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78
Polynomial time approximation schemes for Euclidean TSP and other geometric problems
 In Proceedings of the 37th IEEE Symposium on Foundations of Computer Science (FOCS’96
, 1996
"... 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 � 2, 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 a ..."
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Cited by 320 (3 self)
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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 � 2, 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 � d, the running time increases to O(n(log n) (O(�dc))d�1). For every fixed c, d the running time is n � poly(log n), that is nearly linear in n. The algorithm can be derandomized, but this increases the running time by a factor O(n d). The previous best approximation algorithm for the problem (due to Christofides) achieves a 3/2approximation in polynomial time. We also give similar approximation schemes for some other NPhard Euclidean problems: Minimum Steiner Tree, kTSP, and kMST. (The running times of the algorithm for kTSP and kMST involve an additional multiplicative factor k.) The previous best approximation algorithms for all these problems achieved a constantfactor approximation. We also give efficient approximation schemes for Euclidean MinCost 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 �p for p � 1 or other Minkowski norms). They also have simple parallel (i.e., NC) implementations.
Supporting Dynamic Data Structures on DistributedMemory Machines
, 1995
"... this article, we describe an execution model for supporting programs that use pointerbased dynamic data structures. This model uses a simple mechanism for migrating a thread of control based on the layout of heapallocated data and introduces parallelism using a technique based on futures and lazy ..."
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Cited by 149 (8 self)
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this article, we describe an execution model for supporting programs that use pointerbased dynamic data structures. This model uses a simple mechanism for migrating a thread of control based on the layout of heapallocated data and introduces parallelism using a technique based on futures and lazy task creation. We intend to exploit this execution model using compiler analyses and automatic parallelization techniques. We have implemented a prototype system, which we call Olden, that runs on the Intel iPSC/860 and the Thinking Machines CM5. We discuss our implementation and report on experiments with five benchmarks.
Nearly Linear Time Approximation Schemes for Euclidean TSP and other Geometric Problems
, 1997
"... We present a randomized polynomial time approximation scheme for Euclidean TSP in ! 2 that is substantially more efficient than our earlier scheme in [2] (and the scheme of Mitchell [21]). For any fixed c ? 1 and any set of n nodes in the plane, the new scheme finds a (1+ 1 c )approximation to ..."
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Cited by 91 (4 self)
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We present a randomized polynomial time approximation scheme for Euclidean TSP in ! 2 that is substantially more efficient than our earlier scheme in [2] (and the scheme of Mitchell [21]). For any fixed c ? 1 and any set of n nodes in the plane, the new scheme finds a (1+ 1 c )approximation to the optimum traveling salesman tour in O(n(log n) O(c) ) time. (Our earlier scheme ran in n O(c) time.) For points in ! d the algorithm runs in O(n(log n) (O( p dc)) d\Gamma1 ) time. This time is polynomial (actually nearly linear) for every fixed c; d. Designing such a polynomialtime algorithm was an open problem (our earlier algorithm in [2] ran in superpolynomial time for d 3). The algorithm generalizes to the same set of Euclidean problems handled by the previous algorithm, including Steiner Tree, kTSP, kMST, etc, although for kTSP and kMST the running time gets multiplied by k. We also use our ideas to design nearlylinear time approximation schemes for Euclidean vers...
Software Caching and Computation Migration in Olden
, 1995
"... The goal of the Olden project is to build a system that provides parallelism for general purpose C programs with minimal programmer annotations. We focus on programs using dynamic structures such as trees, lists, and DAGs. We demonstrate that providing both software caching and computation migratio ..."
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Cited by 91 (0 self)
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The goal of the Olden project is to build a system that provides parallelism for general purpose C programs with minimal programmer annotations. We focus on programs using dynamic structures such as trees, lists, and DAGs. We demonstrate that providing both software caching and computation migration can improve the performance of these programs, and provide a compiletime heuristic that selects between them for each pointer dereference. We have implemented a prototype system on the Thinking Machines CM5. We describe our implementation and report on experiments with ten benchmarks.
A "Memetic" Approach for the Traveling Salesman Problem Implementation of a Computational Ecology for Combinatorial Optimization on MessagePassing Systems
 IN PROCEEDINGS OF THE INTERNATIONAL CONFERENCE ON PARALLEL COMPUTING AND TRANSPUTER APPLICATIONS
, 1992
"... In this paper we present an approach for global combinatorial optimization applied to the TSP which combines local search heuristics with a populationbased strategy. Due to its intrinsic parallelism and the inherent asynchronicity of the method it is specially appealing for MIMD messagepassing par ..."
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Cited by 62 (8 self)
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In this paper we present an approach for global combinatorial optimization applied to the TSP which combines local search heuristics with a populationbased strategy. Due to its intrinsic parallelism and the inherent asynchronicity of the method it is specially appealing for MIMD messagepassing parallel computers, such as those constructed from transputers. The approach is similar to that used by Muhlenbein [14] [15] [16], Brown et al. [1], GorgesSchleuter [3] and work performed by the Dynamics of Computation Group at Xerox PARC [4]. We consider them as prototype examples of "memetic" algorithms in the sense described in Ref. [12] (see also Ref. [5]). A preliminary description of our work can also be found in Ref. [17].
Approximating Geometrical Graphs Via Spanners and Banyans
, 1998
"... The main result of this paper is an improvement of Arora's method to find (1+ ffl) approximations for geometric NPhard problems including the Euclidean Traveling Salesman Problem and the Euclidean Steiner Minimum Tree problems. For fixed dimension d and ffl, our algorithms run in O(N log N) time. ..."
