Framed Capacity Inequalities (FCIs) Overview

• FCIs are advanced constraints that generalize classic rounded capacity inequalities by partitioning customer sets, enabling tighter operational bounds in CVRP.
• They integrate combinatorial partitioning and bin-packing methods to enhance LP relaxation and reduce optimality gaps, as demonstrated in empirical benchmarks.
• FCIs extend beyond routing, appearing in network coding, fractional geometry, and lattice theory, highlighting their broad applicability in capacity optimization.

Framed Capacity Inequalities (FCIs) appear in multiple research settings, with a particularly explicit formulation in the Capacitated Vehicle Routing Problem (CVRP), where they generalize Rounded Capacity Inequalities (RCIs) by replacing a single customer subset with a partition of a customer set and thereby incorporating the interplay between multiple disjoint subsets of customers. In the supplied research record, the same label is also used for field-independent capacity outer bounds in network coding, Brunn–Minkowski-type inequalities for fractional capacity, sharp geometric capacity inequalities in convex cones and asymptotically flat manifolds, and lattice-theoretic inequalities formulated through X-difference operators (Sim et al., 1 Oct 2025).

1. CVRP formulation and relation to rounded capacity inequalities

In the context of the CVRP, FCIs are introduced as a generalization of the classic RCIs. RCIs consider a single subset $S \subseteq V_C$ and impose

$$x(\delta(S)) \geq 2 \left\lceil \frac{\sum_{i \in S} q_i}{Q} \right\rceil.$$

By contrast, FCIs consider partitions of a customer set. Let $\Omega = \{ S_i : i \in I \}$ be a partition of a subset $H \subseteq V_C$, where $V_C$ is the set of customers. For each subset $S_i$, demand may be split into multiple bin-packing items if needed. The bin-packing value $r(\Omega)$ is the minimum number of vehicles needed to collectively serve all items. The inequality is written as

$$x(\delta(H)) + \sum_{i \in I} x(\delta(S_i)) \geq 2r(\Omega) + 2 \sum_{i \in I} \left\lceil \frac{d(S_i)}{Q} \right\rceil.$$

Here, $x(\delta(H))$ is the sum of edge variables crossing the boundary of $H$, $$x(\delta(S)) \geq 2 \left\lceil \frac{\sum_{i \in S} q_i}{Q} \right\rceil.$$0 is the sum of edges crossing each partition subset, $$x(\delta(S)) \geq 2 \left\lceil \frac{\sum_{i \in S} q_i}{Q} \right\rceil.$$1 is total demand in $$x(\delta(S)) \geq 2 \left\lceil \frac{\sum_{i \in S} q_i}{Q} \right\rceil.$$2, $$x(\delta(S)) \geq 2 \left\lceil \frac{\sum_{i \in S} q_i}{Q} \right\rceil.$$3 is vehicle capacity, and $$x(\delta(S)) \geq 2 \left\lceil \frac{\sum_{i \in S} q_i}{Q} \right\rceil.$$4 is the minimal number of vehicles by bin packing the partition. Because FCIs consider multiple mutually disjoint subsets at once, they yield potentially stronger, tighter constraints than RCIs: combining subsets can identify infeasible fractional solutions that RCIs alone might miss. At the same time, separation is substantially harder, since FCIs require solving a bin-packing problem over subset demands at each separation attempt, making separation NP-hard and making enumeration over general partitions completely intractable (Sim et al., 1 Oct 2025).

2. Separation, graph coarsening, and the GraphCHiP algorithm

The recent CVRP literature treats FCI discovery as a separation problem embedded in a neural cutting-plane pipeline. The starting observation is that a deep learning-based separation method can learn to find high-quality cuts, but the model produces fewer cuts than expected because it is insufficiently sensitive to generate a diverse set of generated subsets. The proposed alternative is to enhance the performance of a trained model at inference time through a new test-time search with stochasticity. First, stochastic edge selection is introduced into the graph coarsening procedure, replacing the previously proposed greedy approach. Second, the Graph Coarsening History-based Partitioning algorithm, GraphCHiP, leverages coarsening history to identify not only RCIs but also, for the first time, FCIs (Sim et al., 1 Oct 2025).

