736 research outputs found
Phase Transitions and Backbones of the Asymmetric Traveling Salesman Problem
In recent years, there has been much interest in phase transitions of
combinatorial problems. Phase transitions have been successfully used to
analyze combinatorial optimization problems, characterize their typical-case
features and locate the hardest problem instances. In this paper, we study
phase transitions of the asymmetric Traveling Salesman Problem (ATSP), an
NP-hard combinatorial optimization problem that has many real-world
applications. Using random instances of up to 1,500 cities in which intercity
distances are uniformly distributed, we empirically show that many properties
of the problem, including the optimal tour cost and backbone size, experience
sharp transitions as the precision of intercity distances increases across a
critical value. Our experimental results on the costs of the ATSP tours and
assignment problem agree with the theoretical result that the asymptotic cost
of assignment problem is pi ^2 /6 the number of cities goes to infinity. In
addition, we show that the average computational cost of the well-known
branch-and-bound subtour elimination algorithm for the problem also exhibits a
thrashing behavior, transitioning from easy to difficult as the distance
precision increases. These results answer positively an open question regarding
the existence of phase transitions in the ATSP, and provide guidance on how
difficult ATSP problem instances should be generated
Iterative Patching and the Asymmetric Traveling Salesman Problem
Although Branch and Bound (BnB) methods are among the most widely used techniques for solving hard problems, it is still a challenge to make these methods smarter. In this paper, we investigate iterative patching, a technique in which a fixed patching procedure is applied at each node of the BnB search tree for the Asymmetric Traveling Salesman Problem. Computational experiments show that iterative patching results in general in search trees that are smaller than the usual classical BnB trees, and that solution times are lower for usual random and sparse instances. Furthermore, it turns out that, on average, iterative patching with the Contract-or-Patch procedure of Glover, Gutin, Yeo and Zverovich (2001) and the Karp-Steele procedure are the fastest, and that ?iterative? Modified Karp-Steele patching generates the smallest search trees.
On the number of -cycles in the assignment problem for random matrices
We continue the study of the assignment problem for a random cost matrix. We
analyse the number of -cycles for the solution and their dependence on the
symmetry of the random matrix. We observe that for a symmetric matrix one and
two-cycles are dominant in the optimal solution. In the antisymmetric case the
situation is the opposite and the one and two-cycles are suppressed. We solve
the model for a pure random matrix (without correlations between its entries)
and give analytic arguments to explain the numerical results in the symmetric
and antisymmetric case. We show that the results can be explained to great
accuracy by a simple ansatz that connects the expected number of -cycles to
that of one and two cycles.Comment: To appear in Journal of Statistical Mechanic
On the expected efficiency of branch and bound for the asymmetric TSP
Let the costs for an instance of the asymmetric traveling
salesperson problem be independent uniform random variables. We
consider the efficiency of branch and bound algorithms that use the assignment
relaxation as a lower bound. We show that w.h.p. the number of steps taken in
any such branch and bound algorithm is for some small
absolute constant
Stochastic Dynamic Vehicle Routing in the Euclidean Plane: The Multiple-Server, Capacitated Vehicle Case
In a previous paper [12], we introduced a new model for stochastic and dynamic vehicle routing called the dynamic traveling repairman problem (DTRP), in which a vehicle traveling at constant velocity in a Euclidean region must service demands whose time of arrival, location and on-site service are stochastic. The objective is to find a policy to service demands over an infinite horizon that minimizes the expected system time (wait plus service) of the demands. We showed that the stability condition did not depend on the geometry of the service region (i.e. size, shape, etc.). In addition, we established bounds on the optimal system time and proposed an optimal policy in light traffic and several policies that have system times within a constant factor of the lower bounds in heavy traffic. We showed that the leading behavior of the optimal system time had a particularly simple form which increases much more rapidly with traffic intensity than the system time in traditional queues (e.g. M/G/1). In this paper, we extend these results in several directions. First, we propose new bounds and policies for the problem of m identical vehicles with unlimited capacity and show that in heavy traffic the system time is reduced by a factor of 1/m2 over the single server case. Policies based on dividing the service region into m equal subregion
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