H. Crowder and M. W. Padberg, "Solving Large--Scale Symmetric Traveling Salesman Problems to Optimality," Management Science, vol. 26, pp. 495--509, 1980.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....second paper will deal with more complex inequalities such as Clique Tree, Star or Path and Ladder inequalities and give computational results. 1 Introduction Branch and cut has been very successful in the past years in solving large traveling salesman problems to optimality (see [31] 28] [9]) One important part of a branch and cut code is the separation procedure. The only references available on this subject dealing with the traveling salesman problem are [30] and [1] 1 Recently L. Fleisher and E. Tardos [12] gave a polynomial algorithm to find maximally violated combs in planar ....

H. Crowder and Manfred W. Padberg. Solving large scale symmetric traveling salesman problesm to optimality. Management Science, 26:495--509, 1980.

....cut, we move on to consider the next component of the support graph. A variant was also implemented, the 2 connected components heuristic, which begins with the 2 edge connected components of the support graph after removal of the depot. The shrinking heuristic was first suggested for the TSP in [7]. This procedure considers the edges of the support graph. If the two endnodes of any edge determine a capacity constraint violated by x (i.e. the set S = fi; jg induces a violated capacity constraint) we CUT. If not, we define weights e = x e for each edge e of the support graph. Now ....

H. Crowder and M. Padberg, Solving Large-scale Symmetric Traveling Salesman Problems to Optimality, Management Science 26 (1980), 495.

....cut, we move on to consider the next component of the support graph. A variant was also implemented, the 2 connected components heuristic, which begins with the 2 edge connected components of the support graph after removal of the depot. The shrinking heuristic was first suggested for the TSP in [7]. This procedure considers the edges of the support graph. If the two endnodes of any edge determine a capacity constraint violated by x (i.e. the set S = i, j induces a violated capacity constraint) we CUT. If not, we define weights # e = x e for each edge e of the support graph. Now ....

H. Crowder and M. Padberg, Solving Large-scale Symmetric Traveling Salesman Problems to Optimality, Management Science 26 (1980), 495.

.... et al. 1985a) Junger (1985) Barahona and Mahjoub (1989,1992) Chopra (1992) Junger and Mutzel (1993) Leung and Lee (1994) Goemans and Hall (1996) Bauer (1997) Tenary problems: Chopra (1989a) Traveling salesman problems: Dantzig et al. 1954, 1959) Grotschel and Padberg (1979) Crowder and Padberg (1980), Grotschel (1980) Padberg and Hong (1980) Cornu ejols and Pulleyblank (1982) Grotschel and Pulleyblank (1986) Padberg and Rinaldi (1987,1990,1991) Fischetti (1991a,1992) Grotschel and Holland (1991) Naddef and Rinaldi (1991,1992) Reinelt (1991) Naddef (1992) Clochard and Naddef (1993) ....

H. Crowder and M.W. Padberg (1980) "Solving large-scale symmetric traveling salesman problems to optimality", Management Science 26 459--509.

....heuristic algorithms are known for the corresponding separation problem. For this reason, cuttingplane approaches tend to use, in addition to the subtour elimination constraints, the comb inequalities or the clique tree inequalities, see Gr otschel [13] Padberg and Hong [28] Crowder and Padberg [8], Gr otschel and Holland [14] Padberg and Rinaldi [26, 27] J unger et al. 19] and Applegate et al. 1] An exception is a recent implementation of Clochard and Naddef [6] which is based on the path inequalities. The motivation of this paper is best expressed by the following quote from Naddef ....

H. Crowder and M. Padberg. Solving large-scale symmetric traveling salesman problems to optimality. Management Science, 26:495--509, 1980.

....champion heuristic, it was believed that the exact optimum was at 1 or more above the Held Karp bound, but C L O has lowered this number. Finally, we tested the C L O algorithm on large, specific instances solved to optimality by other groups and available [9] These instances were: 1) LIN 318 [24]; 2) AT T 532 [25] and (3) RAT 783 [26] The numbers denote the number of cities, and the references give the authors who first solved the problem to optimality using branch and cut methods. We found that C L O was able to find the optimum solution to LIN 318 in minutes, and the solution to ....

