Routing, Modulation & Spectrum Assignment

• Routing, Modulation and Spectrum Assignment (RMSA) is a problem in elastic optical networks that jointly selects routes, modulation formats, and contiguous spectrum blocks under strict continuity, contiguity, and non-overlap constraints.
• It balances spectral efficiency and transmission reach by choosing optimal modulation formats, ensuring that higher-order modulation reduces slot usage while considering physical-layer limitations.
• Approaches range from exact ILP formulations and graph-theoretic models to advanced learning methods like deep reinforcement learning, diffusion models, and graph transformers aimed at minimizing blocking probability and enhancing network performance.

Routing, Modulation and Spectrum Assignment (RMSA) is the joint resource-allocation problem in elastic optical networks (EONs) of selecting a route, a modulation format, and a contiguous spectrum block for each lightpath while respecting spectrum continuity, spectrum contiguity, non-overlap, and physical-layer reach or QoT constraints. In dynamic settings, requests arrive over time and must be accepted or blocked immediately; in static settings, a set of demands is optimized jointly. Across the literature, RMSA is treated as NP-hard, and the main objectives include minimizing blocking probability, maximizing supported traffic load, minimizing spectrum usage, and, in some formulations, minimizing regenerators or satisfying additional QoE or latency constraints (Chen et al., 2019, Doherty et al., 3 May 2026, David et al., 2023).

1. Core problem statement and resource model

In an EON, each fiber link carries an optical spectrum divided into fine-grained frequency slots or frequency-slot units rather than fixed 50 GHz wavelengths. A demand specifies a source–destination pair and a bitrate, and RMSA must determine three coupled decisions: a path, a modulation format, and a block of slots on every traversed link. The same frequency interval must be reserved on every link of the chosen path, the slots reserved for a lightpath must be consecutive, and the chosen slots must not overlap with slots already used by other lightpaths on any common link (Doherty et al., 3 May 2026, Chen et al., 2019).

The modulation decision couples spectral efficiency and reach. Higher-order modulation uses fewer slots for a given bitrate but has a shorter feasible distance; lower-order modulation uses more slots and reaches farther. In the distance-dependent model used in large-scale dynamic RMSA benchmarking, a modulation $m$ is feasible on path $p$ only if $d(p) \le R_m$, where $d(p)$ is path length and $R_m$ is maximum reach (Doherty et al., 3 May 2026). In DeepRMSA, the required number of slots is written as

$$n = \left\lceil \frac{b}{m \cdot C_{grid}^{BPSK}} \right\rceil,$$

with $m \in \{1,2,3,4\}$ encoding BPSK, QPSK, 8-QAM, and 16-QAM (Chen et al., 2019). In dynamic multi-modulation allocation, the slot count for a segment with bitrate $b$ and modulation $M$ is

$$N_{\text{slots}}(b,M)=\left\lceil \frac{b}{C_M} \right\rceil,$$

with representative capacities and reaches reported for BPSK, QPSK, 8QAM, 16QAM, 32QAM, and 64QAM (Costa et al., 2020).

The standard EON constraints are usually stated as routing, continuity, contiguity, and non-overlap; some works also include guard bands, regeneration, or exact QoT conditions. In the dynamic RMSA environment used by Optical RL-Gym, the action for each request is a complete RMSA decision: choose one route, choose a modulation format allowed for the route length, and choose a contiguous block of spectrum slots satisfying continuity, contiguity, and non-overlap (Li et al., 2024). In static full-RMSA formulations with regeneration, a demand may be split into transparent segments, each with its own modulation, while continuity is enforced within each segment and contiguity is enforced through starting-slot variables and pairwise non-overlap constraints (David et al., 2023).

2. Mathematical formulations, objectives, and performance metrics

Two broad formulations dominate the literature. In static planning, all demands are known in advance and the aim is often to minimize maximum spectrum usage or maximize admitted demands subject to full continuity and contiguity constraints. In dynamic provisioning, demands arrive and depart over time; each arrival must be accepted or blocked immediately, and the dominant objective is long-term blocking minimization (Chen et al., 2019, David et al., 2023).

Dynamic RMSA is commonly written as a sequential decision process. In OpticGAI’s RMSA formulation, at each request arrival time $p$0, the agent observes a state $p$1, chooses an action $p$2, receives an immediate reward $p$3, and seeks to maximize

$p$4

subject to continuity, contiguity, non-overlap, and reach feasibility (Li et al., 2024). DeepRMSA uses the immediate reward

$p$5

and the discounted objective

$p$6

to minimize long-term blocking probability (Chen et al., 2019). In the transformer-based large-scale dynamic RMSA setting, the reward is typically $p$7 for successful allocation and $p$8 for blocking, again aligning cumulative reward maximization with blocking minimization (Doherty et al., 3 May 2026).

