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

TL;DR

This work investigates the line-based Dial-a-Ride Problem (LiDARP) and a novel MinTurn variant, aiming to maximize served requests while handling a fixed stop line with vehicle turn constraints. It establishes a detailed boundary between polynomial-time solvability and NP-hardness by varying features such as time windows, shortcuts, and service promises, and contributes fixed-parameter tractable (FPT) and XP algorithms parameterized by the number of vehicles $k$, capacity $c$, line stops $h$, and time horizon $t$. Through reductions from 3-Partition, the authors prove strong NP-hardness (and in some cases inapproximability within factor 3) for MinTurn and LiDARP in the presence of time windows, and they extend the results to no-time-window settings via specialized constructions. The paper advances practical implications for ride-pooling on fixed lines, offering exact parameterized methods and identifying tractable special cases, complemented by an event-based graph framework and Multiset-Multicover techniques. Overall, the results combine theoretical hardness with actionable algorithms, enabling targeted approaches for autonomous and scheduled line-based routing problems.

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.

Figures (3)

• Figure 1: A MinTurn instance constructed from a 3-Partition instance with $m=3$ and $T=5$, as well as $s_1=3$ and $s_n=2$. The arrows represent the requests, with the white tipped arrows being value requests.
• Figure 2: The layout of the stops constructed for a 3-Partition instance. The line is the continuous path, while shortcuts are represented by (dotted) lines. The distances between neighboring stops are given by (blue) labels, with the exception of distances of $1$, which are omitted.
• Figure 3: A tour serving all requests of an instance constructed in the reduction from 3-Partition to the MinTurn problem with $m=2$. The value requests in $P_{\textrm{V}}$, represented by (blue) dashed arrows, are served in the intervals between the separator requests, represented by (red) solid arrows. The subtours used to return from serving a request are represented by lighter lines.

Theorems & Definitions (48)

• lemma 1: lem:route-to-tour
• lemma 2: lem:route-joining
• theorem 1: thm:poly-lidarp
• lemma 3
• proof
• lemma 4
• proof
• theorem 2: thm:mt-poly-variants
• definition 1: 3-Partition garey_computers_1979
• theorem 3: thm:sp-st-nphard