Dial a ride from k-forest

Anupam Gupta, Mohammad Taghi Hajiaghayi, Viswanath Nagarajan, R. Ravi

Research output: Chapter in Book/Report/Conference proceedingConference contribution


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 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.

Original languageEnglish (US)
Title of host publicationAlgorithms - ESA 2007 - 15th Annual European Symposium, Proceedings
PublisherSpringer Verlag
Number of pages12
ISBN (Print)9783540755197
StatePublished - 2007
Event15th Annual European Symposium on Algorithms, ESA 2007 - Eilat, Israel
Duration: Oct 8 2007Oct 10 2007

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume4698 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Conference15th Annual European Symposium on Algorithms, ESA 2007

ASJC Scopus subject areas

  • Theoretical Computer Science
  • General Computer Science


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