Multicommodity Capacitated Network Design

• MCND is a combinatorial optimization model that designs and expands networks by balancing installation costs with multiple concurrent demand requirements.
• It leverages advanced techniques such as LP rounding, Benders decomposition, and polyhedral combinatorics to provide scalable and near-optimal solutions.
• Practical applications include transportation, telecommunications, logistics, and data center networks, where robust, cost-effective infrastructure is essential.

The multicommodity capacitated network design (MCND) problem encompasses a class of combinatorial optimization models fundamental to the design, expansion, and operation of infrastructure networks—transportation, telecommunications, logistics, and computer systems. MCND captures the strategic planning challenge of determining where and how much capacity to install in a network so that multiple, concurrent end-to-end demands can be satisfied at minimum total cost while observing various capacity and design constraints. Modern research in MCND focuses on both the theoretical aspects—such as polyhedral combinatorics, approximation algorithms, and complexity—as well as on practical and scalable solution approaches for large-scale, real-world instances.

1. Formal Problem Statement and Variants

Let $G = (V, E)$ be a graph (directed or undirected), with node set $V$ and edge (or arc) set $E$. A set $\mathcal{K}$ of commodities $(k)$ is given, each with origin $O(k)$, destination $D(k)$, and demand $d^k>0$. Each edge $(i, j)$ has capacity $u_{ij}$ and may incur a fixed charge $f_{ij}$ (for opening) as well as a per-unit flow cost $c_{ij}$.

The typical mixed-integer programming formulation is:

$$\begin{align*} \min\ & \sum_{(i,j) \in E} f_{ij} y_{ij} + \sum_{k \in \mathcal{K}} \sum_{(i,j) \in E} c_{ij} x_{ij}^k \ \text{s.t.}\ & \sum_{j: (i,j) \in E} x_{ij}^k - \sum_{j: (j,i) \in E} x_{ji}^k = \begin{cases} d^k, & i=O(k) \ -d^k, & i=D(k) \ 0, & \text{otherwise} \end{cases} \ & \sum_{k \in \mathcal{K}} x_{ij}^k \leq u_{ij} y_{ij},\ \forall (i,j)\in E \ & y_{ij} \in \{0,1\},\ x_{ij}^k \geq 0 \end{align*}$$

(Larsen et al., 2023, Rocca et al., 2024, Kazemi et al., 2021)

In this canonical form, MCND is splittable (flow for each commodity can be split among multiple paths). Unsplittable variants—where each commodity must be routed on a single path—require path-based binary variables and significantly complicate the polyhedral structure (Greening et al., 31 Dec 2025). Additional variants include multi-facility (discrete/variable capacity increments per edge) (Atamturk et al., 2017), node-capacitated models (Krishnaswamy et al., 2014), traffic-oblivious subgraph preservation (Chimani et al., 23 Apr 2025), and service-level constraints (Gudapati et al., 2021).

2. Complexity and Approximability

MCND is NP-hard—both for splittable and unsplittable flow settings—even in restricted graph classes or with uniform capacities (Chakrabarty et al., 2010, Charikar et al., 2018). Key complexity results include:

• Polynomial-time approximations:
  • For standard MCND (capacitated multicommodity flow with fixed charges), the best possible approximation in directed graphs is $O(\ln n)$ unless P=NP, via reduction from Set Cover (Charikar et al., 2018).
  • In node-capacitated MCND, the best known algorithm achieves a bicriteria $(O(\log^4 n),\, O(\log^{10} n))$-approximation: cost within $O(\log^4 n)$ of optimal, node capacities violated by at most $O(\log^{10} n)$ (Krishnaswamy et al., 2014). For single-sink, the bounds improve by one logarithmic factor.
  • For global connectivity (all-pairs demand $R$), an $O(\log n)$-approximation is achieved via LP rounding and randomized sampling (Chakrabarty et al., 2010).
  • Allowing multiple copies of edges relaxes hardness, enabling $O(\log k)$-approximations, where $k$ is the number of terminal pairs (Chakrabarty et al., 2010).
  • For unsplittable variants with path-based or arc-based reformulations, new classes of valid inequalities yield 20–85% reductions in duality gaps, dramatically improving IP bounding (Greening et al., 31 Dec 2025).
• Hardness and inapproximability:
  • Directed single-pair instances are hard to approximate within $2^{\log^{1-\delta} n}$ for any $\delta>0$ (Chakrabarty et al., 2010).
  • For the survivable network design variant and multiway cut demands, polylogarithmic gaps remain unresolved.
  • Hardness results remain valid even for specialized versions (see reductions from Label Cover and Set Cover in (Chakrabarty et al., 2010, Chimani et al., 23 Apr 2025)).

3. Model Refinements and Polyhedral Structure

Modern MCND research leverages advanced polyhedral combinatorics for tighter relaxations and improved branch-and-cut performance. Three fundamental cutting-plane families dominate the literature:

• Arc-set relaxations (per-arc knapsack cuts): For each edge, link flow to installed capacity, derive residual-capacity MIR cuts, and c-strong inequalities. These are effective both for splittable and unsplittable flows (Atamturk et al., 2017).
• Cut-set relaxations: Multi-commodity cuts (partitioning nodes) yield inequalities that project out flow variables, including metric inequalities and flow-cut-set inequalities. These can be separated efficiently via shortest-path subproblems and are essential for global capacity constraints (Atamturk et al., 2017).
• Partition (shrinking) techniques: Collapsing network partitions and lifting valid inequalities from the reduced network to the original increases cut strength, though computational payoff diminishes for larger partitions (Atamturk et al., 2017).

