Dial a ride from k-forest

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×V and a "target" k < m, the goal is to find a minimum cost subgraph that connects at least k demand pairs. In this paper, we give an O(min{√n, √k})-approximation algorithm for k-forest, improving on the previous best ratio of O(min{n^2/3, √m}log n) by Segev and Segev [20]. 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 prove that an α-approximation algorithm for the k-forest problem implies an O(α·log² n)-approximation algorithm for Dial-a-Ride. Using our results for k-forest, we get an O(min{√n, √k} ·log² n)-approximation algorithm for Dial-a-Ride. The only previous result known for Dial-a-Ride was an O(√k log n)-approximation by Charikar and Raghavachari [5]; 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 the following generalizations: (i) Non-uniform Dial-a-Ride, where the cost of traversing each edge is an arbitrary non-decreasing function of the number of objects in the vehicle; and (ii) Weighted Dial-a-Ride, where demands are allowed to have different weights. The reduction is essential, as it is unclear how to extend the techniques of Charikar and Raghavachari to these Dial-a-Ride generalizations.