Showing papers on "Approximation algorithm published in 1991"

Approximating clique is almost NP-complete

Abstract: The computational complexity of approximating omega (G), the size of the largest clique in a graph G, within a given factor is considered. It is shown that if certain approximation procedures exist, then EXPTIME=NEXPTIME and NP=P. >

When trees collide: an approximation algorithm for the generalized Steiner problem on networks

Abstract: We give the first approximation algorithm for the {\em generalized network Steiner tree problem}, a problem in network design. An instance consists of a network with link-costs and, for each pair ${i,j}$ of nodes, an edge-connectivity requirement. The goal is to find a minimum-cost network using the available links and satisfying the requirements. Our algorithm outputs a solution whose cost is within $ 2 \log R $ of optimal, where $R$ is the highest requirement value. In the course of proving the performance guarantee, we prove a combinatorial min-max approximate equality relating minimum-cost networks to maximum packings of certain kinds of cuts. As a consequence of the proof of this theorem, we obtain an approximation algorithm for optimally packing these cuts; we show that this algorithm has application to estimating the reliability of a probabilistic network.

Faster scaling algorithms for general graph matching problems

Abstract: An algorithm for minimum-cost matching on a general graph with integral edge costs is presented. The algorithm runs in time close to the fastest known bound for maximum-cardinality matching. Specifically, let n, m, and N denote the number of vertices, number of edges, and largest magnitude of a cost, respectively. The best known time bound for maximum-cardinal ity matching M 0( Am). The new algorithm for minimum-cost matching has time bound 0( in a ( m, n )Iog n m log ( nN)). A slight modification of the new algorithm finds a maximum-cardinality matching in 0( fire) time. Other applications of the new algorlthm are given, mchrding an efficient implementa- tion of Christofides' traveling salesman approximation algorithm and efficient solutions to update problems that require the linear programming duals for matching.

Fast approximation algorithms for fractional packing and covering problems

Abstract: Fast algorithms that find approximate solutions for a general class of problems, which are called fractional packing and covering problems, are presented. The only previously known algorithms for solving these problems are based on general linear programming techniques. The techniques developed greatly outperform the general methods in many applications, and are extensions of a method previously applied to find approximate solutions to multicommodity flow problems. The algorithms are based on a Lagrangian relaxation technique, and an important result is a theoretical analysis of the running time of a Lagrangian relaxation based algorithm. Several applications of the algorithms are presented. >

A packing problem with applications to lettering of maps

Abstract: The following packing problem arises in connection with lettering of maps: Given n distinct points pl, p2, . . . . pn in the plane, determine the supremum uoPi of all reals U, such that there are n pan-wise dtsjomt, axis-parallel, closed squares Ql, Q2, . . . . Qn of side-length u, where each pi ts a corner of Qi. Note that — by using afine transformation — the problem is equivalent to the case when we want largest homothetic cop~es of a jized rectangle or parallelogram tnstead of equal ly-szzed squares. In the cartographic application, the points are items (groundwater-drillho les etc.) and the squares are places for labels associated with these items (sulphate concentration etc.). An algorithm is presented, that in O(n log n] time either produces a solution, that is guaranteed to be at least half as large as the supremum. This is optimal, m the sense that the corresponding decision problem is NP complete, no po[ynomzal approximation algorithm with a guaranteed factor ezceedmg ~ exwts, provided that P # AfP; and there M also a lower bound of C2(n log n) for the running time.

Fast approximation algorithms for multicommodity flow problems

Abstract: All previously known algorithms for solving the multicommodity flow problem with capacities are based on linear programming. The best of these algorithms uses a fast matrix multiplication algorithm and takes O(k3.5n3m0.5 log(nDU)) time for the multicommodity flow problem with integer demands and at least O(k2.5n2m0.5 log(n��1DU)) time to find an approximate solution, where k is the number of commodities, n and m denote the number of nodes and edges in the network, D is the largest demand, and U is the largest edge capacity. As a consequence, even multicommodity flow problems with just a few commodities are believed to be much harder than single-commodity maximum-flow or minimum-cost flow problems. In this paper, we describe the first polynomial-time combinatorial algorithms for approximately solving the multicommodity flow problem. The running time of our randomized algorithm is (up to log factors) the same as the time needed to solve k single-commodity flow problems, thus giving the surprising result that approximately computing a k-commodity maximum-flow is not much harder than computing about k single-commodity maximum-flows in isolation. In fact, we prove that a (simple) k-commodity flow problem can be approximately solved by approximately solving O(k log2n) single-commodity minimum-cost flow problems. Our k-commodity algorithm runs in O (knm log4n) time with high probability. We also describe a deterministic algorithm that uses an O(k)-factor more time. Given any multicommodity flow problem as input, both algorithms are guaranteed to provide a feasible solution to a modified flow problem in which all capacities are increased by a (1 + �)-factor, or to provide a proof that there is no feasible solution to the original problem. We also describe faster approximation algorithms for multicommodity flow problems with a special structure, such as those that arise in "sparsest cut" problems and uniform concurrent flow problems.

