Fixed-Charge Dualities (FCD)

• Fixed-Charge Dualities are defined by the introduction of an indivisible activation cost that alters dual structures, decomposition methods, and computational strategies across various fields.
• Algorithmic techniques leveraging FCD transform concave cost problems and network flows by translating fixed-charge objectives into tractable primal-dual and dynamic programming frameworks.
• FCD research unifies approaches across combinatorial optimization, field theory, and categorical topology, driving innovations in approximation guarantees, non-perturbative analysis, and symmetry-protected phase classifications.

Fixed-Charge Dualities (FCD) encompass a suite of structural, algorithmic, and analytic concepts that arise in optimization, algebraic, and physical systems where a distinguished “charge” or threshold introduces new types of dualities, decompositions, and computational strategies. FCD can denote duality relations between combinatorial optimization problems with fixed-charge objectives, the analytic structure of large-charge expansions in field theory, and categorical dualities for symmetry-protected topological phases. This entry provides a comprehensive account of the mathematical foundations, algorithmic techniques, physical implications, and categorical structures underlying FCD.

1. Foundational Principles of Fixed-Charge Dualities

A fixed-charge cost structure arises when an objective or constraint incurs an indivisible “activation” threshold in addition to linear costs. In combinatorial optimization, this is formalized in problems where each action (e.g., opening a facility, sending flow on an edge, ordering inventory) incurs a fixed cost if any positive activity occurs, plus a variable cost proportional to the magnitude. The fixed-charge duality principle relates the analysis of such problems to their linear-cost counterparts via duality transformations and piecewise-linear approximations.

In the context of optimization algorithms, the fixed-charge duality principle enables the transfer of primal-dual algorithms from linear or fixed-charge settings to concave cost problems by exploiting the separation of fixed charges and variable charges. This is achieved by expressing a concave cost as a minimum over affine functions (“tangent-based” decomposition), thereby revealing a dual structure analogous to classical fixed-charge formulations (Magnanti et al., 2012).

In field theory and algebraic models, FCD arises in expansions where a global charge Q is held fixed, producing expansions whose analytic properties (convergence, divergence, summability) depend critically on the charge sector, often admitting dual descriptions and non-perturbative physics (Antipin et al., 2022). In categorical settings, such as SPT phase classification, FCD refers to braided autoequivalences that preserve the charge decomposition of the symmetry category, enabling symmetric entanglers even in non-invertible symmetry settings (You, 4 Sep 2025).

2. Algorithmic Techniques Leveraging Fixed-Charge Dualities

Primal-Dual Lifting for Concave Cost Problems

The FCD algorithmic technique consists of “lifting” strongly polynomial primal-dual algorithms designed for fixed-charge combinatorial problems to solve richer problems with separable concave costs. This is accomplished by representing every concave cost function $\phi_j(\xi)$ as a minimum over affine tangents:

$$\phi_j(\xi) = \min_p [f_{jp} + s_{jp} \xi], \quad \text{if } \xi > 0; \quad \phi_j(0) = 0,$$

where $s_{jp}$ is the derivative at $p$, and $f_{jp}$ ensures that the tangent matches the function at $p$ (Magnanti et al., 2012). The reformulation produces an instance of the classical fixed-charge problem with an exponentially or infinitely many “facilities” indexed by tangents.

This enables implicit simulation of the primal-dual algorithm without explicit enumeration of the tangents, preserving integrality, approximation ratio, and strong polynomiality where known. In facility location, this technique yields a 1.61-approximation for concave cost objectives. For inventory lot-sizing with general concave ordering costs, the “lifted” algorithm remains exact.

Extended Formulations and Dynamic Programming in Transportation

For fixed-charge transportation problems, FCD manifests via dynamic programming recurrences on tree-structured networks. For a tree, flow decomposes naturally at edges; the recurrences define optimal subtree costs ($\beta_{ij}^l$ for flow $l$ on edge $(i,j)$ and $\alpha_{ijk}$ for cumulative flow $k$ from child $j$) and are propagated upward to compute the global optimum (Angulo et al., 2015):

$$\alpha_{ijk} = \begin{cases} \beta_{ijk}, & j = f(i)\ \min_{0 \leq k' \leq k,\; k-k' \leq a_{ij}} \{\alpha_{i(j-1)k'} + \beta_{ij,\,k-k'} \}, & j > f(i) \end{cases}$$

The dynamic program yields a pseudo-polynomial time algorithm for trees and an LP extended formulation whose dual variables relate to fixed-charge duality, capturing the trade-off between opening and using edges. In general graphs, unary expansion-based formulations leverage single-node convexifications to facilitate cutting plane generation and stronger MIP relaxations.

Polyhedral Strengthening via Path Inequalities

The FCD paradigm in network flows extends to explicit polyhedral descriptions, notably through “path cover” and “path pack” inequalities (Atamturk et al., 2017). These inequalities reflect submodular structure along sequential paths:

• Path cover inequalities dominate flow cover cuts for single-node relaxations and are expressed as

$$y(S^+) + \sum_{t\in S^+} (c_t - \lambda_j)^+(1 - x_t) \leq d_{1n} + c(S^-) + \sum_{t\in L^-} \min\{c_t, \lambda_j\} x_t + y(E^- \setminus (L^- \cup S^-)),$$

• Coefficients $\lambda_j$ are computed via sequences of dynamic programming recurrences that encode marginal value functions and minimum cuts in linear time.
• Necessary and sufficient facet conditions are characterized in terms of forward and backward independence, ensuring inequalities are sharp for the convex hull.

3. Analytic Structures and Dualities in Fixed-Charge Expansions

Fixed-charge expansions in conformal field theory analyze operator dimensions or energies in sectors with large fixed global charge $Q$. These expansions, depending on the model and dimension, exhibit divergent or convergent series behaviors, with deep physical significance (Antipin et al., 2022).

• In $d=3-\epsilon$ $O(N)$ models, the fixed-charge expansion (in the double scaling limit $Q \to \infty$, $\epsilon\to 0$) is non-Borel summable, doubly factorial divergent, with optimal truncation order $\sim \sqrt{Q}$. Singularities in the Borel plane are associated with worldline instantons and the vanishing of a radial mode mass.
• In $d=4-\epsilon$ dimensions, the expansion for the same models yields convergent series at next order, but resummation introduces a branch cut for negative $\epsilon$ relevant to stability of the large-charge sector.
• In QED$_3$, the fixed-charge sector displays only single-factorial divergence and is Borel summable, with physically distinct analytic structure.

Resurgence theory is applied to extract singularities and non-perturbative corrections from asymptotic series, emphasizing the connection between analytic properties and physical dualities in the fixed-charge regime.

4. Practical Applications and Real-World Implications

FCD plays a crucial role in real-world decision-making and reliability planning. For network infrastructure, FCD principles underpin multi-objective planning under uncertainty, such as in CO$_2$ capture and storage pipeline design (Olson et al., 29 Jul 2024).

• The multi-objective fixed-charge network flow problem considers both the minimum-cost initial flow and the repair cost after possible edge failure.
• Iterative three-stage MILP algorithms are developed to generate the Pareto front of solutions, where each point optimally trades off initial investment against post-failure repair expense.
• The approach was evaluated using realistic infrastructure data, demonstrating that slightly higher initial costs yield significant reductions in repair costs, providing explicit guidance to decision-makers.

The dynamics of fixed-charge dualities directly inform the structure, computational tractability, and risk management of diverse applications, from facility location and supply chain management to critical infrastructure planning.

5. Categorical and Topological Manifestations

In categorical and topological frameworks, FCD encapsulates dualities preserving “charge” objects in fusion categories and topological quantum field theories, extending to the classification and entanglement of SPT phases (You, 4 Sep 2025).

• In the symTFT framework, a duality is an FCD if the forgetful functor $F:Z(\mathcal{C}) \rightarrow \mathcal{C}$ satisfies $F(\mathcal{D}(\mu)) = F(\mu)$ for all anyons $\mu$ (i.e., any duality preserves the charge decomposition).
• This structure enables the construction of symmetric entanglers (finite-depth circuits commuting with non-invertible symmetries) between SPT phases connected by an FCD, even when stacking is not defined.
• The paper constructs explicit matrix product unitaries to connect SPT phases with $\mathrm{Rep}(A_4)$ symmetry, demonstrating that block-diagonal MPOs respecting charge classes act as symmetry-commuting entanglers.

This suggests a generalizable notion of phase “stacking” or connection in settings lacking invertible symmetries, with FCD providing the requisite invariance.

6. Computational and Theoretical Impact

The injection of FCD into combinatorial optimization, field theory, and categorical topological analysis delivers several noteworthy consequences:

• Preserves exact or constant factor approximation guarantees when lifting algorithms from fixed-charge to more general concave or non-linear cost structures.
• Facilitates pseudo-polynomial algorithms on structured graphs (e.g., trees in transportation problems) and enables the derivation of strong polyhedral and extended formulations.
• Refines theoretical understanding of divergence and non-perturbative phenomena in quantum and classical models and their physical or operational consequences.
• Explains entanglement constructions and mappings in symmetry-protected topological matter, with categorical invariance framing connections even in the absence of group stacking.

A plausible implication is that FCD, by “charge-preserving” under dualities, provides a powerful and unified perspective for designing algorithms, interpreting expansions, and realizing symmetry constructions across a wide spectrum of mathematical, physical, and applied domains.