GraphCHiP operates by tracing the history of supernode formation during neural graph coarsening rather than using only the final merged customer subset. At each coarsening step $$x(\delta(S)) \geq 2 \left\lceil \frac{\sum_{i \in S} q_i}{Q} \right\rceil.$$5, the node map $$x(\delta(S)) \geq 2 \left\lceil \frac{\sum_{i \in S} q_i}{Q} \right\rceil.$$6 records which original customers are merged into which supernodes. The current supernodes are then taken as candidate partition components. For a given coarsened subset $$x(\delta(S)) \geq 2 \left\lceil \frac{\sum_{i \in S} q_i}{Q} \right\rceil.$$7, the method backtracks to form a partition $$x(\delta(S)) \geq 2 \left\lceil \frac{\sum_{i \in S} q_i}{Q} \right\rceil.$$8 of original nodes by including all supernodes currently in $$x(\delta(S)) \geq 2 \left\lceil \frac{\sum_{i \in S} q_i}{Q} \right\rceil.$$9 or their constituents, and adding remaining nodes as singleton sets. Partitions where any subset violates an RCI are discarded so that the search focuses on genuine FCI structure. Bin packing is then approximated and potentially exactly solved over partition subset demands to compute $\Omega = \{ S_i : i \in I \}$0, and violated FCIs are added as cuts to the LP.

Several features of the method are emphasized. Algorithmic pseudocode is provided as Algorithm 4. The procedure iterates efficiently over coarsening steps, is logarithmic in graph size at the coarsening level, uses filtering heuristics to avoid redundant partitions and subsets, and is polynomial-time and scalable because the candidate partition space is restricted by the coarsening history rather than by explicit partition enumeration. The worst-case number of checks is stated as $\Omega = \{ S_i : i \in I \}$1 (Sim et al., 1 Oct 2025).

3. Computational behavior in CVRP optimization

Empirical results on randomly generated CVRP instances show that RCIs found via GraphCHiP and stochastic coarsening improve the dual gap relative to both classical heuristics and the original deterministic NeuralSEP. For FCIs, the reported pattern is more selective: on most random CVRP instances, violated FCIs are rare, which is attributed to the specifics of support graphs and demand distributions. The paper therefore treats FCIs as especially relevant when cross-partition demand structure is present and RCI-based gap closure has largely been exhausted (Sim et al., 1 Oct 2025).

A detailed case study is the CVRPLIB instance X-n153-k22, described as having skewed demand and rich partition structure. On this instance, GraphCHiP combined with stochastic NeuralSEP found $\Omega = \{ S_i : i \in I \}$2 FCIs, whereas classical CVRPSEP found only $\Omega = \{ S_i : i \in I \}$3. The best lower bound and smallest gap were both achieved with FCIs generated by GraphCHiP, and the optimality gap improved by $\Omega = \{ S_i : i \in I \}$4 over RCI-only neural separation.

Method	Lower Bound	Gap
CVRPSEP RCI	19983.51	5.83%
$\Omega = \{ S_i : i \in I \}$5-NeuralSEP$\Omega = \{ S_i : i \in I \}$6 RCI	19861.37	6.40%
CVRPSEP RCI + CVRPSEP FCI	19984.29	5.82%
$\Omega = \{ S_i : i \in I \}$7-NeuralSEP$\Omega = \{ S_i : i \in I \}$8 RCI + GraphCHiP FCI	20153.95	5.02%

The same experiment reports $\Omega = \{ S_i : i \in I \}$9 FCI cuts for the GraphCHiP configuration and $H \subseteq V_C$0 for CVRPSEP RCI + CVRPSEP FCI. The interpretation given is that adding FCIs tightens the LP relaxation and thus reduces the dual gap; it is also noted that this may potentially speed up branch-and-bound, although that was not shown for the complete tree in the paper. Within the CVRP literature represented here, the method is described as the first learning-based approach for FCI discovery, whereas previous work such as CVRPSEP provided heuristics based on greedy or tree search (Sim et al., 1 Oct 2025).