For unsplittable MCND, new families of "single-arc commodity packing" (SAC-Pack) and generalized SAC-Pack inequalities provide substantial improvements. The BIN reformulation—using multiple-choice binary variables for capacity—enables further strengthening by combining with metric cuts and dynamic root node cut-generation (Greening et al., 31 Dec 2025).

Partial aggregation of commodities can drastically accelerate LP and MIP solving, interpolating between full disaggregation (tightest but slow) and full aggregation (fastest but weak) (Kazemi et al., 2021). The PAe-5 variant (disaggregate each commodity on 5 shortest-path arcs) achieves a robust trade-off, with 85% speedup over full disaggregation and <2% loss in bound quality.

4. Approximation and Algorithmic Frameworks

A spectrum of algorithmic schemes are present:

• LP rounding and random sampling:
  • Rounding fractional edge variables with probabilistically-defined thresholds or sampling enables O(log n) approximations in capacitated undirected designs (Chakrabarty et al., 2010).
• Single-sink confluent flows and clustering:
  • Iterated single-sink clustering, set-cover reductions, and hierarchical merging construct nearly node-disjoint subtrees each supporting a bounded total demand (Krishnaswamy et al., 2014).
• Randomized “hallucination” for inter-cluster flows:
  • Sampling commodity demands (with logarithmic scaling) combined with cut-sparsification preserves feasibility and enables recursive routing, culminating in the polylogarithmic result for node-capacitated design (Krishnaswamy et al., 2014).
• Benders decomposition for chance-constrained stochastic MCND:
  • Scenario-based Benders with the FlowMIS subproblem formulation generates dominant feasibility cuts, achieving order-of-magnitude speedups on large scenario sets (Wouda et al., 2024).
• Lagrangian relaxation and subgradient methods:
  • Path-based Lagrangian relaxations combined with modern nonsmooth optimization (deflected subgradients, Polyak step rules) yield tight dual bounds and effective lower bounds for DCMNDP (Ennaifer et al., 2017).
• Path-based formulations and branch-cut-and-price algorithms:
  • Enumerative or column-generation based approaches are critical for models with per-service constraints or unsplittable flow restrictions. Labeling algorithms generate exponentially-many feasible paths subject to end-to-end metrics; when full enumeration is impractical, dynamic column generation inside branch-and-price dominates (Gudapati et al., 2021, Greening et al., 31 Dec 2025).

5. Extensions: Stochasticity, Robustness, and Algorithmic Benchmarks

MCND has been extended to stochastic environments (random demands/capacities/costs), robust design, chance constraints, and energy-aware settings:

• Stochastic MCND is modeled via two-stage recourse: open capacities here-and-now, optimize flows per scenario. High-performance C++ instance generators (Mulgen++, StoGen++) facilitate reproducible benchmarking (Larsen et al., 2023).
• Chance-constrained MCND imposes a master knapsack constraint on scenario-violation variables, with subproblem feasibility interfacing via Benders decomposition (Wouda et al., 2024).
• Energy-efficient routing via node-capacity design translates MCND approximations into strong guarantees for minimization of routing power, under convex cost models (Krishnaswamy et al., 2014).
• Traffic-oblivious design (MMCFS) asks for a minimum subgraph preserving routability for all possible traffic matrices scaled by the retention ratio $\alpha$, with approximability tightly characterized by max$(2, 1/\alpha)$ (Chimani et al., 23 Apr 2025).

Algorithm performance is strongly instance-dependent; partial aggregation, path-based models, or advanced cut generation outperform standard arc-flow MIPs on large, dense, or service-constrained networks (Kazemi et al., 2021, Gudapati et al., 2021, Greening et al., 31 Dec 2025). Recent studies show that machine learning-based arc selection can further improve heuristic and local search performance for large MCNDs (Rocca et al., 2024).

6. Applications and Practical Considerations

MCND models support real-world engineered systems, especially where cost, capacity, and survivability interact:

• Telecommunication and backbone networks: Facility installation (add/drop multiplexers, routers), capacity expansion, and survivability.
• Supply chain/logistics: Freight consolidation, fulfillment center and transportation network design, parcel consolidation (e-commerce) (Greening et al., 31 Dec 2025).
• Data center and cloud networking: Functions virtualization, dynamic middleboxes, energy efficiency in routing (Charikar et al., 2018, Krishnaswamy et al., 2014).
• Chance-constrained and robust design: Satisfying end-to-end guarantees under traffic, demand, or component uncertainty (Wouda et al., 2024, Larsen et al., 2023).

Robust empirical evaluation is facilitated by pseudo-random instance generators, which support both deterministic and scenario-based stochastic MCNDs, with extensive parameterization for network size, topology, and statistical properties (Larsen et al., 2023).

7. Open Directions and Research Frontiers

Key research challenges include:

• Closing approximation gaps: Improving constants or log-factors in undirected MCND; establishing tight hardness results for the general case (Chakrabarty et al., 2010).
• Polyhedral combinatorics: Developing efficient separation or lifting methods for multicommodity valid inequalities in large-scale models, particularly in the unsplittable regime (Greening et al., 31 Dec 2025, Atamturk et al., 2017).
• Scalable decomposition and ML-augmented methods: Integrating reinforcement or supervised learning into branch-and-bound, cut selection, or variable fixing heuristics (Rocca et al., 2024).
• Stochastic and robust MCND: Systematizing strong feasibility cuts, scenario reduction, and chance-constrained decomposition for massive-scale, uncertainty-aware design problems (Wouda et al., 2024, Larsen et al., 2023).
• Service and reliability constraints: Extending column-and-cut generation, and path-based strategies, for end-to-end metrics critical in emerging 5G, IoT, and cloud networking (Gudapati et al., 2021).

MCND continues to drive both methodological advances in combinatorial optimization and practical innovation in networked system design, blending deep polyhedral analysis with algorithm engineering and large-scale computational experimentation.