H. Crowder and M.W. Padberg. Solving large-scale symmetric traveling salesman problems to optimality. Management Science, 26:495, 1984.

.... and Schulz (1994) Subgraph polytopes: Balas and Pulleyblank (1983) Barahona et al. 1985) Barahona and Mahjoub (1989,1992) Chopra (1992) Junger and Mutzel (1993) Tenary problems:Chopra (1989a) Traveling salesman problems: Dantzig et al. 1954, 1959) Grotschel and Padberg (1979) Crowder and Padberg (1980), Grotschel (1980) Padberg and Hong (1980) Cornu ejols and Pulleyblank (1982) Grotschel and Pulleyblank (1986) Padberg and Rinaldi (1987,1990,1991) Fischetti (1991a,1992) Grotschel and Holland (1991) Naddef and Rinaldi (1991,1992) Reinelt (1991) Naddef (1992) Clochard and Naddef (1993) ....

H. Crowder and M.W. Padberg (1980) "Solving large-scale symmetric traveling salesman problems to optimality", Management Science 26 459--509.

....future paper. In our present implementation of a branch and cut algorithm we have used only the inequalities presented in the last section. As we have observed before, although there exists an exponential number of inequalities of type (4) resp. 4 0 ) they can be separated efficiently (see [CP80]) We have implemented a separation heuristic for these inequalities, based on contractions of the graph, as described in [PG85] In the case of inequalities (4 0 ) these are then lifted as indicated in Lemma 4.2. It is interesting to note that, if the graph is acyclic, then there are no ....

H. Crowder, M.W. Padberg, "Solving large-scale symmetric traveling salesman problems to optimality", Management Science 26, 495-509 (1980).

....the quality of the generated feasible solutions, since otherwise, the number of generated subproblems becomes rapidly very large. Earlier approaches for the solution of hard combinatorial optimization problems combined problem specific cutting planes with a commercial branch and bound software (Crowder and Padberg (1980) and Gr otschel and Holland (1991) A relaxation of the integer programming formulation is attacked with a cutting plane algorithm, which generates preferably facet defining violated constraints. If the cutting plane procedure terminates with a solution which is not the incidence vector of a ....

....has been formulated in section 1. Applications are, e.g. the drilling of printed circuit boards, diffractometer control in x ray crystallography, and vehicle routing. References: Dantzig, Fulkerson and Johnson (1954) Miliotis (1976, 1978) Gr otschel (1977, 1980) Padberg and Hong (1980) Crowder and Padberg (1980), Padberg and Gr otschel (1985) Fleischmann (1985) Padberg and Rinaldi (1987, 1989, 1991) Gr otschel and Holland (1991) J unger, Reinelt and Thienel (1992) Clochard and Naddef (1993) J unger, Reinelt and Rinaldi (1994) Vehicle routing problem The vehicle routing problem is a multiple ....

H. Crowder and M.W. Padberg (1980), Solving large-scale symmetric traveling salesman problems to optimality, Management Science 26, 495--509.

....optimum is subject to debate, but the Held Karp lower bound shows that the average excess length is at most 0. 84 (see [13] and Table 1) Finally, we tested the C L O algorithm on large, specific instances solved to optimality by other groups and available [6] These instances were: 1) LIN 318 [21]; 2) AT T 532 [22] and (3) RAT 783 [23] The numbers denote the number of cities, and the references give the authors who first solved the problem to optimality using branch and cut methods. We found that C L O was able to find the optimum solution to LIN 318 in minutes, and the solution to ....

H. Crowder and M. W. Padberg. Solving large-scale symmetric traveling salesman problems to optimality. Management Science, 26:495, 1984.

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H. Crowder and M. W. Padberg, "Solving Large--Scale Symmetric Traveling Salesman Problems to Optimality," Management Science, vol. 26, pp. 495--509, 1980.

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Crowder, H., and Padberg, M.: Solving Large Scale Symmetric Traveling Salesman Problems to Optimality. Management Science 26, 495, 1980

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H. Crowder and M.W. Padberg (1980) "Solving large-scale symmetric traveling salesman problems to optimality", Management Science 26 459--509.

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