Dial a Ride from k-forest

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-$k$-subgraph problems. Formally, given a metric space on $n$ vertices $V$, with $m$ demand pairs $\subseteq V \times V$ and a ``target'' $k\le m$, the goal is to find a minimum cost subgraph that connects at least $k$ demand pairs. In this paper, we give an $O(\min\{\sqrt{n},\sqrt{k}\})$-approximation algorithm for $k$-forest, improving on the previous best ratio of $O(n^{2/3}\log n)$ 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 $n$ 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 $\alpha$-approximation algorithm for the $k$-forest problem implies an $O(\alpha\cdot\log^2n)$-approximation algorithm for Dial-a-Ride. Using our results for $k$-forest, we get an $O(\min\{\sqrt{n},\sqrt{k}\}\cdot\log^2 n)$- approximation algorithm for Dial-a-Ride. The only previous result known for Dial-a-Ride was an $O(\sqrt{k}\log n)$-approximation by Charikar & 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.