Road Network-Aware Cluster-Then-Route Heuristic

• The paper introduces a heuristic that decomposes routing problems by clustering directly on road graphs, accounting for directed edges and real-world infrastructure.
• It employs graph-based similarity measures and adaptive clustering methods to define subproblems for trajectories, service edges, and collective travel.
• Empirical results highlight trade-offs in operational metrics such as makespan, emissions, and vehicle count across applications like winter maintenance and electric-vehicle routing.

Searching arXiv for recent and foundational papers relevant to road network-aware cluster-then-route heuristics. A road network-aware cluster-then-route heuristic is a decomposition strategy in which clustering and routing are both defined on a road graph rather than in unconstrained Euclidean space. In the literature, the clustered objects vary by application: trajectories and road segments in network-constrained movement analysis, required service edges in winter maintenance, customers in robust electric-vehicle routing, and users or stops in collective travel. The common structure is to partition the problem into graph-respecting subproblems and then compute routes, tours, or shared paths within those partitions, often with feedback from downstream routing quality to upstream clustering decisions (Mahrsi et al., 2012, Xie et al., 27 Feb 2026, Pustilnik et al., 2024, Reza et al., 2017, Fink et al., 2020).

1. Conceptual and graph-theoretic basis

The literature suggests that the term denotes a heuristic family rather than a single canonical algorithm. Its defining property is not merely “cluster first, route second,” but the use of road-network structure in the clustering stage itself. Depending on the model, the road network is represented either as a directed graph $G=(V,E)$, where directionality, segment length, speed limits, lane counts, and adjacency are explicit, or as an undirected graph when roads must be traversed in both directions by the same vehicle.

In trajectory-oriented formulations, a network-constrained trajectory is a sequence of connected directed edges. In service-routing formulations, required edges or customers are assigned to depots, vehicles, or clusters subject to capacity, connectivity, or robustness constraints. In collective travel, shortest-path costs $SPC(u,v)$ on the road graph define both the access/egress cost of users and the vehicle’s shared route. In winter maintenance, each vehicle plan is a connected subgraph rooted at a depot, with maintenance priorities and method compatibility embedded in the feasible set.

The significance of this graph-first viewpoint is that similarity, contiguity, and route cost are all defined with respect to actual infrastructure. This excludes many false positives that arise from geometric proximity alone, such as trajectories on parallel but disjoint roads or customer groupings that are spatially close yet separated by long network detours.

2. Clustering mechanisms that respect the road network

A foundational network-based clustering model represents each trajectory as a bag of directed road segments and weights segments with a TF-IDF-style scheme adapted to road networks. For a segment $e$ in trajectory $T$, the spatial segment frequency is

$$\mathrm{ssf}_{e,T}= \frac{n_{e,T}\cdot \mathrm{length}(e)} {\sum_{e'\in T} n_{e',T}\cdot \mathrm{length}(e')},$$

and the inverse trajectory frequency is

$$\mathrm{itf}_e=\log \frac{|\mathcal{T}|}{|\{T_i: e\in T_i\}|}.$$

The combined weight is $$\omega_{e,T}=\mathrm{ssf}_{e,T}\cdot \mathrm{itf}_e$$. Trajectories are then compared by cosine similarity of their segment-weight vectors. This makes long and discriminative shared segments matter more than ubiquitous arterial segments, and similarity is zero when no directed segment is shared (Mahrsi et al., 2012).

This graph-induced similarity is used to build a weighted similarity graph whose vertices are trajectories and whose edges exist only when similarity is positive. Clustering is then performed by modularity-based community detection with recursive significance testing, yielding a hierarchy of trajectory clusters. The same paper dualizes the construction for segment clustering: segments are represented as bags of trajectories, segment-segment similarity is computed in trajectory space, and two segment graphs are considered. The “loose” graph links segments that co-occur in at least one trajectory; the “strict” graph adds an adjacency requirement in the road network. The former captures functional co-usage, whereas the latter captures contiguous co-used subgraphs.

A different clustering mechanism appears in robust electric-vehicle routing. There, clustering is based on a generalized cost matrix

$$Z = T_\mu + \Phi(P_T)T_\sigma + \frac{1}{R}\left(E_\mu + \Phi(P_E)E_\sigma\right),$$

which combines expected travel time, travel-time variability, expected energy consumption, and energy variability. Customer groups $G_1,\dots,G_M$ are formed subject to constraints such as $|G_i|\le N_{\max}$, $SPC(u,v)$0, one depot per group, and full coverage of depots and customers. The clustering objective is expressed through matrix functionals such as the Frobenius norm or spectral quantities of the group submatrices $SPC(u,v)$1 (Pustilnik et al., 2024).