Load adjustment in adaptive real-time systems

Abstract: A framework is given for discussing how to adjust load in order to handle periodic processes whose timing parameters vary with time. The schedulability of adjustable periodic processes by a preemptive fixed priority scheduler is formulated in terms of a configuration selection problem. Specifically, two process transformations are introduced for the purpose of deriving a bound for the achievable utilization factor of processes whose periods are related by harmonics. This result is then generalized so that the bound is applicable to any process set and an efficient algorithm to calculate the bound is provided. When the list of allowable configurations is implicitly given by a set of scalable periodic processes, the corresponding period assignment problem is shown to be NP-complete. The authors present an approximation algorithm for the period assignment problem for which some encouraging experimental results are included. >

Improved approximation algorithms for shop scheduling problems

Abstract: In the job shop scheduling problem, there are $m$ machines and $n$ jobs. A job consists of a sequence of operations, each of which must be processed on a specified machine, and the aim is to complete all jobs as quickly as possible. This problem is strongly ${\cal NP}$-hard even for very restrictive special cases. The authors give the first randomized and deterministic polynomial-time algorithms that yield polylogarithmic approximations to the optimal length schedule. These algorithms also extend to the more general case where a job is given not by a linear ordering of the machines on which it must be processed but by an arbitrary partial order. Comparable bounds can also be obtained when there are $m'$ types of machines, a specified number of machines of each type, and each operation must be processed on one of the machines of a specified type, as well as for the problem of scheduling unrelated parallel machines subject to chain precedence constraints.

Finding k-cuts within twice the optimal

Abstract: Two simple approximation algorithms are presented for the minimum k-cut problem. Each algorithm finds a k-cut having weight within a factor of (2-2/k) of the optimal. One of the algorithms is particularly efficient, requiring a total of only n-1 maximum flow computations for finding a set of near-optimal k-cuts, one for each value of k between 2 and n. >

Routing of multipoint connections

Abstract: The author addresses the problem of routing connections in a large-scale packet-switched network supporting multipoint communications. He gives a formal definition of several versions of the multipoint problem, including both static and dynamic versions. He looks at the Steiner tree problem as an example of the static problem and considers the experimental performance of two approximation algorithms for this problem. A weighted greedy algorithm is considered for a version of the dynamic problem which allows endpoints to come and go during the life of a connection. One of the static algorithms serves as a reference to measure the performance of the proposed weighted greedy algorithm in a series of experiments. >

Balancing free square-root algorithm for computing singular perturbation approximations

Abstract: The author proposes a novel algorithm with enhanced numerical robustness for computing singular perturbation approximations of linear, continuous or discrete systems. This algorithm circumvents the computation of possibly ill-conditioned balancing transformations. Instead, well-conditioned projection matrices are determined for computing state-space representations suitable for applying the singular perturbation formulas. The projection matrices are computed using the Cholesky (square-root) factors of the gramians. The proposed algorithm is intended for efficient computer implementation. It can handle both minimal and nonminimal systems. >

Linear and nonlinear algorithms for identification in H ∞ with error bounds

Abstract: In this paper, a linear and a nonlinear algorithm are presented for the problem of system "identification in H∞", posed by Helmicki, Jacobson and Nett. We derive some error bounds for the linear algorithm which indicate that if the model error is not too high, then this algorithm has good guaranteed error properties. The linear algorithm requires only FFT (fast Fourier transform) computations. A nonlinear algorithm, which requires an additional step of solving a Nehari best approximation problem, is also presented that has the robust convergence property.

