Capacitated Arc Routing Problem (CARP)

• CARP is a complex arc-routing problem where equal-capacity vehicles must service all customer edges exactly once while minimizing total routing costs.
• Efficient algorithms employ approximation ratios, such as the improved 5/2 - Θ(1/√k) bound, and use decompositions and divide-and-conquer methods for large-scale instances.
• Recent advances integrate learning-based solvers and hyper-heuristics to handle stochastic, dynamic, and time-dependent variations in practical service applications.

Searching arXiv for recent and foundational CARP papers to ground the article. Capacitated Arc Routing Problem (CARP), introduced by Golden and Wong in 1981, is an arc-routing problem on a graph with edge costs and edge demands in which equal-capacity vehicles must satisfy all demands at minimum total routing cost. In the classical setting, vehicles are based at a depot, each required edge is serviced exactly once, route demand does not exceed vehicle capacity, and edges may be traversed multiple times when deadheading is needed. CARP is NP-hard and is central in operations research because it abstracts network service operations such as urban waste collection, road gritting, snow removal, mail delivery, meter reading, and related edge-service logistics (Zhao et al., 7 Aug 2025, Bevern et al., 2014, Guo et al., 2024, Tong et al., 2021).

1. Core formulation and modeling assumptions

A widely used metric formulation considers an undirected complete graph $G = (V \cup \{v_0\}, E)$ with depot $v_0$, edge costs $c: E \to \mathbb{R}_{\ge 0}$, a subset of customer edges $E^* \subseteq E$, demands $d(e) \ge 0$, vehicle capacity $k \in \mathbb{Z}_{\ge 1}$, and an unlimited fleet of identical vehicles stationed at the depot. The objective is to find a set of depot-anchored routes such that each customer is served exactly once, the total demand on each route does not exceed $k$, and the sum of route costs is minimized (Zhao et al., 7 Aug 2025).

The equal-demand special case sets $d(e)=1$ for all required edges. This case is structurally important in the approximation literature because it permits refined lower bounds and sharper ratios. In that setting, Jansen’s approach uses an iterated tour partitioning scheme: first compute a Rural Postman Problem (RPP) tour covering all required edges, then partition the tour into feasible fragments, reconnect each fragment to the depot, and choose the best partitioning strategy (Zhao et al., 7 Aug 2025).

CARP is closely related to node-routing models but differs in the serviced objects and in the role of shortest-path connections. The problem generalizes the Capacitated Vehicle Routing Problem (CVRP), the Traveling Salesman Problem (TSP), and other combinatorial optimization problems, while retaining the arc-routing feature that service is attached to edges rather than vertices (Zhao et al., 7 Aug 2025). This distinction becomes decisive in both approximation theory and in application-driven extensions.

2. Complexity and approximation theory

A basic complexity dichotomy appears in the equal-demand metric setting: for $k=1$ or $2$, CARP is polynomial-time solvable, while for $v_0$0 it is APX-hard (Zhao et al., 7 Aug 2025). For many years, the standard approximation benchmark for equal-demand CARP was Jansen’s ratio

$v_0$1

which remained unimproved for three decades according to Zhao and Xiao (Zhao et al., 7 Aug 2025).

A central theoretical development concerns the role of the triangle inequality. Earlier influential work claimed that CARP without triangle inequality is NP-hard to approximate within any constant factor. That claim was corrected by the result that any factor-$v_0$2 approximation for CARP with triangle inequality yields a factor-$v_0$3 approximation for general CARP. The reduction replaces edge costs by shortest-path costs while treating zero-demand and positive-demand edges differently: $v_0$4 The associated theorem states that CARP is polynomial-time self-reducible, mapping any instance $v_0$5 to an instance $v_0$6 such that $v_0$7 satisfies the triangle inequality and a factor-$v_0$8 approximate solution for $v_0$9 is polynomially transformable into a factor-$c: E \to \mathbb{R}_{\ge 0}$0 approximate solution for $c: E \to \mathbb{R}_{\ge 0}$1 (Bevern et al., 2014).

This result has two immediate consequences. First, the earlier inapproximability claim for non-metric CARP is formally refuted. Second, approximation guarantees developed for triangle-inequality CARP automatically transfer to the general case. In particular, the paper states the corollary that there is a polynomial-time factor $c: E \to \mathbb{R}_{\ge 0}$2 approximation for CARP even if the edge-cost function does not respect the triangle inequality, obtained by combining the reduction with the $c: E \to \mathbb{R}_{\ge 0}$3-factor algorithms of Jansen and Wøhlk (Bevern et al., 2014).

