Resource-Constrained Shortest Path

• RCSPP is a combinatorial optimization problem that seeks the minimum-cost, resource-feasible route in a directed graph.
• It relies on dynamic programming labeling, dominance rules, and bidirectional search to ensure efficient path evaluation under resource constraints.
• Advanced methods such as FPTAS, parallel computing, and machine learning enhance its scalability for practical applications in routing and scheduling.

The Resource-Constrained Shortest Path Problem (RCSPP) is a fundamental combinatorial optimization problem defined on a directed graph $G=(V,E)$, where each arc carries a non-negative cost and one or more resource consumption metrics. The objective is to find an elementary (loop-free) $s$–$d$ path of minimum total cost such that the consumption of each resource does not exceed a prescribed budget. RCSPP arises in applications where optimal paths must respect supplementary constraints, such as time windows, vehicle capacities, or energy limits in routing, scheduling, and transportation networks. The general problem is NP-hard and admits numerous exact and approximate algorithmic approaches, reflecting its centrality in discrete optimization and combinatorial research (Salani et al., 2023).

1. Formal Mathematical Framework

Let $G=(V,E)$ be a directed graph with node set $V$ and arc set $E$. Each arc $(i,j)\in E$ is assigned:

• A cost $c_{ij}\geq0$.
• $k$ non-negative resource consumption functions $r_j:E\rightarrow\mathbb{R}_+$ for $j=1,\dots,k$.

A path $P$ from source $s\in V$ to sink $d\in V$ comprises a sequence of unique nodes $\left(i_0=s, i_1, \dots, i_p=d\right)$, where $(i_{\ell-1},i_\ell)\in E$ and $i_\ell\neq i_m$ for $\ell\neq m$ (elementarity). The total cost and resource consumptions are: $$C(P) = \sum_{(i,j)\in P} c_{ij}, \qquad R_j(P) = \sum_{(i,j)\in P} r_j(i,j),\quad j=1,\dots,k.$$ The RCSPP seeks

$$\min_{P \text{ is a simple } s\to d \text{ path}} C(P) \quad \text{subject to } R_j(P)\leq B_j \quad \forall j=1,\dots,k,$$

where $B_j$ is the resource bound for resource $j$ (Salani et al., 2023).

In multi-dimensional or nonnegative cost-resource settings, extensions to negative arc costs or arbitrary resource update functions are feasible if negative cycles are absent (Ahmadi et al., 14 Mar 2025).

2. Algorithmic Foundations and Labeling Paradigms

Dynamic programming labeling is the central paradigm for exact RCSPP algorithms. Every partial path is represented as a label, encoding:

• The current node $i$,
• Accumulated cost $c_i$,
• Resource consumption vector $R = (R_1,\dots,R_k)$,
• A binary mask $S$ for elementarity or relaxations.

Label extension propagates a label $l_i$ at node $i$ to neighbor $j$, updating costs and resources: $$c_j' = c_i + c_{ij};\quad R_j' = R + r_{ij};\quad S' = S \land B_j;\quad S'_j := 1$$ for relaxations or elementarity (Salani et al., 2023).

Dominance rules prune labels at each node: label $l$ dominates $l'$ iff $c(l)\leq c(l')$, $R_j(l)\leq R_j(l')$ for all $j$, $S(l)\subseteq S(l')$, with at least one strict inequality—dominated labels are discarded.

Bidirectional algorithms (labeling, A*) join forward and backward label pools at a "halfway" resource threshold for improved efficiency, ensuring feasibility and non-cycling at the meet point (Salani et al., 2023, Ahmadi et al., 18 Dec 2024).

Specialized frameworks integrate labeling with heap-ordered search (A*), exploiting admissible and consistent heuristics from backward shortest-path computations. When costs/resources can be negative, heuristics use Bellman-Ford distances, provided negative cycles are excluded from feasible paths (Ahmadi et al., 14 Mar 2025).

3. Advanced Algorithmic Schemes and Relaxations

Labeling algorithms are extended by state-space relaxations—DSSR, DSSRC, NG-path, NGC, and hybrids—which systematically relax elementarity and resource coupling. These relaxations trade accuracy and speed, enabling the solution of instances ranging from small cyclic graphs to large-scale, sparse, or dense networks (Salani et al., 2023).

Bucket-based parallel labeling techniques exploit resource-discretization to decompose the algorithm into independent bucket processing units, yielding multi-threaded speedups. Bidirectional search with dynamic midpoint and vectorized dominance tests using SIMD instructions further accelerate computations, reaching up to 200× speed-ups on benchmark VRPTW pricing instances (Petersen et al., 3 Nov 2025).

Pulse-style approaches combine best-first/heap expansion with aggressive state-space pruning and time-bucketing, drastically reducing the number of active labels at each node. APULSE exemplifies this approach on large dense terrain graphs, maintaining near-optimality with superior scalability (Soares et al., 23 Nov 2025).

