Dial a Ride from k-Forest

作者：Gupta Anupam^*; Hajiaghayi Mohammadtaghi; Nagarajan Viswanath; Ravi R

来源：ACM Transactions on Algorithms, 2010, 6(2): 41.

DOI：10.1145/1721837.1721857

• Algorithms
• Theory
• Approximation algorithms
• network design
• vehicle routing

The k-forest problem is a common generalization of both the k-MST and the dense-k-subgraph problems. Formally, given a metric space on n vertices V, with m demand pairs. V x V and a "target" k <= m, the goal is to find a minimum cost subgraph that connects at least k pairs. In this paper, we give an O(min{root n . log k, root k})-approximation algorithm for k-forest, improving on the previous best ratio of O( min{n(2/3), root m} log n) by Segev and Segev.
We then apply our algorithm for k-forest to obtain approximation algorithms for several Dial-a-Ride problems. The basic Dial-a-Ride problem is the following: given an n point metric space with m objects each with its own source and destination, and a vehicle capable of carrying at most k objects at any time, find the minimum length tour that uses this vehicle to move each object from its source to destination. We want that the tour be non-preemptive: that is, each object, once picked up at its source, is dropped only at its destination. We prove that an alpha-approximation algorithm for the k-forest problem implies an O(alpha.log(2) n)-approximation algorithm for Dial-a-Ride. Using our results for k-forest, we get an O(min{root n, root k} . log(2) n)-approximation algorithm for Dial-a-Ride. The only previous result known for Dial-a-Ride was an O(root k log n)-approximation by Charikar and Raghavachari; our results give a different proof of a similar approximation guarantee-in fact, when the vehicle capacity k is large, we give a slight improvement on their results. The reduction from Dial-a-Ride to the k-forest problem is fairly robust, and allows us to obtain approximation algorithms (with the same guarantee) for some interesting generalizations of Dial-a-Ride.

• 出版日期2010-3
• 单位IBM