Arc-Level NCP Formulation

• Arc-level NCP formulation is a framework that defines network equilibrium conditions directly on arcs using complementarity constraints on flows, capacities, and dual variables.
• It enables efficient and behaviorally precise modeling in transit assignment, network pricing, and parametric graph analysis by capturing granular interactions.
• Exploiting arcwise decomposition, this approach avoids costly path enumeration and admits single-level reformulations with provable computational properties.

An arc-level NCP (Nonlinear Complementarity Problem) formulation specifies constraints, objective functions, and complementarity conditions directly at the level of network arcs, rather than over aggregated paths or entire routes. This approach underpins efficient, behaviorally precise models in settings such as schedule-based transit assignment, parametric graph analysis, and network pricing. Arc-level NCPs improve tractability, enable granular priority modeling, and frequently admit single-level reformulations with provable computational properties.

1. Conceptual Basis and Definition

Arc-level NCP formulations arise whenever a system's equilibrium or feasibility conditions naturally decompose into expressions over arcs. In the context of equilibrium assignment in transit or networks, this means encoding capacities, costs, and behavioral priorities directly as arcwise constraints and complementarity pairs.

Formally, an arc-level NCP seeks vectors $(x, y)$ such that, for specified arc-level functions $F$, $G$: $$\begin{aligned} &0 \le x \perp F(x, y) \ge 0,\ &0 \le y \perp G(x, y) \ge 0 \end{aligned}$$ with “$\perp$” indicating complementarity ($x_i \ge 0$, $F_i \ge 0$, $x_iF_i = 0$ for all $i$). Variables primarily represent arcwise flows, capacities, or Lagrange multipliers, and $F, G$ encode the system's cost or resource equations at the arc level.

In transit assignment, these variables may be boarding flows $f_{w,b}^r$ on trip arcs, boarding-anxiety Lagrange multipliers $v_A$, and dual prices $\mu_{w,b}$ for demand exhaustion. In bilevel pricing, leader’s tolls $T_e$ and related binary flow and dual variables are defined arcwise, avoiding path enumeration. In parametric graph problems, arc-level weights $w_{ji}(\lambda)$ directly induce global properties (e.g., absence of positive-weight circuits) via arc-level constructs.

2. Structural Elements: Variables, Constraints, and Complementarity

Arc-level NCPs operate on a common triplet: (i) flow/capacity variables per arc, (ii) cost or congestion measures per arc, and (iii) Lagrange multipliers or dual variables enforcing feasibility and exclusivity.

Arc-Level NCP for Transit Assignment

Let $A$ be arcs in an event–activity graph, $f_{w,b}^r$ be equilibrium flows for class $b$ and OD pair $w$ on route $r$, $v_A$ be Lagrange multipliers for arcwise boarding capacity, and $q_A(f)$ the available capacity on $A$ after higher-priority arcs have loaded. The key complementarity for priority enforcement is

$0 \le v_A \perp q_A(f) \ge 0$

which ensures that no lower-priority boarding or transfer can claim capacity before higher-priority groups have been accommodated. Additional complementarity pairs enforce demand exhaustion and equilibrium:

$$\begin{aligned} &0 \le f_{w,b}^r \perp c_{w,b}^r(f) + \sum_{A\in \text{priority}(r)} v_A - \mu_{w,b} \ge 0\ &0 \le \mu_{w,b} \perp \sum_{r \in R_w} f_{w,b}^r - d_{w,b} \ge 0 \end{aligned}$$

where $c_{w,b}^r(f)$ is the generalized cost, $\mu_{w,b}$ is the dual price of demand satisfaction, and the sum over $v_A$ covers those arcs in the route's boarding priority sequence (Feng et al., 12 Jan 2026).

Arc-Level NCP for Parametric Graphs

With a parametric weighted arc function

$$w_{ji}(\lambda) = \max \{P_{ij}+\lambda,\; I_{ij}-\lambda, \; C_{ij}\}$$

the arc-level structure enables determining all $\lambda$ such that the weight of every circuit (sequence of arcs) is non-positive by checking the aggregate arcwise sums over circuits. The problem is thus tractable via arc-level decomposition, with all necessary weights and intersection (“breakpoint”) information extractable directly from the arc parameters (Zorzenon et al., 2021).

Arc-Level NCP for Network Pricing

The "STD" arc-level MIP for the bilevel pricing problem models user reactions by associating binary variables $x_e^k$, $y_e^k$ (whether arc $e$ is used by user $k$), dual node potentials $\lambda_i^k$, leader tolls $T_e$, and auxiliary variables for linearization, all defined arcwise. Constraints enforce arcwise flow conservation, dual feasibility, and optimality by arc, avoiding path enumeration. Complementarity is imposed through duality (e.g., strong duality between follower primal and dual formulations, with arc-level variables) (Bui et al., 2021).

