J. D. C. Little, K. G. Murty, D. W. Sweeney and C. Karel, `An algorithm for the traveling salesman problem', Operations Research, 11, 972--989 ( 1963).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....to start with a good feasible solution (See Clausen and Perregaard [13] To get good starting solutions, we use the simulated annealing method of Burkard and Rendl [8] which leads to excellent (if not optimum) solutions in short times. We investigated carefully the work of Little, et al. [31], utilized by Bazaraa and Kirka [5] Burkard [7] Kaku and Thompson [23] and Pierce and Crowston [35] namely the branching strategy based on the rule of alternative costs . This rule uses the concept of alternate costs first suggested by Lenstra [29] It bases the selection of a ....

Little, J.D.C., K.G. Murty, D.W. Sweeney and K. Caroline, "An algorithm for the Traveling Salesman Problem," Operations Research, vol. 14 (1963): 972-989.

....that returns to the starting point after visiting all cities in the n city network only once. The TSP is known to be NP hard, and exact solutions can only be obtained for tours involving some hundred cities. While the well known branch and bound algorithms of Held and Karp [5] or Little et al. [10] would be among the preferred solution techniques for the TSP in practice 1, we have chosen the method described in Pearl s book [16, p. 10if] because it builds a graph rather than a tree. It does so by successively adding unvisited cities to the end of a temporary partial contiguous tour for as ....

J.D.C. Litfie, K.G. Murty, D.W. Sweeney and G. Karel, "An algorithm for the traveling salesman problem", Operations Research, vol. 11, pp. 972-989,

....practical insight achieved in the study of TSP can often be useful in the solution of other problems in this area. In fact, much progress in combinatorial optimization can be traced back to research on TSP. The now well known computing method, branch and bound, was first used in the context of TSP [3, 4]. It is also worth mentioning that research on TSP was an important driving force in the development of the computational complexity theory in the beginning of the 1970s [5] However, the interest in TSP not only stems from its practical and theoretical importance. The intellectual challenge of ....

J. D. C. Little, K. G. Murty, D. W. Sweeny & C. Karel, An algorithm for the traveling salesman problem, Oper. Res., 11, 972-989 (1963).

....found irrelevant since the optimal value of its LP relaxation turned out to be at least as large as c T x for some previously known element x of S. This scheme is one of the many variants of the branch and bound method. The term branch and bound , coined by Little, Murty, Sweeney, and Karel [44], refers to a general class of algorithms that originated in the work of Bock [7] Croes [16] Eastman [20] Rossman and Twery [65] and Land and Doig [42] in this more general context, relaxations of (1) may come from a universe far wider than that of linear programming relaxations (2) and ....

Little, J.D.C., Murty, K.G., Sweeney, D.W., Karel, C.: An algorithm for the traveling salesman problem. Operations Research 11, 972--989, 1963.

....[0; f0g; 0] exists. An expansion of a node u 2 G yields a child v for each city that remains to be visited in u. This way a TSP tour is constructed by visiting an additional unvisited city, from the present city , in each expansion. The heuristic function used by us is similar to the one given in [9]. While our formulation of TSP reduces it to a graphsearch problem (as will be shown shortly) previous sequential and parallel branch and bound methods employed to address TSP have used a tree search formulation [5, 9, 12, 14] In their formulation, the state of a node is defined by either: 1) ....

....each expansion. The heuristic function used by us is similar to the one given in [9] While our formulation of TSP reduces it to a graphsearch problem (as will be shown shortly) previous sequential and parallel branch and bound methods employed to address TSP have used a tree search formulation [5, 9, 12, 14]. In their formulation, the state of a node is defined by either: 1) a 1 tuple: ordered list of cities visited] or (2) a 2 tuple: set of edges currently in the tour, set of edges excluded from the tour] Consider two nodes u and v, whose states, according to the first tree formulation, are ....

J.D. Little, et. al., "An Algorithm for the Traveling Salesman Problem," Operations. Res., Vol.11, 1963.

