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21
On Syntactic versus Computational Views of Approximability
, 1994
"... We attempt to reconcile the two distinct views of approximation classes: syntactic and computational. Syntactic classes such as MAX SNP permit structural results and have natural complete problems, while computational classes such as APX allow us to work with classes of problems whose approximabilit ..."
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Cited by 126 (11 self)
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We attempt to reconcile the two distinct views of approximation classes: syntactic and computational. Syntactic classes such as MAX SNP permit structural results and have natural complete problems, while computational classes such as APX allow us to work with classes of problems whose approximability is wellunderstood. Our results provide a syntactic characterization of computational classes, and give a computational framework for syntactic classes. We compare the syntactically defined class MAX SNP with the computationally defined class APX, and show that every problem in APX can be “placed" (i.e. has approximation preserving reduction to a problem) in MAX SNP. Our methods introduce a general technique for creating approximationpreserving reductions which show that any “well ” approximable problem can be reduced in an approximationpreserving manner to a problem which is hard to approximate to corresponding factors. We demonstrate this technique by applying it to the classes RMAX(2) and MIN F+n2 (1) which have the clique problem and the set cover problem, respectively, as complete problems. We use the syntactic nature of MAX SNP to define a general paradigm, nonoblivious local search, useful for developing simple yet efficient approximation algorithms. We show that such algorithms can find good approximations for all MAX SNP problems, yielding approximution ratios comparable to the bestknown for a variety of specific MAX SNPhard problem. Nonoblivious local search provably outperforms standard local search in both the degree of approximation achieved and the efficiency of the resulting algorithms.
On the Complexity of Nash Equilibria and Other Fixed Points (Extended Abstract)
 IN PROC. FOCS
, 2007
"... We reexamine what it means to compute Nash equilibria and, more generally, what it means to compute a fixed point of a given Brouwer function, and we investigate the complexity of the associated problems. Specifically, we study the complexity of the following problem: given a finite game, Γ, with 3 ..."
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Cited by 68 (8 self)
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We reexamine what it means to compute Nash equilibria and, more generally, what it means to compute a fixed point of a given Brouwer function, and we investigate the complexity of the associated problems. Specifically, we study the complexity of the following problem: given a finite game, Γ, with 3 or more players, and given ɛ> 0, compute an approximation within ɛ of some (actual) Nash equilibrium. We show that approximation of an actual Nash Equilibrium, even to within any nontrivial constant additive factor ɛ < 1/2 in just one desired coordinate, is at least as hard as the long standing squareroot sum problem, as well as a more general arithmetic circuit decision problem that characterizes Ptime in a unitcost model of computation with arbitrary precision rational arithmetic; thus placing the approximation problem in P, or even NP, would resolve major open problems in the complexity of numerical computation. We show similar results for market equilibria: it is hard to estimate with any nontrivial accuracy the equilibrium prices in an exchange economy with a unique equilibrium, where the economy is given by explicit algebraic formulas for the excess demand functions. We define a class, FIXP, which captures search problems that can be cast as fixed point
A StateOfTheArt Review Of JobShop Scheduling Techniques
, 1998
"... A great deal of research has been focused on solving the jobshop problem (P J ), over the last forty years, resulting in a wide variety of approaches. Recently, much effort has been concentrated on hybrid methods to solve P J as a single technique cannot solve this stubborn problem. As a result muc ..."
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Cited by 32 (0 self)
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A great deal of research has been focused on solving the jobshop problem (P J ), over the last forty years, resulting in a wide variety of approaches. Recently, much effort has been concentrated on hybrid methods to solve P J as a single technique cannot solve this stubborn problem. As a result much effort has recently been concentrated on techniques that combine myopic problem specific methods and a metastrategy which guides the search out of local optima. These approaches currently provide the best results. Such hybrid techniques are known as iterated local search algorithms or metaheuristics. In this paper we seek to assess the work done in the jobshop domain by providing a review of many of the techniques used. The impact of the major contributions is indicated by applying these techniques to a set of standard benchmark problems. It is established that methods such as Tabu Search, Genetic Algorithms, Simulated Annealing should be considered complementary rather than competitive...
The Power of Local Optimization: Approximation Algorithms for Maximumleaf Spanning Tree
 In Proceedings, Thirtieth Annual Allerton Conference on Communication, Control and Computing
, 1996
"... Given an undirected graph G, finding a spanning tree of G with maximum number of leaves is NPcomplete. We use the simple technique of local optimization to provide the first approximation algorithms for this problem. Our algorithms run in polynomial time to produce locally optimal solutions. We pro ..."
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Cited by 25 (3 self)
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Given an undirected graph G, finding a spanning tree of G with maximum number of leaves is NPcomplete. We use the simple technique of local optimization to provide the first approximation algorithms for this problem. Our algorithms run in polynomial time to produce locally optimal solutions. We prove that locally optimal solutions to this problem are globally nearoptimal. In particular, we prove that two such algorithms have performance ratios of 5 and 3. The latter algorithm employs more powerful localimprovement steps than the former and hence has higher running time. This may indicate an interesting tradeoff between the performance ratios and the running times of the series of algorithms we describe. Keywords: Approximation algorithms, NPcomplete problems, Performance ratio, Local optimization, Communication network design, Combinatorial algorithms. 1 Introduction Given an undirected graph G = (V; E), the Maximum Leaf Spanning Tree problem is to find a spanning tree of G with ...
