Showing papers on "Approximation algorithm published in 1992"

Multivariate stochastic approximation using a simultaneous perturbation gradient approximation

Abstract: The problem of finding a root of the multivariate gradient equation that arises in function minimization is considered. When only noisy measurements of the function are available, a stochastic approximation (SA) algorithm for the general Kiefer-Wolfowitz type is appropriate for estimating the root. The paper presents an SA algorithm that is based on a simultaneous perturbation gradient approximation instead of the standard finite-difference approximation of Keifer-Wolfowitz type procedures. Theory and numerical experience indicate that the algorithm can be significantly more efficient than the standard algorithms in large-dimensional problems. >

Proof verification and hardness of approximation problems

Abstract: The class PCP(f(n),g(n)) consists of all languages L for which there exists a polynomial-time probabilistic oracle machine that used O(f(n)) random bits, queries O(g(n)) bits of its oracle and behaves as follows: If x in L then there exists an oracle y such that the machine accepts for all random choices but if x not in L then for every oracle y the machine rejects with high probability. Arora and Safra (1992) characterized NP as PCP(log n, (loglogn)/sup O(1)/). The authors improve on their result by showing that NP=PCP(logn, 1). The result has the following consequences: (1) MAXSNP-hard problems (e.g. metric TSP, MAX-SAT, MAX-CUT) do not have polynomial time approximation schemes unless P=NP; and (2) for some epsilon >0 the size of the maximal clique in a graph cannot be approximated within a factor of n/sup epsilon / unless P=NP. >

Job Shop Scheduling by Simulated Annealing

Abstract: We describe an approximation algorithm for the problem of finding the minimum makespan in a job shop. The algorithm is based on simulated annealing, a generalization of the well known iterative improvement approach to combinatorial optimization problems. The generalization involves the acceptance of cost-increasing transitions with a nonzero probability to avoid getting stuck in local minima. We prove that our algorithm asymptotically converges in probability to a globally minimal solution, despite the fact that the Markov chains generated by the algorithm are generally not irreducible. Computational experiments show that our algorithm can find shorter makespans than two recent approximation approaches that are more tailored to the job shop scheduling problem. This is, however, at the cost of large running times.

The Steiner Tree Problem

Abstract: Euclidean Steiner Problem. Introduction. Historical Background. Some Basic Notions. Some Basic Properties. Full Steiner Trees. Steiner Hulls and Decompositions. The Number of Steiner Topologies. Computational Complexity. Physical Models. References. Exact Algorithms. The Melzak Algorithm. A Linear-Time FST Algorithm. Two Ideas on the Melzak Algorithm. A Numberical Algorithm. Pruning. The GEOSTEINER Algorithm. The Negative Edge Algorithm. The Luminary Algorithm. References. The Steiner Ratio. Lower Bounds of rho. The Small n Case. The Variational Approach. The Steiner Ratio Conjecture as a Maximin Problem. Critical Structures. A Proof of the Steiner Ratio Conjecture. References. Heuristics. Minimal Spanning Trees. Improving the MST. Greedy Trees. An Annealing Algorithm. A Partitioning Algorithm. Few's Algorithms. A Graph Approximation Algorithm. k-Size Quasi-Steiner Trees. Other Heuristics. References. Special Terminal-Sets. Four Terminals. Cocircular Terminals. Co-path Terminals. Terminals on Lattice Points. Two Related Results. References. Generalizations. d-Dimensional Euclidean Spaces. Cost of Edge. Terminal Clusters and New Terminals. k-SMT. Obstacles. References. Steiner Problem in Networks. Introduction. Applications. Definitions. Trivial Special Cases. Problem Reformulations. Complexity. References. Reductions. Exclusion Tests. Inclusion Tests. Integration of Tests. Effectiveness of Reductions. References. Exact Algorithms. Spanning Tree Enumeration Algorithm. Degree-Constrained Tree Enumeration Algorithm. Topology Enumeration Algorithm. Dynamic Programming Algorithm. Branch-and-Bound Algorithm. Mathematical Programming Formulations. Linear Relaxations. Lagrangean Relaxations. Benders' Decomposition Algorithm. Set Covering Algorithm. Summary and Computational Experience. References. Heuristics. Path Heurisitics. Tree Heuristics. Vertex Heuristics. Contraction Heuristic. Dual Ascent Heuristic. Set Covering Heuristic. Summary and Computational Experience. References. Polynomially Sovable Cases. Series-Parallel Networks. Halin Networks. k-Planar Networks. Strongly Chordal Graphs. References. Generalizations. Steiner Trees in Directed Networks. Weighted Steiner Tree Problem. Steiner Forest Problem. Hierarchical Steiner Tree Problem. Degree-Dependent Steiner Tree Problem. Group Steiner Tree Problem. Multiple Steiner Trees Problem. Multiconnected Steiner Network Problem. Steiner Problem in Probabilistic Networks. Realization of Distance Matrices. Other Steiner-Like Problems. References. Rectilinear Steiner Problem. Introduction. Definitions. Basic Properties. A Characterization of RSMTs. Problem Reductions. Extremal Results. Computational Complexity. Exact Algorithms. References. Heuristic Algorithms. Heuristics Using a Given RMST. Heuristics Based on MST Algorithms. Computational Geometry Paradigms. Other Heuristics. References. Polynomially Solvable Cases. Terminals on a Rectangular Boundary. Rectilinearly Convex Boundary.