....[0; f0g; 0] exists. An expansion of a node u 2 G yields a child v for each city that remains to be visited in u. This way a TSP tour is constructed by visiting an additional unvisited city, from the present city , in each expansion. The heuristic function used by us is similar to the one given in [5]. While our formulation of TSP reduces it to a graphsearch problem (as will be shown shortly) previous sequential and parallel branch and bound methods employed to address TSP [3, 5, 7, 10] have used a tree search formulation. In their formulation, the state of a node is defined by either: 1) a ....

....city, from the present city , in each expansion. The heuristic function used by us is similar to the one given in [5] While our formulation of TSP reduces it to a graphsearch problem (as will be shown shortly) previous sequential and parallel branch and bound methods employed to address TSP [3, 5, 7, 10] have used a tree search formulation. In their formulation, the state of a node is defined by either: 1) a 1 tuple: ordered list of cities visited] or (2) a 2 tuple: set of edges currently in the tour, set of edges excluded from the tour] Consider two nodes u and v, whose states, according to ....

J.D. Little, et. al., "An Algorithm for the Traveling Salesman Problem," Operations Research, Vol.11, 1963.

....the start city from the set of unvisited cities and the present city, and S o is the sum of the costs of the least expensive outgoing edge from each unvisited city and the present city to the set of unvisited cities and the start city. The other heuristic function is the well known LMSK heuristic [16]. 4 While our formulation of TSP reduces it to a graph search problem (as will be shown shortly) previous sequential and parallel branch and bound methods employed to address TSP have used a tree search formulation [12, 16, 23, 29] In their formulation, the state of a node is defined by ....

....start city. The other heuristic function is the well known LMSK heuristic [16] 4 While our formulation of TSP reduces it to a graph search problem (as will be shown shortly) previous sequential and parallel branch and bound methods employed to address TSP have used a tree search formulation [12, 16, 23, 29]. In their formulation, the state of a node is defined by either: 1) a 1 tuple: ordered list of cities visited] or (2) a 2 tuple: set of edges currently in the tour, set of edges excluded from the tour] Consider two nodes u and v, whose states according to the first tree formulation are ....

[Article contains additional citation context not shown here]

J.D. Little, et. al., "An Algorithm for the Traveling Salesman Problem," Operations Research, Vol.11, 1963.

.... results in Section 8, where we test the efficacy of our schemes 4 It turns out that the savings in the number of nodes formed more than compensates the extra memory used to store the h 0 costs of unformed children [15] 4 using the above heuristic as well as the much stronger LMSK heuristic [13], further corroborate this assertion. 3 Parallelization of A Here we describe the generic high level approach we have used to parallelize A on distributedmemory machines. Each processor executes an almost independent SEL SEQ A on its own OPEN list. The starting nodes required for a ....

....time per node expanded. We can represent t e and W as polynomials in N , where N is the number of variables in the COP to be solved. For instance, for both the heuristics used in our parallel A algorithms to solve TSP (viz. our simple heuristic of Section 2 and the LMSK heuristic [13]) y = 2, and the average case complexity of solving TSP is polynomial in N [24] Thus let t e = Theta(N y ) and W = Theta(N x ) so that W = Theta(t x=y e ) Hence for isoefficiency: W = Theta(W wt o ) Theta(W: t m t e ) 4) t e = O(d) 5) W = O(d x=y ) 6) Thus: Theorem ....

J.D. Little, et. al., "An Algorithm for the Traveling Salesman Problem," Operations Research, Vol.11, 1963.

....j x j u j j 2 N x j 2 Z j 2 I x j 2 j 2 C; where I is the set of integer variables, C is the set of continuous variables, and N = I [C. The lower and upper bounds l j and u j may take on the values of plus or minus infinity. The term branch and bound was originally coined by Little et al. [27] in their study of such an algorithm to solve the traveling salesman problem. However, the idea of 2 using a branch and bound algorithm for integer programming using linear programming relaxations was proposed somewhat earlier by Land and Doig [26] The process involves keeping a list of linear ....