Combining Helpful Sets and Parallel Simulated Annealing for the GraphPartitioning Problem
 INT. J. PARALLEL ALGORITHMS AND APPLICATIONS
, 1996
"... In this paper we present a new algorithm for the kpartitioning problem which achieves an improved solution quality compared to known heuristics. We apply the principle of so called "helpful sets", which has shown to be very efficient for graph bisection, to the direct kpartitioning prob ..."
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Cited by 14 (4 self)
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In this paper we present a new algorithm for the kpartitioning problem which achieves an improved solution quality compared to known heuristics. We apply the principle of so called "helpful sets", which has shown to be very efficient for graph bisection, to the direct kpartitioning problem. The principle is extended in several ways. We introduce a new abstraction technique which shrinks the graph during runtime in a dynamic way leading to shorter computation times and improved solutions qualities. The use of stochastic methods provides further improvements in terms of solution quality. Additionally we present a parallel implementation of the new heuristic. The parallel algorithm delivers the same solution quality as the sequential one while providing reasonable parallel efficiency on MIMDsystems of moderate size. All results are verified by experiments for various graphs and processor numbers.
Fitness Landscapes And Performance Of MetaHeuristics
 MetaHeuristics: Advances and Trends in Local Search Paradigms for Optimization
, 1999
"... We perform a statistical analysis of the structure of the search space of some planar, euclidian instances of the traveling salesman problem. We want to depict this structure from the point of view of iterated local search algorithms. ..."
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Cited by 11 (0 self)
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We perform a statistical analysis of the structure of the search space of some planar, euclidian instances of the traveling salesman problem. We want to depict this structure from the point of view of iterated local search algorithms.
The npcompleteness column: Finding needles in haystacks
 ACM Transactions on Algorithms
, 2007
"... Abstract. This is the 26th edition of a column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that which M. R. Garey and I used in our book “Computers and Intractability: A Guide to the Theory of NPCompleteness, ” W. H. Freeman & Co., New York, 197 ..."
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Cited by 8 (0 self)
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Abstract. This is the 26th edition of a column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that which M. R. Garey and I used in our book “Computers and Intractability: A Guide to the Theory of NPCompleteness, ” W. H. Freeman & Co., New York, 1979, hereinafter referred to as “[G&J]. ” Previous columns, the first 23 of which appeared in J. Algorithms, will be referred to by a combination of their sequence number and year of appearance, e.g., “Column 1 [1981]. ” Full bibliographic details on the previous columns, as well as downloadable unofficial versions of them, can be found at
A Parallel LocalSearch Algorithm for the kPartitioning Problem
 IN PROCEEDINGS OF 28TH HAWAII INTERNATIONAL CONFERENCE ON SYSTEM SCIENCE
, 1995
"... In this paper we present a new algorithm for the k partitioning problem which achieves an improved solution quality compared to known heuristics. We apply the principle of so called "helpful sets", which has shown to be very efficient for graph bisection, to the direct kpartitioning pr ..."
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Cited by 7 (1 self)
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In this paper we present a new algorithm for the k partitioning problem which achieves an improved solution quality compared to known heuristics. We apply the principle of so called "helpful sets", which has shown to be very efficient for graph bisection, to the direct kpartitioning problem. The principle is extended in several ways. We introduce a new abstraction technique which shrinks the graph during runtime in a dynamic way leading to shorter computation times and improved solutions qualities. The use of stochastic methods provides further improvements in terms of solution quality. Additionally we present a parallel implementation of the new heuristic. The parallel algorithm delivers the same solution quality as the sequential one while providing reasonable parallel efficiency on moderately sized MIMDsystems. All results are verified by experiments for various graphs and processor numbers.
MERLIN: Semiorderindependent hierarchical buffered routing tree generation using local neighborhood search
 in Proceedings of the ACM/IEEE Design Automation Conference
, 1999
"... ABSTRACT This paper presents a solution to the problem of performancedriven buffered routing tree generation in electronic circuits. Using a novel bottomup construction algorithm and a local neighborhood search strategy, this method finds the best solution of the problem in an exponential size so ..."
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Cited by 3 (0 self)
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ABSTRACT This paper presents a solution to the problem of performancedriven buffered routing tree generation in electronic circuits. Using a novel bottomup construction algorithm and a local neighborhood search strategy, this method finds the best solution of the problem in an exponential size solution subspace in polynomial time. The output is a hierarchical buffered rectilinear Steiner routing tree that connects the driver of a net to its sink nodes. The two variants of the problem, i.e. maximizing the driver required time subject to a total buffer area constraint and minimizing the total buffer area subject to a minimum driver required time constraint, are handled by propagating threedimensional solution curves during the construction phase. Experimental results prove the effectiveness of this technique compared to the other solutions for this problem. I.
Hierarchical Buffered Routing Tree Generation
"... AbstractThis paper presents a solution to the problem of performancedriven buffered routing tree generation for VLSI circuits. Using a novel bottomup construction algorithm and a local neighborhood search strategy, our polynomial time algorithm finds the optimum solution in an exponentialsize s ..."
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Cited by 2 (1 self)
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AbstractThis paper presents a solution to the problem of performancedriven buffered routing tree generation for VLSI circuits. Using a novel bottomup construction algorithm and a local neighborhood search strategy, our polynomial time algorithm finds the optimum solution in an exponentialsize solution subspace. The final output is a buffered rectilinear Steiner routing tree that connects the driver of a net to its sink nodes. The two variants of the problem, i.e., maximizing the required time at the driver subject to a maximum total area constraint and minimizing the total area subject to a minimum required time at the driver constraint, are handled by propagating threedimensional solution curves during the construction phase. Experimental results demonstrate the effectiveness of our algorithm compared to other techniques. I.