Vehicle routing is a class of optimization problems where the objective is to find low cost delivery routes from depots to customers using vehicles of limited capacity. Vehicle routing problems generalize the traveling salesman problem and have many real world applications to businesses with high transportation costs such as waste removal companies, newspaper deliverers, and food and beverage distributors. We study two basic vehicle routing problems: unit demand routing and unsplittable demand routing. In the unit demand problem, items must be delivered from depots to customers using a vehicle of limited capacity and each customer requires delivery of a single item. The unsplittable demand problem is a generalization, where customers can have different demands for the number of items, but each customer's entire demand must be delivered all together by one route. Both problems are NP-Hard and do not admit better than constant factor approximation algorithms in the metric setting. However in many practical settings the input to the problem has Euclidean structure. We show how to exploit this to design arbitrarily good approximation algorithms. We design a quasi-polynomial time approximation scheme for the Euclidean unit demand problem in constant dimensions, and asymptotic polynomial time approximation schemes for the unsplittable demand problem in one dimension.
"Approximation Schemes for Euclidean Vehicle Routing Problems."
Computer Science Theses and Dissertations.
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