, when distance is measured using any geometric norm (such as lp for p ≥ 1 or other Minkowski norms). They also have simple parallel (i.e., NC) implementations.

Approximation Algorithms for Connected Dominating Sets

Abstract: The dominating set problem in graphs asks for a minimum size subset of vertices with the following property: each vertex is required to be either in the dominating set, or adjacent to some vertex i

Implementation of the simultaneous perturbation algorithm for stochastic optimization

Abstract: The need for solving multivariate optimization problems is pervasive in engineering and the physical and social sciences. The simultaneous perturbation stochastic approximation (SPSA) algorithm has recently attracted considerable attention for challenging optimization problems where it is difficult or impossible to directly obtain a gradient of the objective function with respect to the parameters being optimized. SPSA is based on an easily implemented and highly efficient gradient approximation that relies on measurements of the objective function, not on measurements of the gradient of the objective function. The gradient approximation is based on only two function measurements (regardless of the dimension of the gradient vector). This contrasts with standard finite-difference approaches, which require a number of function measurements proportional to the dimension of the gradient vector. This paper presents a simple step-by-step guide to implementation of SPSA in generic optimization problems and offers some practical suggestions for choosing certain algorithm coefficients.

An intuitive justification and a simplified implementation of the MAP decoder for convolutional codes

Abstract: An intuitive shortcut to understanding the maximum a posteriori (MAP) decoder is presented based on an approximation. This is shown to correspond to a dual-maxima computation combined with forward and backward recursions of Viterbi algorithm computations. The logarithmic version of the MAP algorithm can similarly be reduced to the same form by applying the same approximation. Conversely, if a correction term is added to the approximation, the exact MAP algorithm is recovered. It is also shown how the MAP decoder memory can be drastically reduced at the cost of a modest increase in processing speed.

Greedy strikes back: improved facility location algorithms

Abstract: A fundamental facility location problem is to choose the location of facilities, such as industrial plants and warehouses, to minimize the cost of satisfying the demand for some commodity. There are associated costs for locating the facilities, as well as transportation costs for distributing the commodities. We assume that the transportation costs form a metric. This problem is commonly referred to as theuncapacitated facility locationproblem. Application to bank account location and clustering, as well as many related pieces of work, are discussed by Cornuejols, Nemhauser, and Wolsey. Recently, the first constant factor approximation algorithm for this problem was obtained by Shmoys, Tardos, and Aardal. We show that a simple greedy heuristic combined with the algorithm by Shmoys, Tardos, and Aardal, can be used to obtain an approximation guarantee of 2.408. We discuss a few variants of the problem, demonstrating better approximation factors for restricted versions of the problem. We also show that the problem is max SNP-hard. However, the inapproximability constants derived from the max SNP hardness are very close to one. By relating this problem to Set Cover, we prove a lower bound of 1.463 on the best possible approximation ratio, assumingNP?DTIMEnO(loglogn)].

Approximation Algorithms for Connected Dominating Sets

Abstract: The dominating set problem in graphs asks for a minimum size subset of vertices with the following property: each vertex is required to be either in the dominating set, or adjacent to some vertex in the dominating set. We focus on the related question of finding a connected dominating set of minimum size, where the graph induced by vertices in the dominating set is required to be connected as well. This problem arises in network testing, as well as in wireless communication. Two polynomial time algorithms that achieve approximation factors of 2H(Δ)+2 and H(Δ)+2 are presented, where Δ is the maximum degree and H is the harmonic function. This question also arises in relation to the traveling tourist problem, where one is looking for the shortest tour such that each vertex is either visited or has at least one of its neighbors visited. We also consider a generalization of the problem to the weighted case, and give an algorithm with an approximation factor of (cn+1) \ln n where cn ln k is the approximation factor for the node weighted Steiner tree problem (currently cn = 1.6103 ). We also consider the more general problem of finding a connected dominating set of a specified subset of vertices and provide a polynomial time algorithm with a (c+1) H(Δ) +c-1 approximation factor, where c is the Steiner approximation ratio for graphs (currently c = 1.644 ).

Approximation of Free-Discontinuity Problems

Abstract: Functions of bounded variation.- Special functions of bounded variation.- Examples of approximation.- A general approach to approximation.- Non-local approximation.