Collective travel introduces a third pattern: clustering emerges through path-coherence. When many shortest paths from nearby origins to distant destinations share long common subpaths, group search on the road network reveals natural convergence nodes that act as effective start-stops, end-stops, or intermediate shared-route attachment points. Here clustering is not an external preprocessing step but an implicit consequence of shortest-path structure on the graph.

3. Routing within clusters and route-informed clustering

In bi-level winter road maintenance, the upper level assigns each treatment-required edge $SPC(u,v)$2 to a depot $SPC(u,v)$3, producing depot-based clusters $SPC(u,v)$4. The lower level then solves a multi-objective VRP or capacitated arc-routing problem within each cluster. The formulation uses an assignment vector $SPC(u,v)$5 and lower-level feasible routes $SPC(u,v)$6, with upper-level objective $SPC(u,v)$7, where $SPC(u,v)$8 is makespan and $SPC(u,v)$9 total emissions. The clustering policy is initialized by KDTree nearest-depot assignment and refined with PPO, using normalized per-segment features including coordinates, length, speed limit, lane count, and distances to depots. Routing is then performed by a constraint-aware nearest-neighbor heuristic with feasibility checks for salt capacity, route distance, 120-minute duration, directionality, and adjacency (Xie et al., 27 Feb 2026).

Robust electric-vehicle routing follows a more classical cluster-first–route-second structure. After constrained clustering in the matrix $e$0, charging stations are added to each group by nearest-in-$e$1 cost, and each cluster is solved as a Robust ECTSP. The route-level model explicitly enforces SoC chance constraints, charging decisions, cumulative load, and a robust objective of the form $e$2. In this setting, robustness is only approximated in clustering and enforced exactly in the route stage (Pustilnik et al., 2024).

Collective travel shows that cluster-then-route can also be interleaved. Given fixed start and end stops, a polynomial-time heuristic incrementally builds a vehicle route on the road graph while repeatedly reassigning each passenger’s boarding and alighting points to the best nodes on the current route prefix. The exact version uses a state space $e$3, where $e$4 and $e$5 records which query endpoints have already been served; the heuristic replaces exponential subset search by Dijkstra-style graph expansion with greedy reassignment of users to stops (Reza et al., 2017).

Large-scale undirected winter maintenance uses a different decomposition. Each vehicle plan is a connected tree rooted at its depot. Local improvement is performed by exchanging branches of these trees between pairs of vehicles that share vertices. The exchange is formulated as a bin-packing problem in which vehicles are bins of capacity $e$6 and rooted branches are items. The route stage is then simply the traversal of the resulting connected plan tree, which is feasible because every road must be traversed in both directions by the same car (Fink et al., 2020).

4. Representative formulations across application domains

Paper	Clustered entity	Route or subproblem stage
(Mahrsi et al., 2012)	Trajectories or directed segments	Community-detected corridor or subgraph structure
(Xie et al., 27 Feb 2026)	Required edges assigned to depots	Multi-objective VRP / capacitated arc routing within each depot cluster
(Pustilnik et al., 2024)	Customers grouped by robust network cost matrix $e$7	Robust ECTSP with charging and SoC chance constraints
(Reza et al., 2017)	Users attached to shared stops or route segments	Shared vehicle route with user access and egress costs
(Fink et al., 2020)	Connected road branches assigned to vehicles	Tree-based maintenance tour under length and priority constraints

These formulations differ in what is clustered, but they share a common decomposition logic. In movement mining, clusters identify corridor-like similarity structure before any downstream routing interpretation. In VRP and arc-routing settings, clustering defines subproblems that are smaller and operationally meaningful. In collective travel, the cluster object is not a territory or depot region but a set of users attached to common stops along a shared path.

The contrast between strict and loose segment clustering, hard partitioning by depot, and branch exchange between overlapping vehicle plans also shows that road network-aware clustering does not imply a single topological notion of “region.” In some formulations, clusters are contiguous subgraphs; in others, they are functional co-usage sets or depot territories.

5. Empirical behavior and comparative findings

Synthetic trajectory experiments on the Oldenburg map used a network with 6105 nodes and 14070 directed edges, together with 1000 trajectories traversing 7890 unique segments. For segment clustering, the loose similarity graph had 572,903 edges and produced a hierarchy with 7 levels, from 16 top-level clusters to 1,222 bottom-level clusters. The strict graph had 8,674 edges and produced a 3-level hierarchy, from 92 top-level clusters to 216 bottom-level clusters. The qualitative interpretation was that loose clusters were broad and only became localized deeper in the hierarchy, whereas strict clusters were already localized and spatially contiguous at the top level. Hierarchical agglomerative clustering with average linkage was reported to produce poor quality, disproportionate clusters in this setting (Mahrsi et al., 2012).