....are minimal usually just changing one variable s bound. Depth first search has another advantage over bestfirst search in finding feasible solutions since feasible solutions tend to be found deep in the search tree. Depth first search was the strategy proposed by Dakin [10] and Little et al. [27], primarily due to the small memory capabilities of computers at that time. Despite its advantages, depth first search can lead to extremely large search trees. This stems from the fact that we may evaluate a good many nodes that would have been fathomed had a better value of z L been known. For ....

J. D. C. Little, K. G. Murty, D. W. Sweeney, and C. Karel. An algorithm for the traveling salesman problem. Operations Research, 21:972--989, 1963.

....sort. In addition to an array of locks, used to allow portions of data to be locked at a time, it uses four barriers to synchronize its computation phases. The details of the parallel algorithm appear in [RSRM93] TSP. The Traveling Sales Person application is implemented using the LMSK algorithm [JLK63] a branch and bound algorithm that proceeds by dynamic construction of a search tree, at the root of which is a description of the initial problem. Independent subproblems are generated by selection of specific edges from the graph and creation of children of the root. Once a tour is found for a ....

D. Sweeney J.D. Little, K. Murty and C. Karel. An algorithm for the traveling salesman problem. Operations Research, 11, 1963.

....system services like file or I O servers. The sample parallel program used in our research is a client server structured application, a parallel branch and bound algorithm solving the Traveling Salesperson problem (TSP) We employ the algorithm of Little, Murty, Sweeney and Karel (LMSK algorithm)[23], and we use a parallelization first described in [32] The resulting parallel algorithm essentially conducts a search in a dynamically constructed search space, where two abstractions are shared among searcher threads: 1) a global best tour value, which is used for pruning the search space, and ....

....space may be pruned by deletion of all leaf nodes with lower bounds greater than or equal to the value of the found tour. When all leaf nodes have been expanded or pruned, the lowest of all the found tours is the solution of the problem. A more complete description of the algorithm can be found in [23]. 2.2 Parallel Implementation of the LMSK Algorithm Our parallel LMSK algorithm is implemented as a collection of asynchronous, cooperating searcher threads each of which independently executes the algorithm s main procedure. The resulting code is outlined in Figure 1 (it implements the LMSK ....

D. Sweeney J.D. Little, K. Murty and C. Karel. An algorithm for the traveling salesman problem. Operations Research, 11, 1963.

....problem is as follows: given a set of K cities with nonnegative cost between each pair of cities, find the cheapest tour. A tour is a path that starting at some initial city visits every city once and only once, and returns to the initial city. We chose the well known method of Little et al. [7] to represent the search space and the lower bound heuristic for the traveling salesman problem. The technique that we used to improve the efficiency of IDA and ITS(0) in the flow shop scheduling problem cannot be used in the traveling salesman problem, because in this problem it is much more ....

J. D. Little, K. G. Murty, D. W. Sweeney, and C. Karel. An algorithm for the traveling salesman problem. Operations Research, 11:972--989, 1963.

....are necessary to achieve scaling performance. 1 Introduction The Branch and Bound (BB) algorithm [11, 14] has been used to solve many problems in science and engineering that otherwise would have no efficient methods of solution. Examples include the well known traveling salesman problem [12], the integer programming problem [3] and state space search [7] The algorithm has been employed in many applications such as floor planning of VLSI circuits [17] placement of electronic components [2] and robot path planning [5] The BB algorithm is computationally intensive, and can benefit ....

....In both algorithms, the idle time degrades the performance of the algorithms. 6 Results The central and concurrent list algorithms have been implemented on a 56 processor KSR1 multiprocessor. The algorithms were used to solve an asymmetric 50 city Travelling Salesman Problem. The LMSK heuristic [12] was used to compute the lower bounds of subproblems. The sequential execution time was about 100 seconds and a little over 8000 subproblems were generated before the optimal solution was determined. The expansion ratio for the two algorithms is shown in Figure 2. The expansion ratio for both ....

J.D.C. Little, K. G. Murty, D. W. Sweeney and C. Karel, "An algorithm for the traveling salesman problem," Operations Research, vol. 11, no. 6, pp. 972--989, 1963.