Approximate graph coloring by semidefinite programming

Abstract: We consider the problem of coloring k-colorable graphs with the fewest possible colors. We present a randomized polynomial time algorithm that colors a 3-colorable graph on n vertices with min{O(D1/3 log1/2 D log n), O(n1/4 log1/2n)} colors where D is the maximum degree of any vertex. Besides giving the best known approximation ratio in terms of n, this marks the first nontrivial approximation result as a function of the maximum degree D. This result can be generalized to k-colorable graphs to obtain a coloring using min{O(D1-2/k log1/2 D log n), O(n1−3/(k+1) log1/2n)} colors. Our results are inspired by the recent work of Goemans and Williamson who used an algorithm for semidefinite optimization problems, which generalize linear programs, to obtain improved approximations for the MAX CUT and MAX 2-SAT problems. An intriguing outcome of our work is a duality relationship established between the value of the optimum solution to our semidefinite program and the Lovasz t-function. We show lower bounds on the gap between the optimum solution of our semidefinite program and the actual chromatic number; by duality this also demonstrates interesting new facts about the t-function.

Approximation algorithms for directed Steiner problems

Abstract: We obtain the first non-trivial approximation algorithms for the Steiner Tree problem and the Generalized Steiner Tree problem in general directed graphs. Essentially no approximation algorithms were known for these problems. For the Directed Steiner Tree problem, we design a family of algorithms which achieve an approximation ratio of O(k^\epsilon) in time O(kn^{1/\epsilon}) for any fixed (\epsilon < 0), where k is the number of terminals to be connected. For the Directed Generalized Steiner Tree Problem, we give an algorithm which achieves an approximation ratio of O(k^{2/3}\log^{1/3} k), where k is the number of pairs to be connected. Related problems including the Group Steiner tree problem, the Node Weighted Steiner tree problem and several others can be reduced in an approximation preserving fashion to the problems we solve, giving the first non-trivial approximations to those as well.

Approximation schemes for Euclidean k-medians and related problems

Abstract: In the k-median problem we are given a set S of n points in a metric space and a positive integer k. We desire to locate k medians in space, such that the sum of the distances from each of the points of S to the nearest median is minimized. This paper gives an approximation scheme for the plane that for any c > 0 produces a solution of cost at most 1+ 1/c times the optimum and runs in time O(n). The approximation scheme also generalizes to some problems related to k-median. Our methodology is to extend Arora’s [1, 2] techniques for the TSP, which hitherto seemed inapplicable to problems such as the k-median problem.

Stochastic Network Interdiction

Abstract: Using limited assets, an interdictor attempts to destroy parts of a capacitated network through which an adversary will subsequently maximize flow. We formulate and solve a stochastic version of the interdictor's problem: Minimize the expected maximum flow through the network when interdiction successes are binary random variables. Extensions are made to handle uncertain arc capacities and other realistic variations. These two-stage stochastic integer programs have applications to interdicting illegal drugs and to reducing the effectiveness of a military force moving materiel, troops, information, etc., through a network in wartime. Two equivalent model formulations allow Jensen's inequality to be used to compute both lower and upper bounds on the objective, and these bounds are improved within a sequential approximation algorithm. Successful computational results are reported on networks with over 100 nodes, 80 interdictable arcs, and 180 total arcs.

Approximating Minimum Feedback Sets and Multicuts in Directed Graphs

Abstract: This paper deals with approximating feedback sets in directed graphs. We consider two related problems: the weighted feedback vertex set (FVS) problem, and the weighted feedback edge set (FES) problem. In the {FVS} (resp. FES) problem, one is given a directed graph with weights (each of which is at least one) on the vertices (resp. edges), and is asked to find a subset of vertices (resp. edges) with minimum total weight that intersects every directed cycle in the graph. These problems are among the classical NP-hard problems and have many applications. We also consider a generalization of these problems: subset-fvs and subset-fes, in which the feedback set has to intersect only a subset of the directed cycles in the graph. This subset consists of all the cycles that go through a distinguished input subset of vertices and edges, denoted by X . This generalization is also NP-hard even when |X|=2 . We present approximation algorithms for the subset-fvs and subset-fes problems. The first algorithm we present achieves an approximation factor of O(log 2 |X|) . The second algorithm achieves an approximation factor of O(min{log τ * log log τ * , log n log log n)} , where τ * is the value of the optimum fractional solution of the problem at hand, and n is the number of vertices in the graph. We also define a multicut problem in a special type of directed networks which we call circular networks, and show that the subset-fes and subset-fvs problems are equivalent to this multicut problem. Another contribution of our paper is a combinatorial algorithm that computes a (1+ɛ) approximation to the fractional optimal feedback vertex set. Computing the approximate solution is much simpler and more efficient than general linear programming methods. All of our algorithms use this approximate solution.

