The Complexity of Counting Turns in the Line-Based Dial-a-Ride Problem

TL;DR

This paper analyzes the computational complexity of the line-based Dial-a-Ride problem and the problem of minimizing turns in optimal solutions, identifying boundaries between tractable and NP-hard cases.

Contribution

It determines the complexity boundaries for LiDARP and MinTurn problems and provides parameterized algorithms for solving them.

Findings

Identified polynomially solvable and NP-hard instances based on parameters

Established complexity boundaries for both problems

Developed parameterized algorithms for solution

Abstract

Dial-a-Ride problems have been proposed to model the challenge to consolidate passenger transportation requests with a fleet of shared vehicles. The line-based Dial-a-Ride problem (LiDARP) is a variant where the passengers are transported along a fixed sequence of stops, with the option of taking shortcuts. In this paper we consider the LiDARP with the objective function to maximize the number of transported requests. We investigate the complexity of two optimization problems: the LiDARP, and the problem to determine the minimum number of turns needed in an optimal LiDARP solution, called the MinTurn problem. Based on a number of instance parameters and characteristics, we are able to state the boundary between polynomially solvable and NP-hard instances for both problems. Furthermore, we provide parameterized algorithms that are able to solve both the LiDARP and MinTurn problem.