Genetic algorithms for numerical optimization

Abstract: Genetic algorithms (GAs) are stochastic adaptive algorithms whose search method is based on simulation of natural genetic inheritance and Darwinian striving for survival. They can be used to find approximate solutions to numerical optimization problems in cases where finding the exact optimum is prohibitively expensive, or where no algorithm is known. However, such applications can encounter problems that sometimes delay, if not prevent, finding the optimal solutions with desired precision. In this paper we describe applications of GAs to numerical optimization, present three novel ways to handle such problems, and give some experimental results.

Approximating Treewidth, Pathwidth, and Minimum Elimination Tree Height

Abstract: We show how the value of various parameters of graphs connected to sparse matrix factorization and other applications can be approximated using an algorithm of Leighton et al. that finds vertex separators of graphs. The approximate values of the parameters, which include minimum front size, treewidth, pathwidth, and minimum elimination tree height, are no more than O(log n) (minimum front size and treewidth) and O(log2n) (pathwidth and minimum elimination tree height) times the optimal values. In addition we examine the existence of bounded approximation algorithms for the parameters, and show that unless P = NP, there are no absolute approximation algorithms for them.

Periodic multiprocessor scheduling

Abstract: A number of scheduling and assignment problems are presented involving the execution of periodic operations in a multiprocessor environment. We consider the computational complexity of these problems and propose approximation algorithms for operations with identical periods as well as for operations with arbitrary integer periods.

Error bounded variable distance offset operator for free form curves and surfaces

Abstract: Most offset approximation algorithms for freeform curves and surfaces may be classified into two main groups. The first approximates the curve using simple primitives such as piecewise arcs and lines and then calculates the (exact) offset operator to this approximation. The second offsets the control polygon/mesh and then attempts to estimate the error of the approximated offset over a region. Most of the current offset algorithms estimate the error using a finite set of samples taken from the region and therefore can not guarantee the offset approximation is within a given tolerance over the whole curve or surface. This paper presents new methods to globally bound the error of the approximated offset of freeform curves and surfaces and then automatically derive new approximations with improved accuracy. These tools can also be used to develop a global error bound for a variable distance offset operation and to detect and trim out loops in the offset.

Numerical Methods for Stochastic Singular Control Problems

Abstract: The paper develops a powerful class of numerical methods for stochastic singular control problems. The basic models used are diffusion or reflected diffusions, but the method is of general applicability. The central idea is that of the Markov chain approximation method, where an approximation to the control problem is found for which an optimal solution is computable, and which is an arbitrarily good approximation to the original problem and its optimal value function. The methods are convenient to program and use (and they have been used with success), and they cover a wide variety of problems. In fact, for the singular problem, they seem to be the only ones currently available. Owing to problems in proving tightness of certain processes that occur in the convergence proofs, the methods of proof used for the nonsingular problems need modifications. Examples of useful approximations, the algorithms, and the convergence proofs are given. To illustrate the power of the methods, two classes of problems are d...

A natural metric for curves—computing the distance for polygonal chains and approximation algorithms

Abstract: The often explored problem to approximate a given polygonal chain has been considered from a computational geometric point of view only for a short time. To model it reasonably we give a natural definition of the distance between curves. Furthermore we give algorithms to calculate this distance between two polygonal chains in the d-dimensional space for arbitrary d. With known methods this yields polynomial time algorithms to approximate polygonal chains. These algorithms find an optimal solution under some constraints. We will show that this solution is only by a constant factor worse than the global optimum.

Approximation and intractability results for the maximum cut problem and its variants

Abstract: The maximum cut problem is known to be an important NP-complete problem with many applications. The authors investigate this problem (which they call the normal maximum cut problem) and a variant of it (which is referred to as the connected maximum cut problem). They show that any n-vertex e-edge graph admits a cut with at least the fraction 1/2+1/2n of its edges, thus improving the ratio 1/2+2/e known before. It is shown that it is NP-complete to decide if a given graph has a normal maximum cut with at least a fraction (1/2+ epsilon ) of its edges, where the positive constant epsilon can be taken smaller than any value chosen. The authors present an approximation algorithm for the normal maximum cut problem on any graph that runs in O((e log e+n log n)/p+log p*log n) parallel time using p(1 >