A general approximation technique for constrained forest problems

Abstract: We present a general approximation technique for a large class of graph problems. Our technique mostly applies to problems of covering, at minimum cost, the vertices of a graph with trees, cycles or paths satisfying certain requirements. In particular, many basic combinatorial optimization problems fit in this framework, including the shortest path, minimum spanning tree, minimum-weight perfect matching, traveling salesman and Steiner tree problems.Our technique produces approximation algorithms that run in O(n2 log n) time and come within a factor of 2 of optimal for most of these problems. For instance, we obtain a 2-approximation algorithm for the minimum-weight perfect matching problem under the triangle inequality. Our running time of O(n2 log n) time compares favorably with the best strongly polynomial exact algorithms running in O(n3) time for dense graphs. A similar result is obtained for the 2-matching problem and its variants.We also derive the first approximation algorithms for many NP-complete problems, including the non-fixed point-to-point connection problem, the exact path partitioning problem and complex location-design problems. Moreover, for the prize-collecting traveling salesman or Steiner tree problems, we obtain 2-approximation algorithms, therefore improving the previously best-known performance guarantees of 2.5 and 3, respectively [4].

Improved Bounds for Mixing Rates of Markov Chains and Multicommodity Flow

Abstract: The paper is concerned with tools for the quantitative analysis of finite Markov chains whose states are combinatorial structures. Chains of this kind have algorithmic applications in many areas, including random sampling, approximate counting, statistical physics and combinatorial optimisation. The efficiency of the resulting algorithms depends crucially on the mixing rate of the chain, i.e., the time taken for it to reach its stationary or equilibrium distribution.The paper presents a new upper bound on the mixing rate, based on the solution to a multicommodity flow problem in the Markov chain viewed as a graph. The bound gives sharper estimates for the mixing rate of several important complex Markov chains. As a result, improved bounds are obtained for the runtimes of randomised approximation algorithms for various problems, including computing the permanent of a 0–1 matrix, counting matchings in graphs, and computing the partition function of a ferromagnetic Ising system. Moreover, solutions to the multicommodity flow problem are shown to capture the mixing rate quite closely: thus, under fairly general conditions, a Markov chain is rapidly mixing if and only if it supports a flow of low cost.

A simple, accurate method to calculate spread-spectrum multiple-access error probabilities

Abstract: The exact calculation of error probabilities for direct-sequence spread-spectrum multiple-access (DS/SSMA) systems has been addressed in the literature. The exact calculation is computationally difficult, so emphasis has been on approximations and bounds. One particularly attractive approximation is to just use a signal-to-noise ratio in a Gaussian approximation, the 'standard approximation'. Unfortunately, that approximation is not generally accurate enough. An improved Gaussian approximation with good accuracy has recently been presented. The author derives an accurate Gaussian approximation which is also computationally very simple. >

Approximating maximum independent sets by excluding subgraphs

Abstract: An approximation algorithm for the maximum independent set problem is given, improving the best performance guarantee known toO(n/(logn)2). We also obtain the same performance guarantee for graph coloring. The results can be combined into a surprisingly strongsimultaneous performance guarantee for the clique and coloring problems. The framework ofsubgraph-excluding algorithms is presented. We survey the known approximation algorithms for the independent set (clique), coloring, and vertex cover problems and show how almost all fit into that framework. We show that among subgraph-excluding algorithms, the ones presented achieve the optimal asymptotic performance guarantees.

e-approximations with minimum packing constraint violation (extended abstract)

Abstract: We present efficient new randomized and deterministic methods for transforming optimal solutions for a type of relaxed integer linear program into provably good solutions for the corresponding NP-hard discrete optimization problem. Without any constraint violation, the e-approximation problem for many problems of this type is itself NP-hard. Our methods provide polynomial-time e-approximations while attempting to minimize the packing constraint violation.Our methods lead to the first known approximation algorithms with provable performance guarantees for the s-median problem, the tree prunning problem, and the generalized assignment problem. These important problems have numerous applications to data compression, vector quantization, memory-based learning, computer graphics, image processing, clustering, regression, network location, scheduling, and communication. We provide evidence via reductions that our approximation algorithms are nearly optimal in terms of the packing constraint violation. We also discuss some recent applications of our techniques to scheduling problems.

