M. Held and R.M. Karp. The traveling-salesman problem and minimum spanning trees: Part II Mathematical Programming 1:6--25, 1971. Home/Search Document Not in Database Summary Related Articles Check |
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CP-based Lagrangian Relaxation for a Multimedia Application - Sellmann, Fahle (2001) (1 citation) (Correct)
....The multimedia application we use to introduce CP based Lagrangian Relaxation is investigated in [16] The goal there was to develop heuristics for an extension of the ARP. Lagrangian relaxation was rst presented in [7] for resource allocation problems. Held and Karp used it for the TSP [13, 14] and it has been applied in many di erent areas since then. We refer to [1] for an introduction. In [10] Focacci et al. introduced a method to strengthen bound propagation using Lagrangian Relaxation. In that approach, Lagrangian multipliers were used to incorporate additional cuts to tighten ....
M. Held and R.M. Karp. The traveling-salesman problem and minimum spanning trees Operations Research 18:11381162, 1970. 13 trees: Part II Mathematical Programming 1:625, 1971.
Correspondences to: Luiz Antonio Nogueira Lorena, LAC/INPE .. - Luiz Antonio Nogueira (2002) (Correct)
....SubG, then the next Lagrangean surrogate relaxations will use this fixed value T as the multiplier and the search SH is not performed. The initial l used is min ij M j i d = l , i N. The step sizes used are: q = p (ub lb) g . The control of parameter p is the Held and Karp [5] classical control. It makes 0 p 2, beginning with p = 2 and halving p whenever lb does not increase for 30 successive iterations. The stopping tests used are: a) p 0.005; b) ub lb 1; c) g = 0. 12 4. The Local Search Heuristics The Lagrangean surrogate approach described in ....
M. Held, R.M. Karp, The Traveling salesman problem and minimum spanning trees. Operations Research 18 (1970) 1138-1162.
Anomalies in Parallel Branch-and-Bound Algorithms - Lai, Sahni (1984) (45 citations) (Correct)
....Science Department, The Ohio State University, Columbus, Ohio. 1 2 1. Introduction Branch and bound is a popular algorithm design technique that has been successfully used in the solution of problems that arise in various fields (e.g. combinatorial optimization, artificial intelligence, etc. [1, 9 15]. We shall briefly describe the branch and bound method as used in the solution of combinatorial optimization problems. Our terminology is from Horowiz and Sahni [10] In a combinatorial optimization problem we are required to find a vector x = x 1 , x 2 , x n ) that optimizes some ....
....A tour is a cycle that includes every vertex (i.e. it is a Hamiltonian cycle) The cost of a tour is the sum of the weights of the edges on the tour. We wish to find a tour of minimum cost. The branch and bound strategy that we used is a simplified version of the one proposed by Held and Karp [9]. Vertex 1 is chosen as the start vertex. There are n 1 possibilities for the next vertex and n 2 for the preceding vertex (assume n 2) This leads to (n 1) n 2) sequences of 3 vertices each. Half of these may be discarded as they are symmetric to other 15 sequences. Any sequence with ....
M. Held and R Karp, "The traveling salesman problem and minimum spanning trees: part II," Math Prog., 1, pp. 6-25, 1971.
A 3-Flip Neighborhood Local Search for the Set Covering.. - Yagiura, KISHIDA, IBARAKI (2003) (Correct)
....method As discussed in Section 2.1, any optimal solution u to LPD SCP is an optimal solution to LD SCP; however, computing such u directly is usually quite expensive, especially for very large scale instances. A common approach to compute an approximate u is the subgradient method [2, 11, 16]. It uses the subgradient vector s(u) associated with a given u, defined by s i (u) 1 a ij x j (u) M. This method generates a sequence u , u , where u is a given initial vector, and u is updated from u by the following formula: max # UB ....
M. Held and R.M. Karp, "The Traveling Salesman Problem and Minimum Spanning Trees: Part II," Mathematical Programming, 1 (1971) 6--25.
Exact Solution of the Quadratic Knapsack Problem - Caprara, Pisinger, Toth (1998) (5 citations) (Correct)
.... (2) 6) Anyway, exact solution of this LP relaxation would be computationally very expensive due to the large number of variables and constraints involved (see also [6] Our approach determines a near optimal multiplier matrix by a standard subgradient optimization procedure; see Held and Karp [15] and Held, Wolfe and Crowder [16] The procedure generates a series ; of matrices, where : 0 and, for k 0, is defined from as follows. Let (x; y) denote an optimal solution of the Lagrangian relaxation associated with . The corresponding subgradient vector is given ....
M. Held and R.M.Karp (1971), "The Traveling Salesman Problem and Minimum Spanning Trees: Part II", Mathematical Programming 1, 6--25.
Flexible and Approximate Computation - Through State-Space Reduction (Correct)
....ffi DFBnB, while the average error is 15.9 (profile=0.841) from DFBnB. We also compared iterative ffi DFBnB against DFBnB on the symmetric TSP (STSP) in which the cost from city i to city j is the same as that of from j to i. In our implementation, we use Held Karp lower bound function [13] to compute node costs. This cost function iteratively computes a Lagrangian relaxation on the STSP, with each step constructing a 1 tree. A 1tree is a minimum spanning tree (MST) 23] on n Gamma 1 cities plus the two shortest edges from the city not in the MST to two cities in the MST. Note that ....
M. Held and R. M. Karp. The traveling salesman problem and minimum spanning trees: Part ii. Mathematical Programming, 1:6--25, 1971.
