In bus routing, the task is to plan a bus route in a network with several agents, each of whom wants to travel from a starting point to a destination. A bus route should account for several factors, including agents' cost for reaching the bus stops, their travel time, or the energy consumption of the buses. We study the complexity of several variants of this problem, focusing on how the objective function and the models for agents' walking costs influence the problem complexity. After observing that even the simplest agent cost model leads to hardness on general networks, we consider networks with tree structure. Our main findings are as follows. First, allowing agent-specific cost models leads to hardness even on extremely limited trees such as stars. Second, consistent agent models (where agents differ only in their starting points and destinations) make the problem easier in some cases. Finally, allowing agents to choose between using the bus and walking directly can make the problem considerably harder. Most of our hardness results show not only classical NP-hardness but also parameterized intractability for the natural parameter k, the number of bus stops.
Figures & tables
Figure 1. Overview of our results.
Figure 2. Example of the input preprocessing in the proof of Theorem 1 . The hollow vertices are the newly added vertices and the dashed edges have weight 0 for both weight functions wa and wb .
Figure 3. Comparison of three k=10 stop selections on the NYC-M15 corridor, restricted to existing stops.
We study a stylized model of infrastructure investment in public transportation. In our model, each agent travels between a pair of terminals in a network captured by a weighted graph, where edge weights represent distances. The central planner can reduce the travel time along a fixed number of edges, with the goal of maximizing the utilitarian or egalitarian welfare. When there is only one agent, we provide a polynomial-time algorithm that combines Dijkstra's algorithm with a dynamic program. We then demonstrate how to use this algorithm as a subroutine to solve the problem for two agents. Generalizing this idea, we present an XP algorithm parameterized by the number of agents n; however, our problem turns out to be W[1]-hard with respect to n. Nevertheless, we establish a fixed-parameter tractability result for the special case where all agents travel to a common hub. If the number of agents is variable, we obtain NP-completeness and inapproximability results. We discuss implications of our results for a related model of railway network design.
Martin Bullinger, Edith Elkind, Kassian Köck
School of Engineering Mathematics and Technology, University of Bristol, Bristol, UK · School of Engineering, Northwestern University, Evanston, USA · School of Computation, Information and Technology, Technical University of Munich, Munich, Germany
Anonymous Multi-Agent Path Finding (AMAPF) admits polynomial-time network-flow algorithms for several objectives, including makespan, total distance, and sum-of-costs (SoC) when agents disappear upon reaching goals. We show that standard goal-staying AMAPF is fundamentally different. We first formulate SoC minimization by augmenting the standard time-expanded flow model with goal-settlement constraints and show that the resulting linear programming relaxation is non-integral. We then prove that minimizing SoC in goal-staying AMAPF is NP-hard via a reduction from 3-SAT. Together with the polynomial-time result for the disappearing variant, this establishes a sharp complexity boundary determined by whether completed agents remain at their goals.
Multi-Agent Route Planning considers selecting vehicles, each associated with a single predefined route, such that route-level coverage utility is maximized while redundant spatial overlaps are limited. This paper gives a formal problem definition, proves NP-hardness by reduction from the Weighted Set Packing problem, and derives a Quadratic Unconstrained Binary Optimization formulation whose coefficients directly encode route utility rewards and pairwise overlap penalties. A single penalty parameter λ controls the coverage--overlap trade-off. We distinguish between a soft regime, which supports multi-objective exploration, and a hard regime, in which the penalty is strong enough to effectively enforce near-disjoint routes. We describe a practical pipeline for generating city instances, constructing candidate routes, building the QUBO matrix, and solving it with a binary quadratic programming baseline (Gurobi), simulated annealing, and D-Wave hybrid quantum annealing. Experiments on Barcelona instances with up to 10,000 vehicles reveal a clear coverage--overlap knee and show that Pareto-optimal solutions are mainly obtained under the hard-penalty regime, while D-Wave hybrid solvers and Gurobi achieve very similar objective values on matching configurations with only minor runtime differences as problem size grows.
Renáta Rusnáková, Martin Chovanec, Juraj Gazda
Department of Computers and Informatics, Faculty of Electrical Engineering and Informatics, Technical University of Košice, Letná 9, 042 00 Košice