PaRoutes Multi-Step Benchmark

• The paper introduces a configurable evaluation suite that rigorously tests multi-step route planning on both simulated grids and actual city networks.
• It defines five minimization objectives—capturing distance, accident delays, elevation, travel time, and smoothness—with explicit mathematical formulations.
• The benchmark employs Pareto-front analysis and the IGD+ indicator to assess algorithmic methods such as NSGA-II, NSGA-III, and d-NSGA-II in diverse settings.

The PaRoutes Multi-Step Benchmark is a rigorously designed evaluation suite for multi-objective, multi-step route planning, originating in computational chemistry for retrosynthetic analysis but extensible to broader multi-step decision-making scenarios. It provides a comprehensive and scalable framework for testing algorithmic performance in long-horizon, many-objective optimization and planning, with detailed formulations that facilitate both simulated and real-world applicability.

1. Formulation and Objectives

The benchmark is constructed around multi-step pathfinding on a directed graph, where each candidate route—typically a sequence of nodes $N = (n_1, n_2, ..., n_k)$—is evaluated according to multiple quantitative objectives. In its canonical instantiation for logistics and pathfinding, five minimization objectives are deployed:

Objective	Mathematical Formulation	Real-world Feature
Total Length	$f_1(N) = \sum_{i=1}^{k-1} d(n_i, n_{i+1})$, $d(u, v)=\lVert u-v \rVert_2$	Path Euclidean distance
Accident Delays	$$f_2(N) = \sum_{i=1}^{k-1} \text{delay}(n_i, n_{i+1})$$	Expected accident or congestion
Elevation Cost	$$f_3(N) = \sum_{i=1}^{k-1} \max(0, h(n_i) - h(n_{i+1}))$$	Aggregated ascent
Travel Time	$$f_4(N) = \sum_{i=1}^{k-1} \frac{2 d(n_i, n_{i+1})}{v(n_i) + v(n_{i+1})}$$	Vehicle speed per grid cell
Smoothness	$$f_5(N) = \sum_{i=2}^{k-1} \arccos\left(\frac{(n_i-n_{i-1}) \cdot (n_{i+1}-n_i)}{\lVert n_i-n_{i-1}\rVert \lVert n_{i+1}-n_i\rVert}\right)$$	Route curvature/angle

These objectives encapsulate both canonical routing metrics (distance, time) and advanced real-world considerations (accidents, elevation, turning smoothness), facilitating realistic scenario construction (Weise et al., 2020).

2. Benchmark Structure and Dataset Construction

PaRoutes employs a grid-based or graph-based abstraction, where nodes encode spatial and contextual features: cell-specific speeds (e.g., highways vs. city roads), impassable obstacles (e.g., lakes), elevation profiles, and accident penalties.

• Configurability: Adjustable grid resolution, obstacle layouts, neighborhood connectivity, and backtracking mechanisms foster a diverse suite of benchmark instances, modulating computational hardness and scenario realism.
• Real-world Transfer: The methodology supports instantiation atop real-world networks (e.g., Berlin’s street graph built from OpenStreetMap), with nodes representing physical intersections and attributes drawn from both open data and statistical reports.

This flexibility in data modeling is intentional, enabling research on both algorithmic fundamentals (in simulation) and real-world transfer (Weise et al., 2020).

3. Evaluation Methodology: Pareto-Front and Algorithmic Assessment

A defining feature of PaRoutes is its exhaustive calculation of the Pareto front—the set of non-dominated solutions across all objectives—using methods such as depth-first search for moderate-scale grids. For larger instances with combinatorial explosion ($>10^{9}$ candidate paths), approximate or partial front computation is necessitated.

Algorithmic performance is assessed using the IGD+ (Inverted Generational Distance Plus) indicator, quantifying the proximity of an algorithm’s outputs to the reference front. The benchmark explicitly supports the evaluation of multi-objective algorithms such as:

• NSGA-II: Employs fast non-dominated sorting with crowding-distance for diversity
• NSGA-III: Utilizes reference points for high-dimensional objective maintenance
• DIR-enhanced NSGA-II (d-NSGA-II): Integrates reference vector-based diversity indicators

Experiments are configured with fixed population size (e.g., 212), crossover/mutation probabilities (0.8/0.2), and hundreds of generations across 31 runs, ensuring statistically robust performance reporting (Weise et al., 2020).

4. Extensions and Real-World Applicability

Beyond synthetic grids, the PaRoutes methodology is employed on city-scale networks using OpenStreetMap data, integrating authentic speed, elevation (via external APIs), and traffic incidence reports. This approach yields solution sets that align with real-world optimal routes (e.g., travel times matching those from established OSM routing engines). The ability to seamlessly transfer benchmarks from simulation to “in-the-wild” networks validates practical relevance.

A notable property is scalability: instances can be dialed from toy illustrative problems to networks comprising thousands of nodes and real traffic/sensor data, supporting both controlled experiments and application-grade assessment (Weise et al., 2020).

5. Algorithmic Innovations and Related Benchmarks

PaRoutes is differentiated by its many-objective scope and realism; the methodology is readily extensible to other planning domains, including but not limited to:

• Retrosynthetic Planning: Used as a benchmark in computer-aided synthesis planning for route enumeration and optimization (Shee et al., 22 May 2024, Prakash et al., 3 Apr 2025, Xuan-Vu et al., 29 Jul 2025).
• Multi-Step Reasoning and Tool-use: Benchmarks such as m&m’s (Ma et al., 17 Mar 2024), ProcBench (Fujisawa et al., 4 Oct 2024), and DABstep (Egg et al., 30 Jun 2025) share the “multi-step” spirit but apply it to tool-chaining, procedural instruction following, and data analysis, respectively—often inspired by PaRoutes’ rigorous, multi-step, multi-objective structure.
• Advanced RL Approaches: PaRoutes is noted as a suitable testbed for innovations in multi-step greedy RL (Tomar et al., 2019), especially as advanced BeLLMan operators (e.g., $\kappa$-greedy, surrogate reward shaping) can mitigate discount-factor bias and facilitate long-horizon improvement in model-free settings.

The benchmark’s modular design allows it to function as a bridge between theoretical algorithm analysis and direct assessment on real-world, high-dimensional planning problems.

6. Summary and Impact

PaRoutes Multi-Step Benchmark delivers a highly configurable, scalable, and realistic test suite for evaluating multi-objective route planning and multi-step algorithmic reasoning. Its objective-driven empirical design, rooted in explicit mathematical formulations, provides a reference for algorithmic development across combinatorial optimization, AI planning, and synthesis design. The transition from simulation to real-world networks, the support for many objectives (including advanced physical and stochastic features), and the integration with evolutionary and RL-based methods establish it as a standard in advanced route planning benchmarking (Weise et al., 2020).

This design—the harnessing of detailed multiple objectives, Pareto-front analysis, and real/simulated network transferability—positions PaRoutes as a foundational asset for evaluating and advancing state-of-the-art in long-horizon multi-step optimization.