The equal-demand metric case has advanced further. Zhao and Xiao improve Jansen’s long-standing ratio by constructing two distinct RPP tours, applying the JITP partitioning algorithm to both, and returning the cheaper solution. One tour is a $c: E \to \mathbb{R}_{\ge 0}$4-approximate RPP tour; the other is a double-tree-style construction based on minimum-cost perfect matching and Kruskal-style component connection. Their analysis yields an approximation ratio of

$c: E \to \mathbb{R}_{\ge 0}$5

which they describe as the first improvement over Jansen’s bound (Zhao et al., 7 Aug 2025).

3. Exact, parameterized, and large-scale algorithmic paradigms

Beyond classical approximation algorithms, CARP admits several structural and parameterized approaches. For mixed and windy CARP, a key parameter is $c: E \to \mathbb{R}_{\ge 0}$6, the number of weakly connected components in the subgraph induced by positive-demand arcs. If the $c: E \to \mathbb{R}_{\ge 0}$7-vertex metric asymmetric TSP admits an $c: E \to \mathbb{R}_{\ge 0}$8-approximation, then the mixed and windy CARP is $c: E \to \mathbb{R}_{\ge 0}$9-approximable, and the paper further states a $E^* \subseteq E$0-approximation in $E^* \subseteq E$1 time. This yields constant-factor approximations for $E^* \subseteq E$2 and $E^* \subseteq E$3-approximations in general, while experiments indicate that practical solution quality depends strongly on $E^* \subseteq E$4 and that $E^* \subseteq E$5 in almost all benchmark instances (Bevern et al., 2015).

Large-scale CARP has motivated divide-and-conquer methods. A notable development is the route cutting off (RCO) decomposition operator, which cuts routes at weak links identified through a task-rank matrix based on shortest-path relationships between task endpoints. RCO was integrated into two state-of-the-art divide-and-conquer frameworks and shown to improve decomposition quality and to yield significantly better results especially when the problem size is very large and the time budget is very tight (Zhang et al., 2019).

Classical metaheuristics remain important. An ant colony optimization scheme specialized for CARP uses a giant-tour representation, a Split procedure to partition the tour into feasible trips, pheromone updates on transitions between required arcs, and a local search based on arc reinsertion and two-opt moves. On benchmark sets from DeArmon, Belenguer and Benavent, and Eglese, the method is reported as the first competitive ant colony scheme for CARP; it competes with CARPET and with the genetic algorithm of Lacomme et al., although it is more time consuming on large-scale instances (Philippe et al., 2022).

4. Learning-based and hyper-heuristic methods

Recent work has attempted to reduce the performance gap between learned solvers and advanced metaheuristics. A prominent example is a neural solver built around the direction-aware attention model (DaAM). The method decomposes each required edge into two directed arcs, embeds directionality through edge-to-arc transformation and multi-dimensional scaling features, makes one-stage decisions about both sequence and direction, and combines supervised pre-training with PPO-based reinforcement fine-tuning. A dynamic-programming path optimizer then reinserts depot returns to improve feasibility and cost. On synthetic Beijing-roadmap datasets, the reported decision quality is within $E^* \subseteq E$6 to $E^* \subseteq E$7 of MAENS across task sizes from 20 to 100 required edges, while maintaining superior efficiency (Guo et al., 2024).

Hyper-heuristic learning has also been developed for uncertain CARP. In UCARP, a collaborative multi-vehicle Solution Construction Procedure (SCP) handles route failure and refill events through split service, dynamic handover of partially served tasks, and opportunistic service during refill trips. On top of this SCP, a Genetic Programming Hyper-Heuristic evolves routing policies from features such as cost from here, remaining capacity, estimated demand, and route-level information. The paper reports that the collaborative GPHH framework significantly outperforms EDASLS and earlier non-collaborative GPHH, especially on instances with larger numbers of tasks and vehicles (MacLachlan et al., 2019).

Reinforcement learning has also been hybridized with constructive heuristics and local search in hierarchical directed arc routing. In the Hierarchical Directed Capacitated Arc Routing Problem (HDCARP), required arcs are partitioned into priority classes, and the objective becomes lexicographic minimization of maximum completion times across classes. The HRDA framework combines heuristic construction, local search, and PPO-guided operator selection; the paper states that setting the number of priority classes to $E^* \subseteq E$8 recovers standard CARP as a special case (Nguyen et al., 1 Jan 2025).

