Dial a Ride from k-forest
Anupam Gupta, MohammadTaghi Hajiaghayi, Viswanath Nagarajan, R. Ravi

TL;DR
This paper presents an improved approximation algorithm for the k-forest problem and applies it to develop better approximation algorithms for Dial-a-Ride problems, achieving near-optimal solutions in complex metric spaces.
Contribution
It introduces an $O(\min\{\sqrt{n},\sqrt{k}\})$-approximation for k-forest and leverages this to improve Dial-a-Ride approximation algorithms.
Findings
Improved approximation ratio for k-forest from $O(n^{2/3}\log n)$ to $O(\min\\{\sqrt{n},\\sqrt{k}\})$.
Derived a new $O(\min\\{\sqrt{n},\\sqrt{k}\\}\log^2 n)$-approximation for Dial-a-Ride.
Provided a different proof and slight improvement over previous Dial-a-Ride approximation results.
Abstract
The k-forest problem is a common generalization of both the k-MST and the dense--subgraph problems. Formally, given a metric space on vertices , with demand pairs and a ``target'' , the goal is to find a minimum cost subgraph that connects at least demand pairs. In this paper, we give an -approximation algorithm for -forest, improving on the previous best ratio of by Segev & 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 point metric space with objects each with its own source and destination, and a vehicle capable of carrying at most objects at any time, find the minimum length tour that uses this vehicle to move each object from its source to…
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