An approximation algorithm for the k -fixed depots problem (original) (raw)
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Approximation algorithms and heuristics for a 2-depot, heterogeneous Hamiltonian path problem
International Journal of Robust and Nonlinear Control, 2011
This article addresses an important routing problem that arises in surveillance applications involving two heterogeneous vehicles. As the addressed routing problem is NP-Hard, we develop an approximation algorithm and heuristics to solve the problem. Our approach involves solving the routing problem in two main steps: Partitioning and Sequencing. Partitioning involves finding a distinct set of targets to be visited by each vehicle. Sequencing provides the order in which each vehicle must visit the subset of targets assigned to it. The problem of partitioning is tackled by solving a linear program (LP) obtained by relaxing some of the constraints of an integer programming model for the problem. We consider two LP models for partitioning. The first LP model is obtained by mainly relaxing both the integrality and degree constraints, whereas the second model relaxes mainly the integrality constraints. Once the targets are partitioned, the sequencing problem can be solved either by Hoogeveen's algorithm or by the Lin-Kernighan heuristic to yield an approximately optimal solution. Computational results show that the algorithms based on the second LP model, on an average, provided better (closer to the optimum) solutions as compared with those based on the first LP model. We also observed that for both the LP models, the average quality of solutions given by the heuristics were found to be within 4% of the optimum, whereas the average quality of solutions obtained from the approximation algorithms were within 8-20% of the optimum depending on the problem size.
Approximation algorithms for a heterogeneous Multiple Depot Hamiltonian Path Problem
Proceedings of the 2011 American Control Conference, 2011
In this article, we present the first approximation algorithm for a routing problem that is frequently encountered in the motion planning of Unmanned Vehicles (UVs). The considered problem is a variant of a Multiple Depot-Terminal Hamiltonian Path Problem and is stated as follows: There is a collection of m UVs equipped with different sensors on-board and there are n targets to be visited by them collectively. There are restrictions on the targets of the following type: (1) A target may be visited by any UV, (2) a target must be visited only by a subset of UVs (with appropriate on-board sensor) and (3) a target may not be visited by a subset of UVs (as the set of onboard sensors on the UV may not be suitable for viewing the targets). The UVs are otherwise identical from the viewpoint of dynamic constraints on their motion and hence, the cost of traveling from a target A to a target B is the same for all vehicles. We will assume that triangle inequality is satisfied by the cost associated with travel, i.e., it is cheaper to travel from a target A to a target B directly than to go via an intermediate target C. The UVs may possibly start from different locations (referred to as depots) and are not required to return to the depot. While there are different objectives that can be considered for this problem, we consider the total cost of travel of all the UVs as an objective to be minimized. The problem considered in this article is a generalized version of single depot-terminal Hamiltonian Path Problem and is NP-hard.
Hamiltonian Approach for Finding Shortest Path
Hamiltonian problem is important branch of graph theory in mathematics and computer science. Need of finding shortest path is increased. The aim of the work is to find the shortest Hamiltonian path from a weighted complete graph. It is easy to understand and very useful in real life such as shortest route on road from home to hospital, social network, pizza delivery, mail delivery, gas delivery, tour travel and school bus travel etc. Due to exact timing delivery, the popularity of enterprise is increased and boom the clients, offerings and earnings. The objective of the work is to find shortest path using Hamiltonian technique from a weighted graph.
Polynomial Algorithms for Shortest Hamiltonian Path and Circuit
The problem of finding shortest Hamiltonian path and shortest Hamiltonian circuit in a weighted complete graph belongs to the class of NP-Complete problems [1]. This well known problem asks for a method or algorithm to locate such path or circuit that passes through every vertex only once in the given weighted complete graph. In this paper we begin with proposing two approximation algorithms for shortest Hamiltonian graphs which essentially consists of applying certain chosen permutations (transpositions or product of transpositions) on the adjacency matrix of given weighted complete graph causing reshuffling of the labels of its vertices. We change the labels of vertices through proper choice of permutations in such a way that in this relabeled graph the Hamiltonian path 1AE2AE3AE….kAE(k+1)AE…AEp becomes approximation to shortest path in the given weighted complete graph under consideration. We then define so called ordered weighted adjacency list for given weighted complete graph ...
A -approximation algorithm for the clustered traveling salesman tour and path problems
Operations Research Letters, 1999
We consider the Ordered Cluster Traveling Salesman Problem OCTSP. In this problem, a vehicle starting and ending at a given depot must visit a set of n points. The points are partitioned into K , K n, prespeci ed clusters. The vehicle must rst visit the points in cluster 1, then the points in cluster 2, : : : , and nally the points in cluster K so that the distance traveled is minimized. We present a 5 3-approximation algorithm for this problem which runs in On 3 time. We show that our algorithm can also be applied to the path version of the OCTSP: the Ordered Cluster Traveling Salesman Path Problem OCTSPP. Here the di erent starting and ending points of the vehicle may o r m a y not be prespeci ed. For this problem, our algorithm is also a 5 3-approximation algorithm.
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.
Linear-time algorithms for the Hamiltonian problems on distance-hereditary graphs
Theoretical Computer Science, 2005
A Hamiltonian path of a graph G is a simple path that contains each vertex of G exactly once. A Hamiltonian cycle of a graph is a simple cycle with the same property. The Hamiltonian path (resp. cycle) problem involves testing whether a Hamiltonian path (resp. cycle) exists in a graph. The 1HP (resp. 2HP) problem is to determine whether a graph has a Hamiltonian path starting from a specified vertex (resp. starting from a specified vertex and ending at the other specified vertex). The Hamiltonian problems include the Hamiltonian path, Hamiltonian cycle, 1HP, and 2HP problems. A graph is a distance-hereditary graph if each pair of vertices is equidistant in every connected induced subgraph containing them. In this paper, we present a unified approach to solving the Hamiltonian problems on distance-hereditary graphs in linear time.