109 research outputs found

    A survey of approximation algorithms for capacitated vehicle routing problems

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    Finding the shortest travelling tour of vehicles with capacity k from the depot to the customers is called the Capacity vehicle routing problem (CVRP). CVRP plays an essential position in logistics systems, and it is the most intensively studied problem in combinatorial optimization. In complexity, CVRP with k \ge 3 is an NP-hard problem, and it is APX-hard as well. We already knew that it could not be approximated in metric space. Moreover, it is the first problem resisting Arora's famous approximation framework. So, whether there is, a polynomial-time (1+ϵ\epsilon)-approximation for the Euclidean CVRP for any ϵ>0\epsilon>0 is still an open problem. This paper will summarize the research progress from history to up-to-date developments. The survey will be updated periodically.Comment: First submissio

    Approximation algorithms for facility location problems (extended abstract)

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    A Water-Filling Primal-Dual Algorithm for Approximating Non-Linear Covering Problems

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    Obtaining strong linear relaxations of capacitated covering problems constitute a significant technical challenge even for simple settings. For one of the most basic cases, the Knapsack-Cover (Min-Knapsack) problem, the relaxation based on knapsack-cover inequalities has an integrality gap of 2. These inequalities are exploited in more general problems, many of which admit primal-dual approximation algorithms. Inspired by problems from power and transport systems, we introduce a general setting in which items can be taken fractionally to cover a given demand. The cost incurred by an item is given by an arbitrary non-decreasing function of the chosen fraction. We generalize the knapsack-cover inequalities to this setting an use them to obtain a (2+?)-approximate primal-dual algorithm. Our procedure has a natural interpretation as a bucket-filling algorithm which effectively overcomes the difficulties implied by having different slopes in the cost functions. More precisely, when some superior segment of an item presents a low slope, it helps to increase the priority of inferior segments. We also present a rounding algorithm with an approximation guarantee of 2. We generalize our algorithm to the Unsplittable Flow-Cover problem on a line, also for the setting of fractional items with non-linear costs. For this problem we obtain a (4+?)-approximation algorithm in polynomial time, almost matching the 4-approximation algorithm known for the classical setting
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