The dominant service-level metric is blocking. Dynamic RMSA studies define blocking probability or service blocking probability as the fraction of blocked requests among total requests (Doherty et al., 3 May 2026, Chen et al., 2019). The transformer study further uses “supported traffic load at target blocking,” usually at $p$9, as the largest offered load at which the algorithm maintains $d(p) \le R_m$0 (Doherty et al., 3 May 2026). Other metrics include episode reward, episode blocking ratio, convergence behavior, link and spectrum utilization, path-length distributions, link usage, and fragmentation patterns across spectrum (Li et al., 2024, Doherty et al., 3 May 2026).

Static formulations may use lexicographic objectives. The exact full-RMSA ILP introduced for EONs uses a primary objective to maximize the number of admitted demands, a secondary objective to minimize the total number of regenerators, and a tertiary objective to minimize total FS usage, implemented in Gurobi’s hierarchical multi-objective mode rather than by large scalar weights (David et al., 2023). In impairment-aware dynamic RMLSA, ABACUS uses

$d(p) \le R_m$1

so that the number of occupied FSUs is minimized while lower-index FSUs are favored to improve clustering and reduce fragmentation (Kiran et al., 2024).

3. Exact methods, graph-theoretic views, and heuristic algorithms

Classical RMSA and closely related RSA formulations have long been associated with integer programming and heuristics. DeepRMSA explicitly states that static RMSA is often formulated as an ILP and that such ILPs are NP-hard and intractable at scale (Chen et al., 2019). The exact full-RMSA paper reports what it describes as the first ILP formulation that jointly considers routing, modulation, spectrum, and regeneration while obeying slot continuity and contiguity end-to-end and simultaneously optimizing admission, regenerators, and FS usage (David et al., 2023). Its revised formulation pre-enumerates feasible per-demand solutions and then uses big-$d(p) \le R_m$2 ordering constraints on starting-slot variables to enforce non-overlap efficiently (David et al., 2023).

Graph-theoretic interpretations remain important. In the RSA-oriented theoretical analysis of spectrum usage in EONs, the routed demands induce a conflict graph whose vertices are requests and whose edges indicate lightpath intersections. The optimal spectrum usage is bounded in terms of the chromatic number $d(p) \le R_m$3, with

$d(p) \le R_m$4

and, under bounded demand widths $d(p) \le R_m$5,

$d(p) \le R_m$6

That work also links the chromatic number to the intersecting probability of routed paths and shows how topology, traffic distribution, and routing scheme enter a quadratic program through conflict coefficients (Wu et al., 2018). This suggests that many “spectrum assignment” outcomes are determined upstream by routing structure and traffic geometry rather than by slot-packing rules alone.

Heuristics remain central because of runtime constraints. KSP-FF and FF-KSP remain strong baselines in dynamic RMSA benchmarking (Doherty et al., 3 May 2026). In fixed-grid routing and wavelength assignment with lightpath reuse, careful benchmarking shows that ordering candidate paths by number of hops rather than total length gives 6% increased throughput for KSP-FF, and the best heuristic remains highly competitive with learned policies (Doherty et al., 20 Feb 2025). In flex-grid RSA based on the availability of consecutive sub-channels, joint routing-and-spectrum heuristics prune paths early using per-path bitmaps and contiguity checks; Type II and Type III algorithms reduce blocking and improve spectrum utilization relative to shortest-path and $d(p) \le R_m$7-shortest-path baselines precisely because they route on the basis of available contiguous slots rather than only on hops or distance (Lohani et al., 2021).

Several heuristic extensions effectively add dimensions beyond classical RMSA. Dynamic multi-modulation allocation allows multiple modulation formats along one end-to-end connection by segmenting a route into virtual hops and then choosing the highest feasible modulation per segment under reach constraints; it reports a reduction in bandwidth blocking of up to two orders of magnitude in an underloaded network scenario, and 86% with higher loads, relative to literature schemes (Costa et al., 2020). Congestion-aware RMCSA in SDM-EONs uses real-time link occupancy $d(p) \le R_m$8, alternative path generation, and caching, and reports lower blocking than SP-, KSP-, K-disjoint-, and adaptive link-weight baselines while maintaining much lower service latency than the adaptive link-weight approach (Heera et al., 2024).

4. Learning-based RMSA: from DeepRMSA to diffusion models and graph transformers

Deep reinforcement learning has become a major RMSA methodology because dynamic RMSA is sequential, combinatorial, and state-dependent. DeepRMSA formulates dynamic RMSA with a compact per-request state, an action space over $d(p) \le R_m$9 candidate routes and $d(p)$0 available FS blocks, and two DNNs for policy and value, trained using modified A3C (Chen et al., 2019). Its episode-based DeepRMSA-EP uses finite-horizon returns, while DeepRMSA-FLX replaces episodes with a sliding window so that each return is computed over exactly $d(p)$1 consecutive requests; on NSFNET and COST239, DeepRMSA-FLX yields blocking probability reductions of more than 20.3% and 14.3% compared with the baselines (Chen et al., 2019).