4. Polyhedral context and field-dependent contrasts

The broader capacity-inequality literature shows that the phrase is not confined to vehicle routing. In network design, capacity constraints are modeled in directed, undirected, and bidirected forms, and polyhedral investigations study how valid inequalities are translated across those models. A key result is that the projections of the undirected and bidirected models onto the space of capacity variables are equal for pairwise symmetric demand matrices. The general translation principle states that a valid inequality for the directed model can be projected onto the bidirected model by appropriately summing coefficients on reverse arcs. While framed capacity inequalities are not named as such in that text, several well-known inequalities from the undirected or bidirected literature are presented as projections or consequences of inequalities valid in the directed model (Atamturk et al., 2017).

In network coding, the terminology is used differently. FCIs are described as general capacity outer bounds, for example using entropy inequalities, that apply regardless of the field. Their limitation is that they may be loose for linear codes over a fixed field. The contrast is made explicit by characteristic-dependent linear rank inequalities for eight variables. One inequality holds for all finite fields whose characteristic is not three and can fail in characteristic three; the other holds for characteristic three and can fail outside characteristic three. On the T8 network, the first inequality gives

$H \subseteq V_C$1

so the linear capacity is at most $H \subseteq V_C$2 for $H \subseteq V_C$3, while a scalar linear solution of capacity $H \subseteq V_C$4 exists over fields of characteristic $H \subseteq V_C$5. On the non-T8 network, the second inequality gives

$H \subseteq V_C$6

so the linear capacity is at most $H \subseteq V_C$7 for characteristic $H \subseteq V_C$8, whereas for other characteristics there exists a linear solution at capacity $H \subseteq V_C$9. In that literature, characteristic-dependent inequalities are therefore stronger than FCIs for specific algebraic settings (Dougherty et al., 2014).

5. Fractional capacity, convex cones, and rigidity

In analysis and geometry, FCIs take the form of inequalities that linearize powers of capacity functionals under geometric operations. For a compact set $V_C$0, the $V_C$1-fractional capacity is defined by

$V_C$2

and the associated capacitary potential solves

$V_C$3

For convex $V_C$4, the super-level sets $V_C$5 are convex and satisfy

$V_C$6

The corresponding Brunn–Minkowski inequality is

$V_C$7

with equality if and only if $V_C$8 and $V_C$9 are homothetic. In this setting, these are described as FCIs because the $S_i$0-powers of the capacity behave linearly under Minkowski sums (Qin et al., 2024).

This framework supports rigidity statements. If $S_i$1 is a bounded convex domain, then

$S_i$2

with equality if and only if $S_i$3 is a ball. A second result states that if the solution $S_i$4 has two homothetic convex level sets, then $S_i$5 is a ball. The proof strategy links optimal concavity of the capacitary potential to equality in the Brunn–Minkowski inequality, and then to homotheticity of level sets (Qin et al., 2024).

A related geometric program studies capacity inequalities for hypersurfaces with boundary in convex cones and for asymptotically flat manifolds with non-compact boundary. There, the paper proves a capacity inequality involving the total mean curvature of free-boundary hypersurfaces in convex cones, together with analogues of Pólya–Szegő, Alexandrov–Fenchel, and Penrose-type inequalities. Equality is characterized by spherical sectors in the cone setting and by the Riemannian half-Schwarzschild manifold in the mass–capacity setting. The inverse mean curvature flow is the core technique: it provides monotonicity of normalized total mean curvature, respects the conical free-boundary geometry, and yields rigidity when the evolving hypersurfaces remain totally umbilical (Cruz, 2017).

6. Finite lattices, Fréchet bounds, and X-difference operators

On a finite lattice $S_i$6, a capacity is a monotone function $S_i$7 with $S_i$8 and $S_i$9. Complete monotonicity is expressed through successive difference operators: $r(\Omega)$0 and recursively

$r(\Omega)$1

If $r(\Omega)$2 for a random up-set $r(\Omega)$3, complete monotonicity corresponds to the existence of a probability measure on the distributive lattice of dual order ideals (Machida, 2011).