On rectilinear link distance

Abstract: In this paper we study two link distance problems for rectilinear paths inside a simple rectilinear polygon P.First, we present a data structure using O(n log n) storage such that a shortest path between two query points can be computed efficiently. If both query points are vertices of P, the query time is O(1 + l), where l is the number of links. If the query points are arbitrary points inside P, then the query time becomes O(log n + l). The resulting path is not only optimal in the rectilinear link metric, but it is optimal in the L1-metric as well. Secondly, it is shown that the rectilinear link diameter of P can be computed in time O(n log n). We also give an approximation algorithm that runs in linear time. This algorithm produces a solution that differs by at most three links from the exact diameter.The solutions are based on a rectilinear version of Chazelle's polygon cutting theorem. This new theorem states that any simple rectilinear polygon can be cut into two rectilinear subpolygons of size at most 34 times the original size, and that such a cut segment can be found in linear time.

Simulating BPP using a general weak random source

Abstract: It is shown how to simulate BPP and approximation algorithms in polynomial time using the output from a delta -source. A delta -source is a weak random source that is asked only once for R bits, and must output an R-bit string according to some distribution that places probability no more than 2/sup - delta R/ on any particular string. Also given are two applications: one to show the difficulty of approximating the size of the maximum clique, and the other to the problem of implicit O(1) probe search. >

From objects to classes: algorithms for optimal objection-oriented design

Abstract: Introduces a novel, axiomatically defined, object-oriented data model called the Demeter kernel model; and secondly, presents abstraction and optimisation algorithms and their relationships for designing classes from objects in the kernel model. The authors analyse several computational problems underlying the class design process, which is divided into two phases; a learning phase and an optimisation phase. This study focuses on approximation algorithms for the optimisation phase and aims to lead to a better understanding and a partial automation of the object-oriented design process. The algorithms and the theory presented have been implemented in the C++Demeter System, a CASE tool for object-oriented design and programming. >

Facility dispersion problems: Heuristics and special cases

Abstract: Facility dispersion problem deals with the location of facilities on a network so as to maximize some function of the distances between facilities. We consider the problem under two different optimality criteria, namely maximizing the minimum distance (MAX-MIN) between any pair of facilities and maximizing the average distance (MAX-AVG) between any pair of facilities. Under either criterion, the problem is known to be NP-hard, even when the distances satisfy the triangle inequality. We consider the question of obtaining near-optimal solutions. For the MAX-MIN criterion, we show that if the distances do not satisfy the triangle inequality, there is no polynomial time relative approximation algorithm unless P=NP. When the distances do satisfy the triangle inequality, we present an efficient heuristic which provides a performance guarantee of 2, thus improving the performance guarantee of 3 proven in [Wh91]. We also prove that obtaining a performance guarantee of less than 2 is NP-hard. For the MAX-AVG criterion, we present a heuristic which provides a performance guarantee of 4, provided that the distances satisfy the triangle inequality. For the 1-dimensional dispersion problem, we provide polynomial time algorithms for obtaining optimal solutions under both MAX-MIN and MAX-AVG criteria. Using the latter algorithm, we obtain a heuristic which provides a performance guarantee of 4(\(\sqrt 2 - 1\)) ≈ 1.657 for the 2-dimensional dispersion problem under the MAX-AVG criterion.

Multiterminal global routing: A deterministic approximation scheme

Abstract: We consider the problem of routing multiterminal nets in a two-dimensional gate-array. Given a gate-array and a set of nets to be routed, we wish to find a routing that uses as little channel space as possible. We present a deterministic approximation algorithm that uses close to the minimum possible channel space. We cast the routing problem as a new form of zero-one multicommodity flow, an integer-programming problem. We solve this integer program approximately by first solving its linear-program relaxation and then rounding any fractions that appear in the solution to the linear program. The running time of the rounding algorithm is exponential in the number of terminals in a net but polynomial in the number of nets and the size of the array. The algorithm is thus best suited to cases where the number of terminals on each net is small.

Planar graph augmentation problems

Abstract: In this paper we investigate the problem of adding a minimum number of edges to a planar graph in such a way that the resulting graph is biconnected and still planar. It is shown that this problem is NP-complete. We present an approximation algorithm for this planar biconnectivity augmentation problem that has performance ratio 3/2 and uses O(n2 log n) time. An O(n3) approximation algorithm with performance ratio 5/4 is presented to make a biconnected planar graph triconnected by adding edges without losing planarity.