G. Reinelt, The traveling salesman, computational solutions for TSP applications, in: Lecture Notes in Computer Science, Vol. 840, Springer, Heidelberg, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....be tackled. For instance, in the traveling salesman problem, the search space S corresponds to all possible tours, that is, all possible permutations of cities. rand would generate a feasible tour and explo would use standard operators like the 2 opt operator or the city insertion operator [21]. Preliminary tests of API on different problems in numerical and combinatorial optimization seem to indicate that API can achieve robust performance for all the tested problems. 6. Conclusion In this paper, we have proposed a new search algorithm inspired by the behavior of Pachyconedyla ....

G. Reinelt, The traveling salesman, computational solutions for TSP applications, in: Lecture Notes in Computer Science, Vol. 840, Springer, Heidelberg, 1994.

....also be tackled. For instance, in the traveling salesman problem, the search space S corresponds to all possible tours, that is, all possible permutations of cities. O rand would generate a feasible tour and O explo would use standard operators like the 2 opt operator or the cityinsertion operator [21]. Preliminary tests of API on di erent problems in numerical and combinatorial optimization seem to indicate that API can achieve robust performance for all the tested problems. 6 Conclusion In this paper, we have proposed a new search algorithm inspired by the behavior of Pachycondyla apicalis ....

G. Reinelt. The traveling salesman, computational solutions for TSP applications. In Lecture Notes in Computer Science, volume 840. Springer Verlag, Heidelberg, Germany, 1994.

....with a shortcut to the optimum solution, like for MST Nobody knows Despite the efforts of many brilliant people, no polynomial algorithm for the TSP has been found. There are some smart and efficient (i.e. polynomial) algorithms that find good solutions but do not guarantee to yield the optimum [12]. According to our definition, the TSP is intractable. Figure 2: Same instance, different problems: A valid configuration of the TRAVELING SALESMAN PROBLEM (left) and the ASSIGNMENT problem (right) Whereas ASSIGNMENT can be solved in polynomial time, the TSP is intractable. Why is the TSP ....

Gerhard Reinelt. The Travelling Salesman. Computational Solutions for TSP Applications, volume 840 of Lecture Notes in Computer Science. Springer-Verlag, Berlin Heidelberg New York, 1994.

....the Locality Problem We formulate the problem of improvement of locality as a Traveling Salesman Problem. This is a classic combinatorial optimization problem, an NP complete in particular. Solving it is equivalent to finding a path of the shortest length possible in a complete weighed graph [10]. In our case the function that indicates the weight of each edge satisfies the triangular inequality, so we have a symmetric TSP. The weight function reflects the distance between rows columns according to the description of locality given previously. There are different heuristics for solving ....

....between rows columns according to the description of locality given previously. There are different heuristics for solving it. We have opted for one based on spanning trees which is divided into two steps: Construction of a minimum spanning tree of the complete graph using the Prim algorithm [10, 5]. Visiting the different vertexes of the tree to establish an order of the nodes using a depth first search [5] To compare the results obtained with this algorithm we have also solved our analogy of TSP with the greedy heuristic called nearest neighbor algorithm [10] Each ordering of the ....

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Gerhard Reinelt. The Traveling Salesman. Computational Solutions for TSP applications. Lecture Notes in Computer Science. Springer--Verlag, 1991.

.... exponents that characterize this transition using finite size scaling, a technique borrowed from statistical physics, and demonstrate our results on two important problems: energy minimization on NK energy functions [5] and tour length minimization for the traveling salesman problem (TSP) [6]. The NK model, a generalization of spin glass models [7] was chosen as one of the few general models capable of generating tunably difficult optimization tasks. The TSP is perhaps the most famous and well studied combinatorial optimization problem and is often used as a test bed for new ideas. ....

Gerhard Reinelt, The Traveling Salesman, computational solutions for TSP applications, (Springer-Verlag, Berlin, Heidelberg, 1994).

