Author:  Xu, Liang 
Title:  Approximability of vehicle routing problems 
Degree:  Ph.D. 
Year:  2012 
Subject:  Vehicle routing problem. Vehicle routing problem  Mathematical models. Hong Kong Polytechnic University  Dissertations 
Department:  Dept. of Logistics and Maritime Studies 
Pages:  xiv, 142 p. : ill. ; 30 cm. 
Language:  English 
InnoPac Record:  http://library.polyu.edu.hk/record=b2507354 
URI:  http://theses.lib.polyu.edu.hk/handle/200/6514 
Abstract:  Vehicle Routing Problems (VRPs) can be described as a class of combinatorial optimization problems that seek to determine a set of feasible routes for a fleet of vehicles to serve customers at different locations, so as to optimize certain objective functions. Due to the growth of the transportation and logistics industries, many new VRPs are emerging in different contexts. Since they are usually NPhard, these problems call for the design of approximation algorithms that achieve constant approximation ratios. Moreover, in the existing literature there are several VRPs whose constantratio approximation algorithms are either unknown or improvable. Therefore, in this thesis we study the approximability of the following three categories of VRPs, by either developing the constantratio approximation algorithms or deriving approximation hardness results. The first category of problems includes the minmax Path Cover Problem (PCP) and its variants, which aim to determine a set of k paths for k vehicles to serve customers in a metric undirected graph, so that the maximum total edge and vertex weight among all paths is minimized. This category of problems was introduced in the literature in the context of "vehicle routing for relief efforts". Since they are relatively new in the literature, approximation algorithms are almost unknown. We consider four variants of the minmax PCP, in which the vehicles have either unlimited or limited capacities, and they start from either a given depot or any depot of a given depot set. We have developed approximation algorithms for these variants, which achieve approximation ratios of max{3  2/k, 2}, 5, max{5  2/k, 4}, and 7, respectively. For these four variants of PCP, we have also proved their first approximation hardness results, by showing that unless P=NP, it is impossible for them to achieve approximation ratios less than 4/3, 3/2, 3/2, and 2, respectively. The second category of problems includes the minmax kTraveling Salesmen Problem on a Tree (kTSPT) and its variants. With k a given positive integer denoting the number of salesmen, which is independent of the input size, the minmax kTSPT aims to determine a set of k tours for the k salesmen to serve all the customers that are located in a tree, such that the k tours all start from and return to the depot, so as to minimize the maximum length of the k tours. In the literature, the question as to whether or not the minmax 2TSPT yields a pseudopolynomial time exact algorithm has remained open for a decade. We have provided a positive answer to this open question by developing a pseudopolynomial time exact algorithm using a dynamic programming approach. Based on this dynamic program, we have further developed a fully polynomial time approximation scheme for the problem. Moreover, we have generalized these algorithms for the minmax kTSPT for any given constant k ≥ 2, and we have extended them to other variants. The third category of problems includes the kdepot Traveling Salesmen Problem (kdepot TSP) and its variants. The kdepot TSP is an extension of the singledepot TSP, and it aims to determine a set of k tours for the k vehicles to serve all the customers on a metric undirected graph so that the total length of the tours is minimized. We have shown that a nontrivial extension of the wellknown Christofides' heuristic has a tight approximation ratio of 2  1/k, which is better than the existing 2approximation algorithm available in the literature. This result is significant when k is small. Moreover, we have demonstrated how this algorithm can be applied to the development of approximation algorithms for other multipledepot VRPs. 
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