The central extension problem is to pass from a marginal capacity on $r(\Omega)$4 to a completely monotone capacity $r(\Omega)$5 on the lattice $r(\Omega)$6 of dual order ideals such that

$r(\Omega)$7

Fréchet bounds describe extremal feasible extensions under these marginal constraints. For $r(\Omega)$8,

$r(\Omega)$9

For the dual order ideal generated by $$x(\delta(H)) + \sum_{i \in I} x(\delta(S_i)) \geq 2r(\Omega) + 2 \sum_{i \in I} \left\lceil \frac{d(S_i)}{Q} \right\rceil.$$0 and $$x(\delta(H)) + \sum_{i \in I} x(\delta(S_i)) \geq 2r(\Omega) + 2 \sum_{i \in I} \left\lceil \frac{d(S_i)}{Q} \right\rceil.$$1, the explicit bound is

$$x(\delta(H)) + \sum_{i \in I} x(\delta(S_i)) \geq 2r(\Omega) + 2 \sum_{i \in I} \left\lceil \frac{d(S_i)}{Q} \right\rceil.$$2

where $$x(\delta(H)) + \sum_{i \in I} x(\delta(S_i)) \geq 2r(\Omega) + 2 \sum_{i \in I} \left\lceil \frac{d(S_i)}{Q} \right\rceil.$$3 is computed via maximal path values over monotone paths from $$x(\delta(H)) + \sum_{i \in I} x(\delta(S_i)) \geq 2r(\Omega) + 2 \sum_{i \in I} \left\lceil \frac{d(S_i)}{Q} \right\rceil.$$4 to $$x(\delta(H)) + \sum_{i \in I} x(\delta(S_i)) \geq 2r(\Omega) + 2 \sum_{i \in I} \left\lceil \frac{d(S_i)}{Q} \right\rceil.$$5.

In this lattice-theoretic setting, FCIs are expressed through the X-difference operator

$$x(\delta(H)) + \sum_{i \in I} x(\delta(S_i)) \geq 2r(\Omega) + 2 \sum_{i \in I} \left\lceil \frac{d(S_i)}{Q} \right\rceil.$$6

with iterates $$x(\delta(H)) + \sum_{i \in I} x(\delta(S_i)) \geq 2r(\Omega) + 2 \sum_{i \in I} \left\lceil \frac{d(S_i)}{Q} \right\rceil.$$7. The inequalities take the form

$$x(\delta(H)) + \sum_{i \in I} x(\delta(S_i)) \geq 2r(\Omega) + 2 \sum_{i \in I} \left\lceil \frac{d(S_i)}{Q} \right\rceil.$$8

These constraints frame the allowed set of extensions. On the subclass of completely monotone capacities, the X-difference operators reduce to standard successive differences, and the Möbius extension is unique and minimal. The same formalism also yields stochastic inequalities for couplings of random sets, namely

$$x(\delta(H)) + \sum_{i \in I} x(\delta(S_i)) \geq 2r(\Omega) + 2 \sum_{i \in I} \left\lceil \frac{d(S_i)}{Q} \right\rceil.$$9

as the feasibility condition for a coupling with $x(\delta(H))$0 (Machida, 2011).

7. Scope, common structure, and recurring themes

Across these domains, FCIs are not a single invariant formula. In CVRP, they are valid inequalities for routing polytopes derived from partitions and bin packing; in network coding, they denote field-independent capacity outer bounds; in fractional and geometric analysis, they are sharp inequalities for capacities under Minkowski addition, curvature, volume, or mass; and in finite lattices, they are path-based inequalities governing the extendability of capacities under marginal constraints.

Even with these differences, recurring themes are visible. FCIs strengthen relaxations or outer bounds by incorporating structure that is invisible to simpler one-set or one-variable constraints. In CVRP that additional structure is the partition $x(\delta(H))$1 and the bin-packing term $x(\delta(H))$2; in network coding it is the distinction between field-independent and characteristic-dependent constraints; in fractional geometry it is equality and rigidity under Brunn–Minkowski-type concavity; and in finite lattices it is the combinatorics of paths, dual order ideals, and Möbius inversion. This suggests a shared role for FCIs as higher-order capacity constraints whose main effect is to rule out configurations that survive more elementary inequalities (Sim et al., 1 Oct 2025).