....near clash constraint as well. The near clash constraint consists of placing the events as separated as possible. It compares mainly the quality of solutions generated by the Brelaz algorithm and the number of constraint violations after a heuristic for solving the TSP (Traveling Salesman Problem) [10] is used to rearrange the slots in such a way that the total number of violations is minimum. 4 Conclusions The intention of the work presented in this article is mainly related to comparing the performance of genetic algorithms and conventional algorithms, specifically the Brelaz and Greedy ....

J.D.C. Little, K.G. Murty, D.W. Sweeney, and C. Karel. An algorithm for the traveling salesman problem. In Operations Research, volume 11. Operations Research Society of America, 1963.

.... one city to another is allowed to grow in proportion to the square of the number of cities (which satisfies the asymptotic optimality conditions in [10] To represent the search space and the lower bound heuristic for the Traveling Salesman Problem, we chose the well known method of Little et al. [11]. The search space in this formulation is a binary tree. We generated two sets of data, which we will call TSP Set 1 and TSP Set 2, and ran both IDA and A on each set. For both sets we selected the number of cities equal to 5, 10, 15, 20, 25, 30, and 35. For each value of the number of cities, ....

J. D. Little, K. G. Murty, D. W. Sweeney, and C. Karel. An algorithm for the traveling salesman problem. Operations Research, 11:972--989, 1963.

....a model. This model predicts, and the experimental results confirm, an eventual drop in speedup caused by an increase in communication overhead. The sequential algorithm uses an old decomposition heuristic based on the assignment problem, but that does not use the special structure of the TSP [36]. The 30 tested instances were generated randomly from a uniform distribution [52] Distributed strategy R. Correa and A. Ferreira: the synchronous and asynchronous implementations described in their paper approach the question of the relation between the order of selections and the possibility ....

J. Little, K. Murty, D. Sweeney, and C. Karel. An algorithm for the traveling salesman problem. Operations Research, 11:972--989, 1963.

....n cities, returning to the starting point, and is required to minimize the total cost of the trip. Every pair of cities i and j has a cost C ij associated with them (if i = j, then C ij is assumed to be of infinite cost) We have implemented the branch bound scheme proposed by Little, et al. [4]. More sophisticated branch bound schemes are available; since our focus is not on the best branch bound scheme, but rather on an efficient prioritized load balancing strategy, Little s scheme is sufficient for our purpose. For a thorough discussion on branch bound schemes and their ....

J. D. C. Little, K. G. Murty, D. W. Sweeney, and C. Karel. An algorithm for the traveling salesman problem. Operations Research, 11:972--989, 1963.

....visit each of N cities. There is an integer cost c(i,j) to travel from city i to city j, where total cost is the sum of the individual costs along the edges of the tour. The problem to be solved is represented as a N N cost matrix encoding the directed graph being traversed. The LMSK algorithm [29], used in our multiprocessor implementation of the TSP application, partitions the original problem into progressively smaller subproblems, which are represented as nodes of a dynamically constructed search tree. The algorithm computes a lower bound on the cost of the best tour in each node, and ....

....space may be pruned by deletion of all leaf nodes with lower bounds greater than or equal to the value of the found tour. When all leaf nodes have been expanded or pruned, the lowest of all the found tours is the solution of the problem. A more complete description of the algorithm can be found in [29]. Parallel Implementation and Locking Behavior. The parallel LMSK algorithm is implemented as a collection of cooperating, asynchronous searcher threads [15] that use two shared abstractions: 1) a work sharing queue ( work queue ) storing the leaf nodes of the tree representing the search space ....

[Article contains additional citation context not shown here]

J.D. Little, K. Murty, D. S., and Karel, C. An algorithm for the traveling salesman problem. Operations Research 11 (1963).

....changing the essential problem. For convenience we choose cost matrices with a mean edge cost of 10, but with varying standard deviations of these costs. To estimate the computational cost of solving TSP problems we used Little s algorithm; the best exact algorithm we could find [Little63] 8 ] It is a kind of backtrack algorithm that efficiently exploits properties of the cost matrix and guarantees to find a minimum cost solution. The results of running Little s algorithm for different numbers of cities with random cost matrices constructed according to a log normal distribution ....