A factor 2 approximation algorithm for the generalized Steiner network problem

Abstract: We present a factor 2 approximation algorithm for finding a minimum-cost subgraph having at least a specified number of edges in each cut. This class of problems includes, among others, the generalized Steiner network problem, which is also known as the survivable network design problem. Our algorithm first solves the linear relaxation of this problem, and then iteratively rounds off the solution. The key idea in rounding off is that in a basic solution of the LP relaxation, at least one edge gets included at least to the extent of half. We include this edge into our integral solution and solve the residual problem.

Approximate medians and other quantiles in one pass and with limited memory

Abstract: We present new algorithms for computing approximate quantiles of large datasets in a single pass. The approximation guarantees are explicit, and apply for arbitrary value distributions and arrival distributions of the dataset. The main memory requirements are smaller than those reported earlier by an order of magnitude.We also discuss methods that couple the approximation algorithms with random sampling to further reduce memory requirements. With sampling, the approximation guarantees are explicit but probabilistic, i.e. they apply with respect to a (user controlled) confidence parameter.We present the algorithms, their theoretical analysis and simulation results on different datasets.

Exact and approximation algorithms for clustering

Abstract: In this paper we present an n^ O(k1-1/d) -time algorithm for solving the k -center problem in \realsd , under L∈fty - and L2 -metrics. The algorithm extends to other metrics, and to the discrete k -center problem. We also describe a simple (1+ɛ) -approximation algorithm for the k -center problem, with running time O(nlog k) + (k/ɛ)^ O(k1-1/d) . Finally, we present an n^ O(k1-1/d) -time algorithm for solving the L -capacitated k -center problem, provided that L=Ω(n/k1-1/d) or L=O(1) .

Improving the interpretability of TSK fuzzy models by combining global learning and local learning

Abstract: The fuzzy inference system proposed by Takagi, Sugeno, and Kang, known as the TSK model in fuzzy system literature, provides a powerful tool for modeling complex nonlinear systems. Unlike conventional modeling where a single model is used to describe the global behavior of a system, TSK modeling is essentially a multimodel approach in which simple submodels (typically linear models) are combined to describe the global behavior of the system. Most existing learning algorithms for identifying the TSK model are based on minimizing the square of the residual between the overall outputs of the real system and the identified model. Although these algorithms can generate a TSK model with good global performance (i.e., the model is capable of approximating the given system with arbitrary accuracy, provided that sufficient rules are used and sufficient training data are available), they cannot guarantee the resulting model to have a good local performance. Often, the submodels in the TSK model may exhibit an erratic local behavior, which is difficult to interpret. Since one of the important motivations of using the TSK model (also other fuzzy models) is to gain insights into the model, it is important to investigate the interpretability issue of the TSK model. We propose a new learning algorithm that integrates global learning and local learning in a single algorithmic framework. This algorithm uses the idea of local weighed regression and local approximation in nonparametric statistics, but remains the component of global fitting in the existing learning algorithms. The algorithm is capable of adjusting its parameters based on the user's preference, generating models with good tradeoff in terms of global fitting and local interpretation. We illustrate the performance of the proposed algorithm using a motorcycle crash modeling example.

Improved approximation algorithms for the uncapacitated facility location problem

Abstract: We consider the uncapacitated facility location problem. In this problem, there is a set of locations at which facilities can be built; a fixed cost fi is incurred if a facility is opened at location i. Furthermore, there is a set of demand locations to be serviced by the opened facilities; if the demand location j is assigned to a facility at location i, then there is an associated service cost proportional to the distance between i and j, cij. The objective is to determine which facilities to open and an assignment of demand points to the opened facilities, so as to minimize the total cost. We assume that the distance function c is symmetric and satisfies the triangle inequality. For this problem we obtain a (1+2/e)-approximation algorithm, where $1+2/e \approx 1.736$, which is a significant improvement on the previously known approximation guarantees. The algorithm works by rounding an optimal fractional solution to a linear programming relaxation. Our techniques use properties of optimal solutions to the linear program, randomized rounding, as well as a generalization of the decomposition techniques of Shmoys, Tardos, and Aardal [Proceedings of the 29th ACM Symposium on Theory of Computing, El Paso, TX, 1997, pp. 265--274].

Optimizing regular path expressions using graph schemas

Abstract: Query languages for data with irregular structure use regular path expressions for navigation. This feature is useful for querying data where parts of the structure is either unknown, unavailable to the user, or changes frequently. Naive execution of regular path expressions is inefficient however, because it ignores any structure in the data. We describe two optimization techniques for queries with regular path expressions. Both rely on graph schemas for specifying partial knowledge about the data's structure. Query pruning uses this structure to restrict navigation to only a fragment of the data; we give an efficient algorithm for rewriting any regular path expression query into a pruned one. Query rewriting using state extents can eliminate or reduce navigation altogether; it is reminiscent of optimizing relational queries using indices. There may be several ways to optimize a query using state extents; we give a polynomial space algorithm that finds all such optimizations. For restricted forms of regular path expressions, the algorithm is provably efficient. We also give an efficient approximation algorithm that works on all regular path expressions.

Minimizing service and operation costs of periodic scheduling

Abstract: We study the problem of scheduling activities of several types under the constraint that, at most, a fixed number of activities can be scheduled in any single time slot. Any given activity type is associated with a service cost and an operating cost that increases linearly with the number of time slots since the last service of this type. The problem is to find an optimal schedule that minimizes the long-run average cost per time slot. Applications of such a model are the scheduling of maintenance service to machines, multi-item replenishment of stock, and minimizing the mean response time in Broadcast Disks. Broadcast Disks recently gained a lot of attention because they were used to model backbone communications in wireless systems, Teletext systems, and Web caching in satellite systems. The first contribution of this paper is the definition of a general model that combines into one several important previous models. We prove that an optimal cyclic schedule for the general problem exists, and we establish the NP-hardness of the problem. Next, we formulate a nonlinear program that relaxes the optimal schedule and serves as a lower bound on the cost of an optimal schedule. We present an efficient algorithm for finding a near-optimal solution to the nonlinear program. We use this solution to obtain several approximation algorithms. 1 A 9/8 approximation for a variant of the problem that models the Broadcast Disks application. The algorithm uses some properties of “Fibonacci sequences.” Using this sequence, we present a 1.57-approximation algorithm for the general problem. 2 A simple randomized algorithm and a simple deterministic greedy algorithm for the problem. We prove that both achieve approximation factor of 2. To the best of our knowledge this is the first worst-case analysis of a widely used greedy heuristic for this problem.

Machine scheduling with availability constraints

Abstract: We will give a survey on results related to scheduling problems where machines are not continuously available for processing. We will deal with single and multi machine problems and analyze their complexity. We survey NP-hardness results, polynomial optimization and approximation algorithms. We also distinguish between on-line and off-line formulations of the problems. Results are concerned with criteria on completion times and due dates.

Approximating a finite metric by a small number of tree metrics

Abstract: Y. Bartal (1996, 1998) gave a randomized polynomial time algorithm that given any n point metric G, constructs a tree T such that the expected stretch (distortion) of any edge is at most O (log n log log n). His result has found several applications and in particular has resulted in approximation algorithms for many graph optimization problems. However approximation algorithms based on his result are inherently randomized. In this paper we derandomize the use of Bartal's algorithm in the design of approximation algorithms. We give an efficient polynomial time algorithm that given a finite n point metric G, constructs O(n log n) trees and a probability distribution /spl mu/ on them such that the expected stretch of any edge of G in a tree chosen according to /spl mu/ is at most O(log n log log n). Our result establishes that finite metrics can be probabilistically approximated by a small number of tree metrics. We obtain the first deterministic approximation algorithms for buy-at-bulk network design and vehicle routing; in addition we subsume results from our earlier work on derandomization. Our main result is obtained by a novel view of probabilistic approximation of metric spaces as a deterministic optimization problem via linear programming.