5. Uncertainty, dynamics, and time dependence

Several important CARP variants arise when edge demands, traversal costs, or service costs are not static. In UCARP, task demands $E^* \subseteq E$9 and traversal costs $d(e) \ge 0$0 are random variables revealed only when service or traversal occurs. A major operational event is route failure, when realized demand exceeds remaining vehicle capacity. The collaborative SCP proposed for UCARP changes the standard recourse model by allowing the remaining part of a failed task to return to the unassigned pool so that any vehicle may complete it, and by allowing refill trips to perform opportunistic partial service on traversed required tasks (MacLachlan et al., 2019).

Stochastic CARP replaces deterministic arc demands with random variables and evaluates routes by expected cost, trip-failure probabilities, and robustness criteria. In one formulation, if trip $d(e) \ge 0$1 requires an extra depot visit with probability $d(e) \ge 0$2 and incurs penalty $d(e) \ge 0$3, then the expected total cost is

$d(e) \ge 0$4

with standard deviation

$d(e) \ge 0$5

This analytic evaluation is embedded in a stochastic NSGA-II framework that jointly considers cost, duration of the longest trip, and standard deviations; the reported experiments show that robust solutions can be obtained in rather short computation times (Gérard et al., 2022).

Dynamic CARP addresses changes during execution rather than uncertainty known only statistically. A formal DCARP model introduces outside vehicles, remaining capacities, and route costs based on current vehicle locations. The generalised optimisation framework with virtual tasks transforms a DCARP instance into a static-like CARP instance by creating depot-to-vehicle virtual tasks whose demands encode used capacity. This permits reuse of existing static CARP metaheuristics through either a restart strategy or a sequence-transfer strategy, and the reported empirical study shows that such reuse significantly improves over prior dynamic optimisation algorithms (Tong et al., 2021).

Time dependence can attach either to network travel times or to service costs. TDCARP places time-dependent travel and service speed functions directly on the network, so the quickest path between locations may change over time. The model uses preprocessed piecewise-linear quickest-path functions, bucket-based query structures, a branch-and-price algorithm, and a Hybrid Genetic Search with indirect encoding and a dynamic-programming decoder. The branch-and-price method finds optimal solutions for instances counting up to 75 services, while the heuristic achieves high-quality solutions up to 375 services (Vidal et al., 2020). In CARPTDSC, by contrast, the time dependence lies in service costs. MAENS-GN separates routing from vehicle departure-time optimization and uses Golden Section Search or Negatively Correlated Search depending on the shape of the route-cost function, while KGMA-GN introduces knowledge-guided initialization and local search filters that reduce unnecessary evaluations and substantially improve runtime (Li et al., 2024, Li et al., 29 Jul 2025).

6. Application-driven extensions and open directions

Application-specific extensions of CARP often modify the service model itself. The splittable capacitated arc routing problem (SCARP), motivated by agricultural spraying, allows the demand on a single edge to be split among multiple vehicles or trips. Mixed-integer formulations, lazy constraints, symmetry elimination, and a heuristic repair method are used to solve SCARP, and the reported experiments show that SCARP can produce cheaper solutions than classical CARP on some benchmark instances because the no-splitting restriction is removed (Wan et al., 2023).

Multi-depot and real-time operational settings have also been studied. Urban water sprinkler routing is formulated as a multi-depot mixed capacitated arc routing problem with real-time demands, where fixed daily plans are adjusted online when new demand links are detected through IoT-supported FRD monitoring. An improved adaptive large neighborhood search with tabu-list, perturbation mechanism, and an acceleration mechanism is reported to generate solutions that are superior or at least comparable to traditional ALNS while requiring significantly lower computation time (Yang et al., 2024).

The winter road maintenance literature reveals a different axis of generalization: graph structure. On trees, and more generally on bounded tree-width graphs, a natural arc-routing model with route-length, capacity, priority, and fairness constraints admits dynamic-programming algorithms. The same work also shows sharp hardness boundaries: the maintaining plan routing problem is strongly NP-complete even on stars, remains NP-complete on planar graphs under restricted settings, and admits sub-exponential algorithms on planar graphs with a small number of maintaining cars through tree-width arguments (Fink et al., 2020).

Two open directions are particularly explicit in the current literature. First, the reduction that transfers approximation guarantees from metric to general CARP extends to directed CARP but does not directly extend to mixed graphs, which remains an open problem (Bevern et al., 2014). Second, in equal-demand metric CARP, the new $d(e) \ge 0$6 ratio raises the question of whether the long-standing $d(e) \ge 0$7 barrier can be lowered further, possibly by adapting more recent CVRP techniques or by developing sharper primal-dual frameworks (Zhao et al., 7 Aug 2025).