A -approximation algorithm for the clustered traveling salesman tour and path problems (original) (raw)
Related papers
Traveling Salesman Problem with Clustering
Journal of Statistical Physics, 2010
In the original traveling salesman problem, the traveling salesman has the task to find the shortest closed tour through a proposed set of nodes, touching each node exactly once and returning to the initial node at the end. For the sake of the tour length to be minimized, nodes close to each other might not be visited one after the other but separated in the tour. However, for some practical applications, it is useful to group nodes to clusters, such that all nodes of a cluster are visited contiguously. Here we present an approach which leads to an automatic clustering with a clustering parameter governing the sizes of the clusters.
Operations Research Letters, 2007
In this paper, we present an algorithm with an approximation factor of 2 for a Generalized, Multiple Depot, Multiple Travelling Salesman Problem (GMTSP) when the costs are symmetric and satisfy the triangle inequality. The algorithm requires finding a degree constrained minimum spanning tree which we compute using a Lagrangian relaxation. c ij = c ji and satisfy the triangle inequality, namely, c ij + c jk c ik for all i, j, k ∈ V . A tour of salesman V i is an ordered set, TOUR i , of at least r + 2, r 1 elements of the form {V i , V i 1 , . . . , V i r , V i }, where V i l , l = 1, . . . , r i corresponds to r i distinct destinations being visited in that sequence by the ith salesman.
Approximation algorithms for the traveling salesman problem
Mathematical Methods of Operations Research (ZOR), 2003
We first prove that the minimum and maximum traveling salesman problems, their metric versions as well as some versions defined on parameterized triangle inequalities (called sharpened and relaxed metric traveling salesman) are all equi-approximable under an approximation measure, called differential-approximation ratio, that measures how the value of an approximate solution is placed in the interval between the worst-and the best-value solutions of an instance. We next show that the 2 OPT, one of the mostknown traveling salesman algorithms, approximately solves all these problems within differential-approximation ratio bounded above by 1/2. We analyze the approximation behavior of 2 OPT when used to approximately solve traveling salesman problem in bipartite graphs and prove that it achieves differential-approximation ratio bounded above by 1/2 also in this case. We also prove that, for any > 0, it is NP-hard to differentially approximate metric traveling salesman within better than 649/650 + and traveling salesman with distances 1 and 2 within better than 741/742 +. Finally, we study the standard approximation of the maximum sharpened and relaxed metric traveling salesman problems. These are versions of maximum metric traveling salesman defined on parameterized triangle inequalities and, to our knowledge, they have not been studied until now.
Literature Review on Travelling Salesman Problem
International Journal of Research, 2018
The Traveling Salesman Problem (TSP) is a classical combinatorial optimization problem, which is simple to state but very difficult to solve. The problem is to find the shortest tour through a set of N vertices so that each vertex is visited exactly once. This problem is known to be NP-hard, and cannot be solved exactly in polynomial time. Many exact and heuristic algorithms have been developed in the field of operations research (OR) to solve this problem. In this paper we provide overview of different approaches used for solving travelling salesman problem.