Hierarchical Structure embedding transliterates $G$ into a $k$-layer, acyclic topology supporting $O(m\log n)$ Lagrangian-searches and shortcutting, reducing overall complexity and runtime by an order of magnitude compared to repeated Dijkstra/LARAC calls, with minimal optimality loss in practice (Erzin et al., 2022).

4. Polyhedral, Monoidal, and Decision Diagram Approaches

Polyhedral models reformulate RCSPP as integer-flow problems on extended networks: $$\min \sum_{a\in A} \ell(a) y_a \quad\text{s.t.}\quad\text{(flow conservation),}\quad \sum_{a\in A} G_{i,a} y_a \le d_i \text{ for all resources,}$$ with $y_a\in\{0,1\}$ indicating arc usage (Lozano et al., 2022).

The monoidal generalization encompasses nonlinear and stochastic variants, defining resources in a lattice-ordered monoid $M$ with order-compatible aggregation. Feasibility, cost, and dominance are encoded as monotone operations on $M$, and polynomial-time procedures (Bellman, Dijkstra) propagate resource bounds, enabling tight lower bound pruning and label dominance testing (Parmentier, 2015, Kruber et al., 2018).

Decision diagram constructions encode the RCSPP in layered acyclic graphs, supporting both MILP/polyhedral and dynamic-programming formulations. Side constraints (interdiction, robust optimization) are incorporated by indicator functions on arcs or state variables, strengthening relaxations and facilitating L-shaped state-augmentation and branch-and-cut (Lozano et al., 2022).

5. Approximation Schemes and Complexity

RCSPP is NP-hard even for single-resource, integral cost/delay graphs. Fully polynomial time approximation schemes (FPTAS) have been developed for broad graph classes:

• For general directed graphs, $(1+\varepsilon)$-approximation in $O(mn(1/\varepsilon+\log\log\log n))$ time (Holzmüller, 2017).
• For planar, undirected integer-delay graphs, $O(mn/\varepsilon)$.
• For DAGs, $O(mn/\varepsilon)$ via linear search and DP.

FPTAS methods round resource increments, scale costs, and propagate Pareto-optimal label sets. Battery-expanded discretization achieves similar guarantees by mapping states in the original graph to expanded nodes stratified by resource levels, tolerating small rounding errors tied to the discretization (Schwan et al., 2016).

Lagrangian relaxations with bucketed Dijkstra or monotonic search yield $O(m\log n)$ runtime for large graphs, with empirical errors below 1–3% (Erzin et al., 2022).

6. Parallelization, Machine Learning, and Practical Implementations

Kernels for RCSPP run efficiently on many-core CPUs and GPUs using wavefront simulation, label-bucket parallelism, and vectorized dominance checks (Matic, 2015, Petersen et al., 3 Nov 2025). Active vertices and edges are tracked temporally in percolation metrics, minimizing global workload and enabling near-linear scaling for lattice graphs.

Data-driven approaches, including supervised machine learning on arc-level features, are effective for RCSPP pricing in column generation frameworks. Learned classifiers reduce the active edge set, yielding up to 40% reductions in runtime without loss of solution quality on public-transit crew scheduling and VRPTW benchmarks. Hybrid schemes combine arc classification with reduced-cost filtering for further acceleration (Morabit et al., 2022).

Open-source software frameworks, such as PathWise, encapsulate these methodologies with modular interfaces for algorithm, resource, and label management. Extensibility to custom resource types, dominance criteria, and high-level parallelism is promoted, alongside support for profiling and data-driven heuristic selection (Salani et al., 2023).

7. Benchmark Applications, Performance, and Future Directions

RCSPP serves as the principal pricing subproblem in column and branch-and-price schemes for VRP, scheduling, and supply-chain routing. Empirical studies on road-network (DIMACS), prize-collecting (SPPRCLIB), and virtual network embedding demonstrate orders-of-magnitude performance improvements from algorithmic, relaxation, and parallelization advances (Salani et al., 2023, Petersen et al., 3 Nov 2025, Soares et al., 23 Nov 2025, Chemodanov et al., 2018).

Quadratic speed-up methods (Neighborhoods Method) apply to large graphs, provably halving the worst-case exponential complexity of naive search. Integrations of machine learning, multi-resource extensions, and hybrid heuristic-guided algorithms (A*/Pulse) are current and future directions for scaling RCSPP to dynamic, data-driven, and stochastic network environments.

The theoretical and practical research on RCSPP confirms its central role in combinatorial optimization, with continued advances in computational tractability, methodological diversity, and real-world applicability (Salani et al., 2023, Ahmadi et al., 18 Dec 2024, Soares et al., 23 Nov 2025, Lozano et al., 2022, Morabit et al., 2022, Kruber et al., 2018).