Subsequent work has emphasized that many published RL methods are weaker than carefully engineered heuristics. The transformer-based large-scale study explicitly reports that previous RL methods are worse than optimized KSP-FF or FF-KSP heuristics in almost all benchmark cases, echoing an earlier systematic study that naïve heuristics plus good engineering often beat many published RL methods (Doherty et al., 3 May 2026). This directly addresses a common misconception that RL automatically dominates heuristic RMSA. The evidence instead suggests that learned policies must be compared against strong heuristic baselines, not only against weaker or mismatched ones (Doherty et al., 3 May 2026, Doherty et al., 20 Feb 2025).

Recent work has therefore concentrated on stronger architectures and training stabilization. OpticGAI introduces an AI-generated policy paradigm in which a diffusion model replaces the usual parametric policy network in DRL. For each request, a reverse denoising process generates an action-scheme vector $d(p)$2, and the final RMSA action is the candidate corresponding to the maximum component of $d(p)$3 (Li et al., 2024). On the 14-node NSFNET DeepRMSAEnv, OpticGAI is reported as the only model to reach a reward value of 42 or higher, to reach reward 40 earlier than all other DRL baselines, and to maintain a slight advantage in blocking rate over PPO and A2C after convergence (Li et al., 2024). The authors argue that diffusion policies can represent complex, multi-modal action distributions, whereas standard feed-forward policies are often unimodal and more prone to local optima (Li et al., 2024).

A distinct line uses graph transformers. The 2026 graph-transformer study reports the first stable RL training of a transformer for dynamic RMSA by combining rotary positional encodings for graph-structured data, off-policy invalid action masking, valid mass regularization, and GPU-accelerated simulation (Doherty et al., 3 May 2026). The network is encoded as a directed graph with one token per directed link; each token includes per-FSU normalized remaining holding times, graph positional features derived from the line graph Laplacian, and current demand features (Doherty et al., 3 May 2026). The action space selects one candidate path and one spectrum sub-band, with first-fit performed inside the selected sub-band (Doherty et al., 3 May 2026). Its ablations show that removing off-policy invalid action masking causes training to fail, while removing damping or valid mass loss causes collapse or severe instability, especially on large topologies (Doherty et al., 3 May 2026). Empirically, it is reported as the first RL method to exceed all benchmarks, increasing the supportable traffic load by up to 13% on benchmark settings and by up to 4% at low blocking probability on large real topologies up to 143 nodes and 362 links (Doherty et al., 3 May 2026).

The broader lesson is not that one architecture has universally solved RMSA, but that RMSA learning performance is tightly coupled to state representation, invalid-action handling, and benchmark strength. A plausible implication is that future progress will depend as much on environment fidelity and training stability as on raw model expressivity.

5. Impairment-aware, service-aware, and architecture-specific generalizations

RMSA has repeatedly been generalized by adding new physical or service dimensions. In SDM-EONs, the problem becomes RMSCA or RMCSA because core selection must be made together with route, modulation, and spectrum. The survey on crosstalk and RMSCA states that with multi-core fibers the core choice becomes an additional decision dimension and that crosstalk thresholds depend on modulation format, with representative thresholds reported for BPSK, QPSK, 8QAM, 16QAM, 32QAM, and 64QAM (Brasileiro et al., 2019). In the congestion-aware and latency-aware RMCSA algorithm for SDM-EONs, each core is an elastic spectrum, $d(p)$4 measures link occupancy, and the controller jointly determines route, modulation, core, and spectrum under continuity, contiguity, core continuity, guard-band, and modulation-reach constraints (Heera et al., 2024).

Fiber heterogeneity yields another extension. In hybrid ultra-low-loss and SSMF EONs, each link contains one SSMF and one ULL fiber, and the problem becomes RFMSA: routing, fiber, modulation format, and spectrum allocation (Ouyang et al., 2024). A node-arc MILP minimizes the maximum FS index used in the network while selecting route, fiber type per used link, modulation, and starting slot (Ouyang et al., 2024). Its SWP-based heuristics show that, in static traffic, the OSNR-aware strategy exhibits performance similar to the MILP regarding the maximum number of FSs used in the entire network, while in dynamic traffic the spectrum-usage-based strategy remarkably surpasses the others in lightpath blocking probability (Ouyang et al., 2024). This suggests that once multiple physical substrates coexist on each link, “routing” and “modulation” alone no longer characterize the decision space adequately.