3. Illustrative Formulations from Core Applications

The arc-level NCP framework appears in multiple foundational models:

Application Area	Arc-Level Variables	Core Complementarity/Constraint
Transit Equilibrium (Feng et al., 12 Jan 2026)	$f_{w,b}^r$, $v_A$, $\mu_{w,b}$	$0 \le v_A \perp q_A(f) \ge 0$, $0 \le f_{w,b}^r \perp \dots$
Parametric Circuits (Zorzenon et al., 2021)	$w_{ji}(\lambda)$, $P_{ij}$, $I_{ij}$, $C_{ij}$	Circuit mean $\mu^*(\lambda) \le 0$ via arcwise aggregation
Bilevel Pricing (Bui et al., 2021)	$x_e^k$, $T_e$, $t_e^k$, $\lambda_i^k$	Arcwise flow balance, dual feasibility, bilinear toll linearization

These arc-level variables and constraints ensure that the global properties of the system (equilibrium, absence of profitable cycles, optimal follower reactions) can be expressed and solved efficiently as collections of arcwise conditions.

4. Algorithmic Properties and Solution Methods

Arc-level NCPs enable algorithmic improvements by leveraging the problem’s arcwise decomposition:

• In schedule-based transit assignment, the arc-level NCP admits a continuously differentiable Mathematical Program with Equilibrium Constraints (MPEC) reformulation by encoding the complementarity $0 \le v_A \perp q_A(f) \ge 0$ via the Fischer–Burmeister function:

$\varphi(a,b) = \sqrt{a^2+b^2} - (a+b)$

This yields a smooth merit function $\Psi$ and enables tractable solution by projected-gradient or SQP (Sequential Quadratic Programming) methods, with the lower-level equilibrium solved as a static arc-level assignment (Feng et al., 12 Jan 2026).

• In parametric graph NCP, the strongly polynomial $O(n^4)$ algorithm batches cycle-mean checks over intervals between $O(n^2)$ “breakpoints”, identified by pairwise arcwise function intersections. Weighted automaton techniques and parametric shortest-path routines exploit the arcwise structure to efficiently enumerate and check all relevant cases (Zorzenon et al., 2021).
• In the network pricing problem, the arc-level MIP grows only polynomially with $|V|, |E|, |K|$ and requires no path enumeration, unlike path-based formulations. Bilinear terms are linearized by introducing auxiliary arcwise variables and big-M constraints (Bui et al., 2021).

5. Theoretical Guarantees and Behavioral Consistency

Arc-level NCP formulations provide provable algorithmic and behavioral properties not generally attained by route-level or aggregate formulations.

• Existence of equilibria is guaranteed under mild continuity and nondegeneracy conditions, as shown for both transit assignment and parametric NCP settings (Feng et al., 12 Jan 2026, Zorzenon et al., 2021).
• Multiple equilibria can exist in general; however, the arc-level refinement eliminates certain “behaviorally questionable” or anomalous solutions. For example, in transit assignment, atomic arcwise enforcement of boarding and continuance priority rules out paradoxes (e.g., two-stage boarding anomalies) that persist under route-level or group-level constraints (Feng et al., 12 Jan 2026).
• In parametric circuit problems, the maximum circuit mean $\mu^*(\lambda)$ is provably convex and piecewise linear in $\lambda$, and the set of feasible $\lambda$ (for which all circuits are non-positive) is exactly the union of intervals captured by the arc-level procedure. The $O(n^4)$ runtime holds tightly for edge-wise encoding (Zorzenon et al., 2021).

6. Applications and Comparative Advantages

Arc-level NCP formulations are critical in several domains:

• Transit Assignment: The arc-level NCP enforces continuance and first-come-first-served (FCFS) priority directly, capturing queuing phenomena and departure-time shifts driven by capacity constraints. It bypasses costly route enumeration, yielding greater scalability and behavioral realism (Feng et al., 12 Jan 2026).
• Network Pricing: Arc-level single-level MIPs dominate the literature for the bilevel pricing problem, as they do not require path enumeration and allow for robust, scalable computations even as network complexity grows (Bui et al., 2021).
• Parametric Weighted Graphs: The arc-level NCP enables efficient, systematic characterization of feasible parameter regimes by decomposing the global problem into a finite, explicitly tractable collection of arc interactions (Zorzenon et al., 2021).

A plausible implication is that arc-level NCP formulations represent a fundamental methodological advance, both theoretically in equilibrium and feasibility analysis, and practically for their computational tractability across large, real-world instances.

7. Illustrative Example: Priority in Transit Assignment

In a transit network with two lines, multiple runs, and several boarding origins, the arc-level NCP ensures that at a given stop and run, (i) dwelling arcs (continuance) always have sufficient capacity (if $q_A > 0$), (ii) boarding arcs respect FCFS ordering (seats allocated in queue order if $q_A > 0$), and (iii) only the lowest-priority transfer arc may be capacity-limited ($q_A = 0$, $v_A > 0$), enforcing strict atomic priority (Feng et al., 12 Jan 2026). This yields a unique, behaviorally consistent equilibrium unattainable through group-averaged or route-level NCPs.

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Arc-level NCP formulations are thus essential in operational research and network optimization, combining theoretical rigor with algorithmic feasibility, and enabling granular, behaviorally-representative models for a wide range of applications.