Approximations with Minimum Packing Constraint Violation

Abstract: We present efficient new randomized and deterministic methods for transforming optimal solutions for a type of relaxed integer linear program into provably good solutions for the corresponding NP-hard discrete optimization problem. Without any constraint violation, the epsilon-approximation problem for many problems of this type is itself NP-hard. Our methods provide polynomial-time epsilon-approximations while attempting to minimize the packing constraint violation. Our methods lead to the first known approximation algorithms with provable performance guarantees for the s-median problem, the tree pruning problem, and the generalized assignment problem. These important problems have numerous applications to data compression, vector quantization, memory-based learning, computer graphics, image processing, clustering, regression, network location, scheduling, protocol testing, and communication. We provide evidence via reductions that our approximation algorithms are nearly optimal in terms of the packing constraint violation. We also discuss some recent applications of our techniques to scheduling problems.

Multicasting for multimedia applications

Abstract: The authors investigate multicast routing for high-bandwidth delay-sensitive applications in a point-to-point network as an optimization problem. They associate an edge cost and an edge delay with each edge in the network. The problem is to construct a tree spanning the destination nodes, such that it has the least cost, and so that the delay on the path from the source to each destination is bounded. Since the problem is computationally intractable, the authors present an efficient approximation algorithm. Experimental results through simulations show that the performance of the heuristic is near optimal. >

Approximation of FIR by IIR digital filters: an algorithm based on balanced model reduction

Abstract: An algorithm for the approximation of finite impulse response (FIR) filters by infinite impulse response (IIR) filters is presented. The algorithm is based on a concept of balanced model reduction. The matrix inversions normally associated with this procedure are notoriously error prone due to ill conditioning of the special matrix forms required. This difficulty is circumvented here by directly formulating a reduced state-space system description which is input/output equivalent to the system that would more conventionally be obtained following the explicit step of constructing an (interim) balanced realization. Examples of FIR by IIR filter approximations are included. >

On the complexity of approximating the independent set problem

Abstract: We show that for some positive constant c it is not feasible to approximate Independent Set (for graphs of n nodes) within a factor of n c , provided Maximum 2-Satisfiability does not have a randomized polynomial time approximation scheme We also study reductions preserving the quality of approximations and exhibit complete problems

Interprocedural aliasing in the presence of pointers

Abstract: An Alias occurs at some program point during execution when two more more names exist for the same location. We've investigated the theoretical difficulty of determining the aliases of a program, developed an approximation algorithm for solving for aliases in C like languages, and explored the precision (i.e., closeness of our approximate solution to the actual solution) and time behavior of this algorithm. Myers (Mye81) explored the theoretical difficulty of solving flow sensitive interprocedural data flow problems in the presence of aliasing. However, he did not make any claims about the difficulty of determining aliases. We isolate various programming language mechanisms that create aliases. The complexity of the alias problem (i.e., determining the aliases for a program) induced by each mechanism and their combinations is considered separately and categorized as ${\cal NP}$-hard, complement ${\cal NP}$-hard, or polynomial time $({\cal P}).$ We proved that if there are two (or more) levels of indirection, regardless of the alias mechanisms used, aliasing is ${\cal NP}$-hard. However, if only one level of indirection is possible, then the alias problem is in ${\cal P}.$ In addition, we show the alias problem in the presence of some mechanisms to be ${\cal P}$-space complete. Therefore, in a language which allows general purpose pointers, the problem of determining which aliases can occur during program execution is ${\cal NP}$-hard. We present an algorithm which safely approximates Interprocedural May Alias (i.e., we fail to report an alias only if it never occurs) in the presence of pointers. We are able to show, relative to our definition of precision, that our algorithm is as precise as possible in the worst case. We also pinpoint the sources of imprecision in our algorithm and can empirically bound its precision. Our algorithm has been implemented in a prototype analysis tool for C programs, and we present a preliminary empirical investigation of algorithm speed and precision.

The swapping problem

Abstract: Each vertex of a graph initially may contain an object of a known type. A final state, specifying the type of object desired at each vertex, is also given. A single vehicle of unit capacity is available for shipping objects among the vertices. The swapping problem is to compute a shortest route such that a vehicle can accomplish the rearrangement of the objects while following this route. We exhibit several structural properties of shortest routes and develop polynomial approximation algorithms that are variations of a well-known “patching” algorithm for the traveling salesman problem. We prove tight constant performance guarantees for these algorithms and note as a side product that these bounds hold and are tight also for the latter problem.

Approximation algorithms for indefinite quadratic programming

Abstract: We considerź-approximation schemes for indefinite quadratic programming. We argue that such an approximation can be found in polynomial time for fixedź andt, wheret denotes the number of negative eigenvalues of the quadratic term. Our algorithm is polynomial in 1/ź for fixedt, and exponential int for fixedź. We next look at the special case of knapsack problems, showing that a more efficient (polynomial int) approximation algorithm exists.