An Ejection Chain Approach for the Generalized Assignment.. - Yagiura, Ibaraki, Glover (1999) (1 citation) (Correct)
....relaxation of (1) b i u i s.t. v j a ij u i c ij , J, u i I , 3) is an optimal solution to the Lagrangian dual [8] However, computing such v # by solving (3) is expensive for large scale instances. A fast method for finding a near optimal v is the subgradient method [8, 12], which is based on the subgradients, s j (v) 1 x ij , J. This approach generates a sequence v , where v is defined arbitrarily and v , k 1 are updated by, e.g. v = v # UB s(v s j (v ) j where UB is an upper bound on the objective ....
M. Held and R.M. Karp, "The Traveling Salesman Problem and Minimum Spanning Trees: Part II," Mathematical Programming, 1 (1971) 6--25.
A Structural EM Algorithm for Phylogenetic Inference - Friedman, Ninio, Pe'er (2002) (11 citations) (Correct)
....This is the Maximum Spanning Tree problem with a slighttwist that some nodes are required to be leaves. As wenowshow, we can easily reduce this problem to a Maximum Spanning Tree problem, and then apply a standard algorithm, e.g. 22] Our transformation is motivated by ideas of Held and Karp [15] in approximating TSP using maximum weighted spanning trees. Let W = fw i#j : i# j =1#: #2N ; 2g, and let c be a constant. We define W to be the matrix with the following entries: i#j = w i#j ; 2c if i N and j N w i#j ; c if i N or j N , but not both w i#j otherwise It is ....
M. Held and R.M. Karp. The traveling-salesman problem and minimum spanning trees. Operations Research, 18:1138--1162, 1970.
Constraint Programming based Lagrangian Relaxation - For The Automatic (Correct)
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M. Held and R.M. Karp. The traveling-salesman problem and minimum spanning trees: Part II Mathematical Programming 1:6--25, 1971.
Constraint Programming based Lagrangian Relaxation - For The Automatic (Correct)
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M. Held and R.M. Karp. The traveling-salesman problem and minimum spanning trees Operations Research 18:1138--1162, 1970.
On the Number of Iterations for Dantzig-Wolfe Optimization and .. - Klein, Young (2002) (6 citations) (Correct)
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M. Held and R. M. Karp. The traveling salesman problem and minimum spanning trees: Part II. Mathematical Programming , 1:6--25, 1971.
On the Number of Iterations for Dantzig-Wolfe Optimization and .. - Klein, Young (2002) (6 citations) (Correct)
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M. Held and R. M. Karp. The traveling salesman problem and minimum spanning trees. Operations Research, 18:1138--1162, 1971
A Generalization of Totally Unimodular and Network Matrices - Kotnyek (2002) (Correct)
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M. Held and R. M. Karp. The traveling-salesman problem and minimum spanning trees. Operations Research, 18:1138--1162, 1970.
A Generalization of Totally Unimodular and Network Matrices - Kotnyek (2002) (Correct)
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M. Held and R. M. Karp. The traveling-salesman problem and minimum spanning trees. Operations Research, 18:1138--1162, 1970.
A Generalization of Totally Unimodular and Network Matrices - Kotnyek (2002) (Correct)
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M. Held and R. M. Karp. The traveling-salesman problem and minimum spanning trees. Operations Research, 18:1138--1162, 1970.
A Lower Bound for the Breakpoint Phylogeny Problem - Bryant (2004) (7 citations) (Correct)
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M. Held, R. Karp, The traveling salesman problem and minimum spanning trees, Oper. Res. 18 (1970) 1138--1162.
A Lower Bound for the Breakpoint Phylogeny Problem - Bryant (2004) (7 citations) (Correct)
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M. Held, R. Karp, The traveling salesman problem and minimum spanning trees: part II, Math. Programming 1 (1970) 16--25.
The TOMLAB Optimization Environment in Matlab - Holmström (1999) (Correct)
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M. Held and R. M. Karp, The Traveling-Salesman problem and minimum spanning trees: Part II, Mathematical Programming, 1 (1971), pp. 6--25.
A Multilevel Lin-Kernighan-Helsgaun Algorithm for the Travelling.. - Walshaw (2001) (Correct)
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M. Held and R. M. Karp. The Traveling Salesman Problem and Minimum Spanning Trees. Oper. Res., 18:1138--1162, 1970.
Evaluation of Algorithms for Local Register Allocation - Liberatore, Farach-Colton.. (1999) (5 citations) (Correct)
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M. Held and R. Karp. The traveling salesman problem and minimum spanning trees. Operations Research, 18:1138--1162, 1970.
A Multilevel Approach to the Travelling Salesman Problem - Walshaw (2000) (1 citation) (Correct)
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M. Held and R. M. Karp. The Traveling Salesman Problem and Minimum Spanning Trees. Oper. Res., 18:1138--1162, 1970.
Improving Table Compression with Combinatorial Optimization - Buchsbaum, Fowler, Giancarlo (2002) (1 citation) (Correct)
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M. Held and R. M. Karp. The traveling salesman problem and minimum spanning trees: Part II. Math. Prog., 1:6--25, 1971.
Improving Table Compression with Combinatorial Optimization - Buchsbaum, Fowler, Giancarlo (2002) (1 citation) (Correct)
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M. Held and R. M. Karp. The traveling salesman problem and minimum spanning trees. OR, 18:1138--62, 1970.
Multilevel Refinement for Combinatorial Optimisation Problems - Walshaw (2001) (3 citations) (Correct)
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M. Held and R. M. Karp. The Traveling Salesman Problem and Minimum Spanning Trees. Oper. Res., 18:1138--1162, 1970.
Implementing the Dantzig-Fulkerson-Johnson Algorithm.. - Applegate, Bixby.. (2003) (2 citations) (Correct)
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Held, M., R. M. Karp. 1971. The traveling-salesman problem and minimum spanning trees: part II. Mathematical Programming 1, 6-25.
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