Solving the clustered traveling salesman problem with ‐relaxed priority rule
International Transactions in Operational Research, 2020
The Clustered Traveling Salesman Problem with a Prespecified Order on the Clusters, a variant of the well-known traveling salesman problem is studied in literature. In this problem, delivery locations are divided into clusters with different urgency levels and more urgent locations must be visited before less urgent ones. However, this could lead to an inefficient route in terms of traveling cost. This priority-oriented constraint can be relaxed by a rule called d-relaxed priority that provides a trade-off between transportation cost and emergency level. Our research proposes two approaches to solve the problem with d-relaxed priority rule. We improve the mathematical formulation proposed in the literature to construct an exact solution method. A meta-heuristic method based on the framework of Iterated Local Search with problem-tailored operators is also introduced to find approximate solutions. Experimental results show the effectiveness of our methods. Keywords. Clustered traveling salesman problem, d-relaxed priority rule, mixed integer programming, iterated local search. the locations are supposed to have the same degrees of urgencies, i.e., they can be visited in any order. However, in a number of real-world routing applications, different levels of priorities at the delivery locations need to be taken into account in routing plans. For example, as a result of a natural disaster such as a storm, earthquake, tsunami, or hurricane, there are demands at many locations for relief supplies such as food, bottled water, blankets, or medical packs. Some locations are in more urgent need of supplies than other locations due to the relative position of the source of disasters, the damage status, or its importance (schools, hospitals, and government institutions should be considered as more important). Locations requiring the same level of urgency can be clusterized into groups. And the priority of a group during the relief process has to be considered, e.g., higher priority groups should be visited before others. In the example above, the priorities indicate the importance (or urgency) of the demand at each location. Typically, priority 1 nodes must be served before priority 2 nodes, priority 2 nodes must be served before priority 3 nodes, and so on. Such a problem is called the Clustered Traveling Salesman Problem with a Prespecified Order on the Clusters (CTSP-PO) and has been studied in [22, 17]. However, this rule is strict with respect to the priority and can lead to an inefficient route in terms of traveling cost. It may be relevant to visit some lower priority nodes while serving higher priority nodes. In [4, 5], the authors proposed a simple, but elegant rule called d-relaxed priority that provides flexibility to the decision maker in terms of capturing trade-offs between total distance and node priorities. In [5] and Chapter 14 of [6], the d-relaxed priority rule is defined as follows. Given a positive number d, at any point of the route, if p is the highest priority class among all unvisited locations, the relaxed rule allows the vehicle to visit locations with priority p, p + 1, ..., p + d before visiting all locations in class p. By changing the value of d, we can flexibly control to focus more on economic aspect or urgency level. Indeed, if we consider the 0-relaxed priority rule (i.e., d = 0), all the higher priority nodes must be visited before lower priority nodes. The problem is a CTSP-PO, the strictest version w.r.t priority. On the other hand, if d is set to g − 1, where g is the number of priorities, the problem becomes a typical TSP, all the node priorities being ignored.
Approximation algorithms for the capacitated traveling salesman problem with pickups and deliveries
Naval Research Logistics, 1999
We consider the Capacitated Traveling Salesman Problem with Pickups and Deliveries (CTSPPD). This problem is characterized by a set of n pickup points and a set of n delivery points. A single product is available at the pickup points which must be brought to the delivery points. A vehicle of limited capacity is available to perform this task. The problem is to determine the tour the vehicle should follow so that the total distance traveled is minimized, each load at a pickup point is picked up, each delivery point receives its shipment and the vehicle capacity is not violated. We present two polynomial-time approximation algorithms for this problem and analyze their worst-case bounds.
Algorithms for the traveling Salesman Problem with Neighborhoods involving a dubins vehicle
Proceedings of the 2011 American Control Conference, 2011
We study the problem of finding the minimumlength curvature constrained closed path through a set of regions in the plane. This problem is referred to as the Dubins Traveling Salesperson Problem with Neighborhoods (DTSPN). Two algorithms are presented that transform this infinite dimensional combinatorial optimization problem into a finite dimensional asymmetric TSP by sampling and applying the appropriate transformations, thus allowing the use of existing approximation algorithms. We show for the case of disjoint regions, the first algorithm needs only to sample each region once to produce a tour within a factor of the length of the optimal tour that is independent of the number of regions. We present a second algorithm that performs no worse than the best existing algorithm and can perform significantly better when the regions overlap.
A heuristic for the traveling salesman problem based on a continuous approximation
Transportation Research Part B: Methodological, 1998
A procedure for solving, suboptimally, the traveling salesman problem is presented. The set of points on the traveling salesman tour is distributed over a region having the shape of a circular or ring sector. The procedure is based on an optimal partition of the sector and reduces the tour construction to a sorting problem. Namely, the tour is constructed by visiting the points in radial or angular order depending on the part of the sector on which they are located. The optimal partition is derived by a continuous approximation of the set of points. It is de®ned by a single parameter and simple analytical expressions for it are obtained. A great number of numerical tests were carried out to evaluate the performance of the procedure. These tests allowed a measure of the dierence from the optimum solution that could be obtained for problems up to a few hundred points. The results show that the euclidean length of the tours produced by the partition procedure grows, on average, like AN p , where A is the region area and N the number of points. The initial tours are improved by means of the Or's algorithm and the ®nal tours obtained are nearly as good as those given by more intrincate improvement heuristics. The whole procedure, that is, the tour construction and improvement heuristics, is rather simple to implement on a computer, which makes it very appealing to use in routing software.