....generator was also evaluated on 26 instances from TSPLIB [15] 1987397 terminals) This library is a collection of instances for the Traveling Salesman Problem (TSP) mainly plane real world Euclidean problem instances. The 26 selected instances are the same as those chosen in a study by Reinelt [16] on heuristics for the TSP; in addition we have chosen the instance pla7397, the largest TSP instance solved to optimality to date. These instances are quite representative for the whole TSPLIB collection. Computational results are presented for each instance in Table 2. For many of the instances ....

G. Reinelt. The Traveling Salesman, Computational Solutions for TSP Applications. Lecture Notes in Computer Science 840. Springer-Verlag, 1994.

....algorithms while requiring a similar running time, taking the scale factors into account. Table II gives computational results for another set of problems and algorithms. Algorithms in columns LK 1, LK 2 and LK 3 concern three variants of a LinKernighan type procedure, which were shown in Reinelt [29] to be the best among 49 local search alternative algorithms for the TSP. Also, we have updated Reinelt s results for problems whose new lower bounds became known after publication. Results in column I LK correspond to 20 consecutive runs of LK 3, starting each run after the first from a perturbed ....

....the previous run. Such an iterated Lin Kernighan process improves upon all the results obtained in a single run of the LK 3. Nevertheless no computation times are given in the reference for this version of the algorithm, but the author has reported that the running time is considerable (Reinelt [29]) Finally, it should be noted that the results obtained by the indicated LinKernighan variants are obtained by starting from an initial solution given by a modified Christofides s [2] heuristic. Less attractive results are obtained when the same algorithms start from a randomly generated ....

G. Reinelt. The Traveling Salesman Problem, Computational Solutions for TSP Applications, volume 840 of Lecture Notes in Computer Science. Springer-Verlag, 1994.

....(like simulated annealing and genetic algorithms) are unable to compete with carefully hand crafted solutions for specific search problems. The Traveling Salesman (TSP) problem is an excellent example of such a situation; the best search algorithms for the TSP problem are hand tailored for it [12]. Linear programming problems are another example; the simplex algorithm is a search algorithm specifically designed to solve cost functions of a particular type. In both of these situations, the procedure followed by the researcher is to: identify salient aspects of f (e.g. it is a TSP problem, ....

Gerhard Reinelt, The Traveling Salesman, computational solutions for TSP applications, Springer Verlag Berlin Heidelberg (1994).

....(like simulated annealing and genetic algorithms) are unable to compete with carefully hand crafted solutions for specific search problems. The Traveling Salesman Problem (TSP) is an excellent example of such a situation; the best search algorithms for the TSP problem are hand tailored for it [12]. Linear programming problems are another example; the simplex algorithm is a search algorithm specifically designed to solve cost functions of a particular type. In both of these situations, the procedure followed by the researcher is to: identify salient aspects of f (e.g. it is a TSP problem, ....

Gerhard Reinelt, The Traveling Salesman, computational solutions for TSP applications, Springer Verlag Berlin Heidelberg (1994).

....good (if not optimal) solution. While heuristics exist for solving the TSP and MTSP in polynomial time within a bounded amount of optimal, they either remain computationally intractable with very large exponents for even relatively large bounds[3] or do not provide a solution regularly superior[26] to that which can be found via randomized search. Simulated annealing [13] was chosen in this work for a randomized algorithm, as it has been shown to be very successful in this application[26] Simulated annealing requires[25] an objective function E, a set of random functions which can change ....

....very large exponents for even relatively large bounds[3] or do not provide a solution regularly superior[26] to that which can be found via randomized search. Simulated annealing [13] was chosen in this work for a randomized algorithm, as it has been shown to be very successful in this application[26]. Simulated annealing requires[25] an objective function E, a set of random functions which can change one solution into another, and an annealing schedule which defining how many random solutions will be produced at a given step and how quickly the system will be cooled. The objective function ....

Reinelt, G. The Travelling Salesman, Computational Solutions for TSP Applications. Springer-Verlag, 1991.