Multiple alignment, communication cost, and graph matching

Abstract: Multiple sequence alignment is an important problem in computational molecular biology. Dynamic programming for optimal multiple alignment requires too much time to be practical. Although many algorithms for suboptimal alignment have been suggested, no “performance guarantees” algorithms have been known until recently. A computationally efficient approximation multiple alignment algorithm with guaranteed error bounds equal to the normalized communication cost of a corresponding graph is given in this paper. Recently, Altschul and Lipman [SIAM J. Appl. Math., 49 (1989), pp. 197–209] used suboptimal alignments for reducing the computational complexity of the optimal alignment algorithm. This paper develops the Altschul–Lipman approach and demonstrates that bounds for optimal multiple alignment of k sequences can be derived from a solution of the maximum weighted matching problem in a k-vertex graph. Fast maximum matching algorithms allow efficient implementation of dynamic bounds for the multiple alignment ...

Algorithms for the Container Loading Problem

Abstract: In this paper we consider the three-dimensional problem of optimal packing of a container with rectangular pieces. We propose an approximation algorithm based on the “forward state strategy” of dynamic programming. A suitable description of packings is developed for the implementation of the approximation algorithm, and some computational experience is reported.

On the Approximability of the Maximum Common Subgraph Problem

Abstract: Some versions of the maximum common subgraph problem are studied and approximation algorithms are given. The maximum bounded common induced subgraph problem is shown to be Max SNP-hard and the maximum unbounded common induced subgraph problem is shown to be as hard to approximate as the maximum independent set problem. The maximum common induced connected subgraph problem is still harder to approximate and is shown to be NPO PB-complete, i.e. complete in the class of optimization problems with optimal value bounded by a polynomial.

Approximation Algorithms for Graph Augmentation

Abstract: We study the problem of increasing the connectivity1 of a graph at an optimal cost. Since the general problem is NP-hard, we focus on efficient approximation schemes that come within a constant factor from the optimal. Previous algorithms either do not take edge costs into consideration, or run slower than our algorithm. Our algorithm takes as input an undirected graph G0 = (V, E0) on n vertices, that is not necessarily connected, and a set Feasible of m weighted edges on V, and outputs a subset Aug of edges which when added to G0 make it 2-connected. The weight of Aug, when G0 is initially connected, is no more than twice the weight of the least weight subset of edges of Feasible that increases the connectivity to 2. The running time of our algorithm is O(m + n logn). We also study the problem of increasing the edge connectivity of any graph G, to k, within a factor of 2 (for any k > 0). The running time of this algorithm is O(nk log n(m + n log n)). We observe that when k is odd we can use different techniques to obtain an approximation factor of 2 for increasing edge connectivity from k to (k+1) in O(kn2) time.

Approximation algorithms for minimum time broadcast (extended abstract)

Abstract: This paper deals with the problem of broadcasting in minimum time. Approximation algorithms are developed for arbitrary graphs, as well as for several restricted graph classes.

A distributed M-ary hypothesis testing problem with correlated observations

Abstract: The distributed M-ary hypothesis testing problem of detecting one of M/sub 0/ events by an (N+1)-person hierarchical team, when the observations are correlated, is examined. In this problem, each of the N subordinate decision makers (DMs) transmits one of a prespecified set of messages based on their data to a primary decision maker who, in turn, combines the messages with his or her own data to make the final team decision. The necessary conditions for the optimal decision rules of the DMs are derived. A nonlinear Gauss-Seidel iterative algorithm is developed for the person-by-person optimal decision rules, and its monotonic convergence to the person-by-person optimum is established. A fast approximation algorithm is proposed for computing certain conditional probabilities arising in the person-by-person optimal decision rules. The algorithms are illustrated with several examples, and implications for distributed organizational designs are pointed out. >

When Cycles Collapse: A General Approximation Technique for Constrained Two-Connectivity Problems

Abstract: We present a general approximation technique for a class of network design problems where we seek a network of minimum cost that satisfies certain communication requirements and is resilient to worst-case single-link failures Our algorithm runs in $O(n^2 \log n)$ time on a graph ith $n$ nodes and outputs a solution of cost at most thrice the optimum We extend our technique to obtain approximation algorithms for augmenting a given network so as to satisfy certain communication requirements and achieve resilience to single-link failures Our technique allows one to find nearly minimum-cost two-connected networks for a variety of connectivity requirements For example, our result generalizes earlier results on finding a minimum-cost two-connected subgraph of a given edge-weighted graph and an earlier result on finding a minimum-cost subgraph two-connecting a specified subset of the nodes Using our technique, we can also approximately solve for the first time a two-connected version of the generalized Steiner network problem and a two-connected version of the non-fixed point-to-point connection problem