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Cited by 61 (0 self)
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The main result of this paper is an improvement of Arora's method to find (1+ ffl) approximations for geometric NPhard problems including the Euclidean Traveling Salesman Problem and the Euclidean Steiner Minimum Tree problems. For fixed dimension d and ffl, our algorithms run in O(N log N) time. An interesting byproduct of our work is the definition and construction of banyans, a generalization of graph spanners. A (1 + ffl)banyan for a set of points A is a set of points A 0 and line segments S with endpoints in A [ A 0 such that a 1 + ffl optimal Steiner Minimum Tree for any subset of A is contained in S. We give a construction for banyans such that the total length of the line segments in S is within a constant factor of the length of the minimum spanning tree of A, and jA 0 j = O(jAj), when ffl and d are fixed. In this abbreviated paper, we only provide proofs of these results in two dimensions. The full paper on WDS's web page (http://www.neci.nj.nec.com/homepages/wds, c...
Experimental Analysis of Heuristics for the STSP
 Local Search in Combinatorial Optimization
, 2001
"... In this and the following chapter, we consider what approaches one should take when one is confronted with a realworld application of the TSP. What algorithms should be used under which circumstances? We ..."
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Cited by 53 (1 self)
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In this and the following chapter, we consider what approaches one should take when one is confronted with a realworld application of the TSP. What algorithms should be used under which circumstances? We
Asymptotically optimal algorithm for job shop scheduling and packet routing
 J. Algorithms
, 1999
"... We propose asymptotically optimal algorithms for the job shop scheduling and packet routing problems. We propose a fluid relaxation for the job shop scheduling problem in which we replace discrete jobs with the flow of a continuous fluid. We compute an optimal solution of the fluid relaxation in clo ..."
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Cited by 51 (3 self)
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We propose asymptotically optimal algorithms for the job shop scheduling and packet routing problems. We propose a fluid relaxation for the job shop scheduling problem in which we replace discrete jobs with the flow of a continuous fluid. We compute an optimal solution of the fluid relaxation in closed form, obtain a lower bound Cmax to the job shop scheduling problem, and construct a feasible schedule from the fluid relaxation with objective value at most C � OŽ C. max ' max, where the constant in the OŽ. � notation is independent of the number of jobs, but it depends on the processing time of the jobs, thus producing an asymptotically optimal schedule as the total number of jobs tends to infinity. If the initially present jobs increase proportionally, then our algorithm produces a schedule with value at most C � OŽ. max 1. For the packet routing problem with fixed paths the previous algorithm applies directly. For the general packet routing problem we propose a linear programming relaxation that provides a lower bound Cmax and an asymptotically optimal algorithm that uses the optimal solution of the relaxation with objective value at most C � OŽ C. max ' max. Unlike asymptotically optimal algorithms that rely on probabilistic assumptions, our proposed algorithms make no probabilistic assumptions and they are asymptotically optimal for all instances with a large number of jobs Ž packets.. In computational experiments our algorithms produce schedules which are within 1 % of optimality even for moderately sized problems.
New scaling algorithms for the assignment and minimum mean cycle problems
, 1992
"... In this paper we suggest new scaling algorithms for the assignment and minimum mean cycle problems. Our assignment algorithm is based on applying scaling to a hybrid version of the recent auction algorithm of Bertsekas and the successive shortest path algorithm. The algorithm proceeds by relaxing th ..."
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Cited by 50 (4 self)
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In this paper we suggest new scaling algorithms for the assignment and minimum mean cycle problems. Our assignment algorithm is based on applying scaling to a hybrid version of the recent auction algorithm of Bertsekas and the successive shortest path algorithm. The algorithm proceeds by relaxing the optimality conditions, and the amount of relaxation is successively reduced to zero. On a network with 2n nodes, m arcs, and integer arc costs bounded by C, the algorithm runs in O(,/n m log(nC)) time and uses very simple data structures. This time bound is comparable to the time taken by Gabow and Tarjan's scaling algorithm, and is better than all other time bounds under the similarity assumption, i.e., C = O(n k) for some k. We next consider the minimum mean cycle problem. The mean cost of a cycle is defined as the cost of the cycle divided by the number of arcs it contains. The minimum mean cycle problem is to identify a cycle whose mean cost is minimum. We show that by using ideas of the assignment algorithm in an approximate binary search procedure, the minimum mean cycle problem can also be solved in O(~/n m log nC) time. Under the similarity assumption, this is the best available time bound to solve the minimum mean cycle problem.
Survivable networks, linear programming relaxations and the parsimonious property
, 1993
"... We consider the survivable network design problem the problem of designing, at minimum cost, a network with edgeconnectivity requirements. As special cases, this problem encompasses the Steiner tree problem, the traveling salesman problem and the kedgeconnected network design problem. We establ ..."
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Cited by 44 (12 self)
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We consider the survivable network design problem the problem of designing, at minimum cost, a network with edgeconnectivity requirements. As special cases, this problem encompasses the Steiner tree problem, the traveling salesman problem and the kedgeconnected network design problem. We establish a property, referred to as the parsimonious property, of the linear programming (LP) relaxation of a classical formulation for the problem. The parsimonious property has numerous consequences. For example, we derive various structural properties of these LP relaxations, we present some algorithmic improvements and we perform tight worstcase analyses of two heuristics for the survivable network design problem.