Little, J. D. C., et al., "An Algorithm for the Traveling Salesman Problem", O.R.S.A., 11, 972-989, 1963.

....taken, rather than any uncertainty in the leaf values. This interesting study is long overdue for follow up work. 4.1 Alpha Beta One early paper on computer chess [ Newell et al. 1958 ] recognized that a full minimax search was not essential to determine the value of the tree. Some years later a little known work by Brudno [ 1963 ] provided a theoretical basis for pruning in minimax search. From these observations, the alpha beta pruning algorithm was developed, and it remains today the mainstay for game tree search. Of course many improvements and enhancements have been added over the years, and some of these will be ....

....memory contention becomes a serious bottleneck. It turns out, that the centralized strategy is only useful in domains where the node expansion time, T exp , is large compared to the time needed for an Open list access, T acc . One such application is the LMSK algorithm (named after its inventors Little, Murty, Sweeney and Karel [ 1963 ] for solving the traveling salesman problem. Almost linear time speedups were achieved [ Kumar et al. 1988 ] only with up to T exp =T acc processors. The saturation point lies well below 20 when the Open list entries are maintained in a global linked list structure. A more sophisticated ....

J.D.M. Little, K.G. Murty, D.W. Sweeney, and C. Karel. An algorithm for the traveling salesman problem. Operations Research, 11:972--989, 1963.

....Problem (TSP) is as follows: given a set of K cities with nonnegative cost between each pair of cities, find the cheapest tour. A tour is a path that starting at some initial city visits every city once and only once, and returns to the initial city. We chose the well known method of Little et al. [10] to represent the search space and the lower bound heuristic for the Traveling Salesman Problem. The search space in this formulation is a binary tree. The technique that we used to improve the efficiency of IDA and ITS(0) in the 3 machine flow shop scheduling problem cannot be used in the ....

J. D. Little, K. G. Murty, D. W. Sweeney, and C. Karel. An algorithm for the traveling salesman problem. Operations Research, 11:972--989, 1963.

....visit n cities, returning to the starting point, and is required to minimize the total cost of the trip. Every pair of cities i and j has a cost C ij associated with them (if i = j, then C ij is assumed to be of infinite cost) In the branch bound scheme originally proposed by Little, et al. [8]. one starts with an initial partial solution, a cost function (C) and an infinite upper bound. A partial solution comprises a set of edges (pairs of cities) that have been included in the circuit, and a set of edges that have been excluded from the circuit. The cost function provides for each ....

J. D. C. Little, K. G. Murty, D. W. Sweeney, and C. Karel. An algorithm for the traveling salesman problem. Operations Research, 11:972--989, 1963.

....techniques. In this problem, a salesman must visit n cities, returning to the starting point, and is required to minimize the total cost of the trip. Every pair of cities i and j has a cost C ij associated with them. We have implemented the branch bound scheme proposed by Little, et al. [6]. In Little s approach, one starts with an initial partial solution, a cost function (C) and an infinite upper bound. A partial solution comprises a set of edges (pairs of cities) that have been included in the circuit, and a set of edges that have been excluded from the circuit. The cost ....

J. D. C. Little, K. G. Murty, D. W. Sweeney, and C. Karel. An algorithm for the traveling salesman problem. Operations Research, 11:972--989, 1963.

....of the Open list and the parallel speedup is limited. Incidentally, for the same reason, the mandatory parallelism is also negligible. The travelling salesman problem (TSP) is a more sizeable application with greater scope to benefit from parallelism. We have implemented the algorithm of (Little et al. 1963) in the A framework, by defining the s 2 and h 2 procedures suitably. In this algorithm, each node of the search tree contains a cost matrix (of city city distances) and a partial tour (initially empty) Every node has two successors: one which adds a promising link (a pair of cities) to the ....

Little, J.D.C., Murty, K.G., Sweeney, D.W., and Karel, C. 1963. An algorithm for the traveling salesman problem. Operations Research 11, pp. 972--989.

No context found.

J. D. C. Little, K. G. Murty, D. W. Sweeney and C. Karel, `An algorithm for the traveling salesman problem', Operations Research, 11, 972--989 ( 1963).

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC