When trees collide: an approximation algorithm for the generalized Steiner problem on networks
STOC '91 Proceedings of the twenty-third annual ACM symposium on Theory of computing
A general approximation technique for constrained forest problems
SODA '92 Proceedings of the third annual ACM-SIAM symposium on Discrete algorithms
Approximation algorithms for directed Steiner problems
Proceedings of the ninth annual ACM-SIAM symposium on Discrete algorithms
The k-traveling repairman problem
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
The Finite Capacity Dial-A-Ride Problem
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
Paths, Trees, and Minimum Latency Tours
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
Approximation Algorithms for Orienteering and Discounted-Reward TSP
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
Approximation algorithms for deadline-TSP and vehicle routing with time-windows
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Saving an epsilon: a 2-approximation for the k-MST problem in graphs
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Tighter Bounds for Graph Steiner Tree Approximation
SIAM Journal on Discrete Mathematics
The prize-collecting generalized steiner tree problem via a new approach of primal-dual schema
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Approximate k-Steiner forests via the Lagrangian relaxation technique with internal preprocessing
ESA'06 Proceedings of the 14th conference on Annual European Symposium - Volume 14
ESA'07 Proceedings of the 15th annual European conference on Algorithms
Hardness of preemptive finite capacity dial-a-ride
APPROX'06/RANDOM'06 Proceedings of the 9th international conference on Approximation Algorithms for Combinatorial Optimization Problems, and 10th international conference on Randomization and Computation
Improved approximating algorithms for Directed Steiner Forest
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Online and offline algorithms for the sorting buffers problem on the line metric
Journal of Discrete Algorithms
ESA'07 Proceedings of the 15th annual European conference on Algorithms
Thresholded covering algorithms for robust and max-min optimization
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
Approximating k-generalized connectivity via collapsing HSTs
Journal of Combinatorial Optimization
Euclidean prize-collecting steiner forest
LATIN'10 Proceedings of the 9th Latin American conference on Theoretical Informatics
An approximation algorithm for the Generalized k-Multicut problem
Discrete Applied Mathematics
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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{n2/3,√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 a-approximation algorithm for the k-forest problem implies an O(αċlog2 n)-approximation algorithm for Dial-a-Ride. Using our results for k-forest, we get an O(min{√n,√k}ċlog2 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 nondecreasing function of the number of objects in the vehicle; and (ii) Weighted Diala-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.