....Note that for the general TSP, 14 3 TSP AND TEAMWORK it is not necessary that c ij = c ji for all i; j 2 f1; ng. However, if this equation does hold, we speak of the symmetric TSP. Since it is the most common variant of the TSP, we consider only the symmetric TSP. For reference, see also [Re94] and [LLRS85] 3.2 Specialists We will now explain the three most important specialists that are integrated in Dott. Specialists are necessary for performing special tasks. For example, experts that use genetic algorithms need a specialist that creates an initial population (specialist FI) ....

Reinelt, G.: The Traveling Salesman Problem. Computational Solutions for TSP Applications., Springer 1994.

....a face of the symmetric travelling salesman polytope Q T (2n) Thus, the result of Billera Sarangarajan [63] also applies to the symmetric TSP polytope. Two recent references that describe the method for how to find a solution for a Traveling Salesman Problem if you really need one are Reinelt [529] and Junger, Reinelt Thienel [514] Page 23: The latest success in the race for the TSP Olympics (i.e. for largest traveling salesman problem ever solved ) is reported in [488] David Applegate, Bob Bixby, Vasek Chv atal and Bill Cook [493] have been able to solve a 7397 city instance to ....

Gerhard Reinelt: The Traveling Salesman. Computational Solutions for TSP Applications, Lecture Notes in Computer Science 840, Springer-Verlag, Berlin 1994.

.... Such methods often combine the characteristics of hill climbing and greedy algorithms, although numerous variations are possible (e.g. simulated annealing) The local search methods we consider in the next section are inspired by the k opt exchange heuristics for the traveling salesman problem [6, 8]. We use k CSP variables, V 1 ; V k , each with domain A = fa 1 ; a n g, with constraints between variables encoded in goal function G that calculates the score of any particular instantiation. The algorithms look at pairs of actions and then swap then according to some heuristic ....

Gerhard Reinelt. The Traveling Salesman, Computational Solutions for TSP Applications. Springer-Verlag, 1994.

....travel distance from one cluster to another. This can be taken into account by the computation of a minimum spanning tree. Indeed, for a tour, Next is a chain (a special kind of tree, of constant degree 1) spanning all vertices 1, n 1 . Hence it is a spanning tree rooted in node 1 (see [2] [21]) Two lower bounds related to minimal spanning trees can be computed : the first one is the weight of an undirected minimal spanning tree (MST) which can be computed by Prim s algorithm in O(n 2 ) 10] or by an incremental version of Kruskal s algorithm (as it is used by [17] The second ....

....with Edmonds algorithm in O(mn) 9] 10] Both these lower bounds may be used for pruning the search tree. 3.4. Results The set of benchmarks that we consider in the next table and in the rest of the paper comes from TSPLIB, a library of benchmarks related to the TSP collected by Reinelt [21]. All problems have symmetrical distance matrices and their number of nodes should be self evident from their names. Problems gr17, 21,24 are problems from M. Gr tschel, problem fri26 comes from Fricker, problems bayg29 and bays29 come from Gr tschel, J nger and Reinelt, problem dantzig42 comes ....

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G. Reinelt. The Travelling Salesman, Computational Solutions for TSP Applications. LNCS 840, Springer, 1994.

....alternating tours which are of length smaller than 1:21 OPT (worst case) and 1:12 OPT (average case) respectively. Our implementation of the matching method uses the Nearest Neighbor and full 3 OPT heuristics to gain a near optimal tour through the red points. Empirical studies of Reinelt [25] indicate that this combination even with limited 3 OPT achieves solutions typically deviating by no more than 4 from the optimum value. However, it is impossible to generally improve the upper bound. Even if we restrict ourselves to con gurations where all red points are con ned in some central ....

G. Reinelt, The Traveling Salesman { Computational Solutions for TSP Applications. Lecture Notes in Computer Science 840, Springer (1994).

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Reinelt, G., The Traveling Salesman, Computational Solutions for TSP Applications, Springer Verlag, 1994.

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