Service-aware objectives also reshape the problem. In video-oriented RSA, the objective combines a network fragmentation cost,

$d(p)$5

with a video-quality cost,

$d(p)$6

and imposes the hard QoE constraint $d(p)$7 for video sessions (Ghazijahani et al., 2019). That work does not optimize modulation, but it explicitly notes that the framework is suitable for extension to full RMSA by adding a discrete modulation variable and modulation-dependent OSNR thresholds (Ghazijahani et al., 2019). In impairment-aware dynamic RMLSA, ABACUS similarly shows that route determination and spectrum assignment should not be separated and that physical-layer impairments can be integrated into a joint ILP while still maintaining reasonable time complexity on NSFNET (Kiran et al., 2024).

There are also adjacent formulations that illuminate RMSA by analogy. Routing and wavelength assignment with lightpath reuse in fixed-grid networks uses a GN-model path capacity and incremental reuse of existing lightpaths; it retains routing and wavelength continuity but omits contiguous variable-width slot assignment and explicit modulation selection (Doherty et al., 20 Feb 2025). Broadband quantum entanglement distribution uses routing plus a max–min fair spectrum-allocation ILP over discrete channels, with routing decoupled from spectrum because all channels experience the same path loss (Bali et al., 2024). These are not full RMSA formulations, but they clarify which parts of RMSA complexity come from modulation discretization, contiguity, and per-link spectrum coupling rather than from routing alone.

6. Benchmarks, practical trade-offs, and open problems

The contemporary RMSA literature identifies three recurring practical tensions: exactness versus scale, heuristic simplicity versus adaptability, and learning performance versus stability. Exact ILPs offer strong baselines and structural insight but remain computationally demanding. The full-RMSA ILP on NSFNET with 80 FSs per link and 100–120 demands solves a majority of instances within a few hours, but some cases time out even with a 30-hour limit (David et al., 2023). This confirms the value of exact formulations for benchmarking, but also why dynamic RMSA in realistic settings continues to rely on heuristics or RL (David et al., 2023).

A second tension concerns whether heuristic baselines are already near-optimal. Large-scale transformer benchmarking finds that optimized KSP-FF and FF-KSP heuristics with sufficiently large $d(p)$8 consistently yield the best blocking performance among heuristics, and previous RL methods are often worse than these heuristics (Doherty et al., 3 May 2026). In fixed-grid lightpath reuse, the best heuristic remains within roughly 1–2% of the new GAT-based RL agent, despite the latter’s much greater training cost (Doherty et al., 20 Feb 2025). This does not imply that learned RMSA is unpromising; rather, it suggests that claims of superiority must be interpreted relative to the actual baseline strength.

A third tension is between model fidelity and deployability. Diffusion-based policies and graph transformers improve representational flexibility, but both add computational complexity. OpticGAI explicitly notes that inference requires $d(p)$9 denoising steps and is more demanding than a single feed-forward pass, though still compatible with centralized SDN control in practical EON-scale networks (Li et al., 2024). The transformer-based approach uses GPU-accelerated simulation and millions of steps per second in some setups, but also relies on carefully stabilized PPO and slot aggregation to make large action spaces tractable (Doherty et al., 3 May 2026). This suggests that hardware-aware implementations and environment design are now part of the RMSA method itself, not just an engineering afterthought.

Open problems recur across otherwise different formulations. Several papers call for stronger generalization across topologies and traffic patterns, movement beyond a single benchmark topology, and validation on real traffic traces or diverse topologies (Li et al., 2024, Doherty et al., 3 May 2026). Explainability is another recurring issue: diffusion-augmented DRL is described as even more of a black box than standard DRL, and production RMSA systems require interpretable reasons for routing and spectrum choices (Li et al., 2024). Physical realism also remains incomplete in many high-performing methods: the graph-transformer work uses a distance-based physical layer rather than full QoT with nonlinear impairments and ROADM penalties (Doherty et al., 3 May 2026), while hybrid-fiber RFMSA and crosstalk-aware RMSCA show how quickly the decision space expands once additional physical dimensions are modeled (Ouyang et al., 2024, Brasileiro et al., 2019).

Overall, RMSA remains the central combinatorial control problem of EONs because it concentrates almost every hard systems interaction in optical networking: path choice, physical reach, spectral efficiency, fragmentation, and online admission control. The literature now spans exact node-arc MILPs, conflict-graph theory, congestion-aware heuristics, cooperative and single-agent DRL, diffusion policies, graph transformers, and architecture-specific generalizations such as RMSCA, RMCSA, and RFMSA. A plausible implication is that future advances will come less from any single paradigm than from tighter integration of physical modeling, state representation, and benchmark methodology.