In winter road maintenance on UK strategic networks, the post-processed graph had 37,007 nodes, 71,505 directed edges, and 1515 required directed edges, with three depots in a $e$8 area. The baseline KDTree+NN method achieved makespan $e$9 minutes and emissions $T$0 kg CO$T$1 using 20 vehicles, whereas KDTree-PPO+NN with 10 PPO iterations achieved makespan $T$2 minutes and emissions $T$3 kg CO$T$4 using 28 vehicles. Because both methods used the same routing heuristic and constraints, the reported performance difference was attributed to clustering quality rather than a change in the routing stage (Xie et al., 27 Feb 2026).

For robust energy-capacitated routing, the clustering heuristic plus RECTSP solver was within roughly 2–5% of the MIP optimum on small random instances. On large deterministic benchmark instances, the clustering-based method was reported to be 3–4 times faster than the fastest methods in the literature for the largest cases, with gaps that decreased as problem size grew, reaching about $T$5 for 916 nodes and $T$6 for 1001 nodes (Pustilnik et al., 2024).

For collective travel on the San Francisco road network with 174,956 nodes and 223,001 bidirectional edges, the fast end-stop algorithm was reported to be up to about 1000 times faster than the baseline in practical settings. The exact intermediate-stop algorithm became practically infeasible beyond roughly 5–6 users, whereas the heuristic handled up to 50 queries in under a second with average relative error typically below 5% (Reza et al., 2017).

For large-scale undirected winter maintenance with 1719 vertices, 2280 edges, 34 depots, and a per-car limit $T$7 km, the heuristic produced a solution with 93 cars and 361 km of deadhead in 2 minutes 36 seconds using 82 MB of memory. Connectivity-based lower bounds implied about 150 km of unavoidable deadhead and a lower bound of 88 cars, so the computed solution exceeded the lower bound by five vehicles (Fink et al., 2020).

6. Limitations, trade-offs, and recurrent misconceptions

A recurrent misconception is that cluster-then-route is primarily a geometric preprocessing step. The literature contradicts this. Trajectory clustering based on shared directed segments, depot assignment using graph-aware edge features, clustering by robust travel-time-and-energy matrices, and shortest-path-based stop assignment all depend on network structure rather than straight-line proximity (Mahrsi et al., 2012, Xie et al., 27 Feb 2026, Pustilnik et al., 2024, Reza et al., 2017).

A second misconception is that better clustering should improve every operational metric simultaneously. The winter-maintenance bi-level study shows otherwise: improved makespan and reduced emissions were accompanied by a larger vehicle count. This indicates that route compactness, workload balance, emissions, and fleet usage need not move in the same direction under a change in cluster boundaries (Xie et al., 27 Feb 2026).

Several methodological limitations are explicit. Similarity-graph construction for trajectory clustering is worst-case $T$8, with modularity optimization bounded by $T$9 in theory. Extremely diverse trajectories yield sparse similarity graphs, and noise or rarely traveled segments can degrade cluster quality (Mahrsi et al., 2012). Exact user-to-stop routing for collective travel has exponential dependence on the number of queries through the $$\mathrm{ssf}_{e,T}= \frac{n_{e,T}\cdot \mathrm{length}(e)} {\sum_{e'\in T} n_{e',T}\cdot \mathrm{length}(e')},$$0-scale state space, which confines exact methods to very small groups (Reza et al., 2017). In robust electric-vehicle routing, energy feasibility is not enforced during clustering itself; it is only represented indirectly through the matrix $$\mathrm{ssf}_{e,T}= \frac{n_{e,T}\cdot \mathrm{length}(e)} {\sum_{e'\in T} n_{e',T}\cdot \mathrm{length}(e')},$$1 and then enforced in the route stage (Pustilnik et al., 2024). In the undirected winter-maintenance bin-packing method, one-way streets are excluded, all roads are assumed to require traversal in both directions by the same car, and travel times are deterministic (Fink et al., 2020).

The comparison between loose and strict segment clustering also clarifies a topological issue. Loose clusters capture functional co-usage and need not be contiguous, whereas strict clusters enforce adjacency and therefore align more naturally with territorial partitioning or subgraph-restricted routing (Mahrsi et al., 2012). This suggests that the appropriate cluster definition depends on whether the target application values co-usage similarity, contiguity, or depot accountability.

Taken together, these studies support a consistent design principle: clustering should be evaluated through downstream route quality, feasibility, and operational structure, not through geometric compactness alone. That principle is the central distinguishing feature of road network-aware cluster-then-route heuristics.