Approximation schemes for scheduling on parallel machines

Abstract: We discuss scheduling problems with m identical machines and n jobs where each job has to be assigned to some machine. The goal is to optimize objective functions that solely depend on the machine completion times. As a main result, we identify some conditions on the objective function, under which the resulting scheduling problems possess a polynomial-time approximation scheme. Our result contains, generalizes, improves, simplifies, and unifies many other results in this area in a natural way.

Successive Galerkin approximation algorithms for nonlinear optimal and robust control

Abstract: Nonlinear optimal control and nonlinear H infinity control are two of the most significant paradigms in nonlinear systems theory. Unfortunately, these problems require the solution of Hamilton-Jacobi equations, which are extremely difficult to solve in practice. To make matters worse, approximation techniques for these equations are inherently prone to the so-called 'curse of dimensionality'. While there have been many attempts to approximate these equations, solutions resulting in closed-loop control with well-defined stability and robustness have remained elusive. This paper describes a recent breakthrough in approximating the Hamilton-Jacobi-Bellman and Hamilton-Jacobi-Isaacs equations. Successive approximation and Galerkin approximation methods are combined to derive a novel algorithm that produces stabilizing, closed-loop control laws with well-defined stability regions. In addition, we show how the structure of the algorithm can be exploited to reduce the amount of computation from exponential to po...

An Outer Approximation Algorithm for Generating AllEfficient Extreme Points in the Outcome Set of a Multiple ObjectiveLinear Programming Problem

Abstract: Various difficulties have been encountered in using decision set-based vector maximization methods to solve a multiple objective linear programming problem (MOLP). Motivated by these difficulties, some researchers in recent years have suggested that outcome set-based approaches should instead be developed and used to solve problem (MOLP). In this article, we present a finite algorithm, called the Outer Approximation Algorithm, for generating the set of all efficient extreme points in the outcome set of problem (MOLP). To our knowledge, the Outer Approximation Algorithm is the first algorithm capable of generating this set. As a by-product, the algorithm also generates the weakly efficient outcome set of problem (MOLP). Because it works in the outcome set rather than in the decision set of problem (MOLP), the Outer Approximation Algorithm has several advantages over decision set-based algorithms. It is also relatively easy to implement. Preliminary computational results for a set of randomly-generated problems are reported. These results tangibly demonstrate the usefulness of using the outcome set approach of the Outer Approximation Algorithm instead of a decision set-based approach.

An improved approximation algorithm for multiway cut

Abstract: Given an undirected graph with edge costs and a subset of k nodes called terminals, a multiway cut is a subset of edges whose removal disconnects each terminal from the rest. Multiway Cut is the problem of finding a multiway cut of minimum cost. Previously, a very simple combinatorial algorithm due to Dahlhaus, Johnson, Papadimitriou, Seymour, and Yannakakis gave a performance guarantee of 2(1?1k). In this paper, we present a new linear programming relaxation for Multiway Cut and a new approximation algorithm based on it. The algorithm breaks the threshold of 2 for approximating Multiway Cut, achieving a performance ratio of at most 1.5?1k. This improves the previous result for every value of k. In particular, for k=3 we get a ratio of 76<43.

Nonlinear Programming and Variational Inequality Problems: A Unified Approach

Abstract: Preface. 1. Introduction. 2. Technical Preliminaries. 3. Instances of the Cost Approximation Algorithm. 4. Merit Functions for Variational Inequality Problems. 5. Convergence of the CA Algorithm for Nonlinear Programs. 6. Convergence of the CA Algorithm for Variational Inequality Problems. 7. Finite Identification of Active Constraints and of Solutions. 8. Parallel and Sequential Decomposition CA Algorithms. 9. A Column Generation/Simplicial Decomposition Algorithm. A. Definitions. References. Index.