This dissertation considered three variants/generalizations of a fundamental rout- ing problem involving multiple vehicles and developed approximation algorithms to address each of them. Developing approximation algorithms are difficult for multi- ple vehicle routing problems because it involves partitioning the targets among the vehicles and then solving a single TSP for each vehicle. This dissertation advances the state of art by developing new algorithms with better approximation factors for each of the considered problems. In particular, a novel 2-approximation algorithm is developed for a multiple terminal, Hamiltonian path problem by using a matroid intersection algorithm. For a special case of this multiple terminal problem where all the vehicles start from the same depot, the dissertation developed a 53-approximation algorithm.
The second routing problem addressed in this dissertation is a multiple vehicle routing problem where the cost of traveling between any two target locations depend on the type of the vehicle. A novel primal-dual algorithm with an approximation ratio of 2 was developed for a special case of this problem where the costs are symmetric, satisfy the triangle inequality and a monotonicity property. This is the first approx- imation result in the literature that provides a bound independent of the number of vehicles for a multiple depot, heterogeneous vehicle problem. Simulation results were also presented to corroborate the performance of the proposed approximation algorithm.
Finally, we develop the first approximation algorithm for a multiple, heteroge- neous vehicle routing problem where the costs are asymmetric and satisfy the triangle inequality. The approximation ratio of this algorithm is mdlog2(n + 1)e where m is
the number of vehicles and n is the number of targets. A primal-dual heuristic was also developed and the performance of all the proposed algorithms were corroborated using simulation results.
The following are the fundamental problems that are open in the area of approx- imation algorithms for multiple vehicle routing problems.
• Currently, there is no constant factor approximation algorithm for a general multiple depot, heterogeneous vehicle routing problem when all the travel costs satisfy the triangle inequality. Due to the difficulty involved in addressing heterogeneous costs, it is even possible that no constant factor approximation algorithms are possible. Any contribution is answering this question will be useful.
• Another important class of multiple vehicle routing problems includes cases where the vehicles may be homogenous but there may be additional vehicle- target assignments that must be satisfied. These problems arise in applications where the vehicles may carry different types of sensors and due to resource constraints, some targets must be visited only by some vehicles. Even though some special cases of this class of problems has constant factor approximation algorithms, the scenario where there are general vehicle-target constraints is still open.
• Currently, simulation results suggest that assigning targets to vehicles by round- ing a tight LP relaxation (section 4) leads to good approximate solutions. However, a better analysis of this rounding technique is required if a good approximation ratio is desired.
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