TA-TSP: Trigger Arc Traveling Salesman Problem

• TA-TSP is a traveling salesman variant where arc feasibility and travel costs are dynamically determined by prior route events and trigger conditions.
• Modeling approaches for TA-TSP include state augmentation of nodes and functional constraint encoding, which integrate route history into feasibility and cost evaluation.
• The framework bridges continuous-time optimal control, discrete augmented graphs, and quantum methodologies, highlighting open avenues for optimization under dynamic constraints.

Searching arXiv for papers relevant to “Trigger Arc Traveling Salesman Problem” and closely related TSP variants. {"query":"all:\"Trigger Arc Traveling Salesman Problem\" OR all:\"TA-TSP\" OR (all:\"traveling salesman\" AND (all:\"trigger arc\" OR all:\"conditional edge\" OR all:\"precedence\"))","max_results":10,"sort_by":"submittedDate","sort_order":"descending"} Searching for broader arXiv work on TSP variants with state-dependent or arc-dependent constraints. {"query":"all:\"traveling salesman\" AND (all:\"graphs of convex sets\" OR all:\"optimal control\" OR all:\"drones\" OR all:\"quantum iterative\")","max_results":10,"sort_by":"submittedDate","sort_order":"descending"} Trigger Arc Traveling Salesman Problem (TA-TSP) is best understood, in the present arXiv literature considered here, as a TSP-style constrained variant in which route feasibility or arc cost depends on prior route events such as trigger-arc, precedence, or conditional-edge rules. None of the cited papers provides a discrete graph-theoretic formulation under the exact name “TA-TSP.” Instead, the available work delineates the concept indirectly: one paper develops a continuous-time functional framework in which arcs and vertices are themselves functionals and constraint activation is trajectory-dependent; another encodes TSP as a visited-set-augmented shortest-path problem whose admissible transitions depend on state; and a third presents a quantum solver for the ordinary symmetric and asymmetric TSP, explicitly noting that trigger-arc or conditional-edge constraints are outside its scope (Ross et al., 2020, Luna et al., 7 Apr 2026, Rodríguez-Almazán et al., 10 Jun 2026).

1. Terminological status and scope

Within the cited material, “TA-TSP” is not a standardized problem statement but a descriptive umbrella for constrained TSP models that go beyond the classical minimum-cost Hamiltonian cycle. The clearest negative delimitation comes from the quantum paper "Quantum iterative approach to the Traveling Salesman Problem" (Rodríguez-Almazán et al., 10 Jun 2026), which states that it treats a standard TSP solver, including both symmetric and asymmetric instances, and does not introduce “trigger-arc / precedence / conditional-edge / TA-TSP-style constrained” structure. Its problem remains: given $m$ cities, find a tour that visits each city exactly once and returns to the start while minimizing total cost.

This contrast is useful because it identifies what TA-TSP would have to add. A trigger-arc interpretation requires route-dependent admissibility or route-dependent cost formation, rather than merely symmetric versus asymmetric edge weights. In that sense, asymmetry alone is insufficient: the asymmetric modification $c'=\frac{c}{1-\Delta y/2}$ in the 4-city example of (Rodríguez-Almazán et al., 10 Jun 2026) still defines an ordinary ATSP, not a trigger-conditioned problem.

A compact comparison of the cited literature is therefore appropriate.

Paper	Relation to TA-TSP	Main relevant mechanism
(Rodríguez-Almazán et al., 10 Jun 2026)	Not TA-TSP directly	QPE + Grover-Long for standard TSP/ATSP
(Ross et al., 2020)	Conceptual precursor	Trajectory-dependent functionals on vertices and arcs
(Luna et al., 7 Apr 2026)	Structural analogue	Visited-set augmentation and state-dependent transitions
(Morandi et al., 2022)	Terminological contrast	Arc retraversal in truck-drone routing

This suggests that TA-TSP is presently better characterized by its modeling requirements than by a settled canonical formulation on arXiv.

2. Core structural ideas behind trigger-dependent TSP models

Across the cited works, the common thread most relevant to TA-TSP is state dependence. In a trigger-arc formulation, the legality or cost of choosing an arc cannot be read from a static edge weight matrix alone; it depends on what has already happened along the route. The functional-optimal-control framework of (Ross et al., 2020) states this most directly by remarking that the objective must account for functional dependence of the sequence of cities and that the values of arc weights are not independent of the path. The augmented-graph construction of (Luna et al., 7 Apr 2026) realizes an analogous idea discretely: the available outgoing transitions depend on the currently visited subset of targets, so the same physical move is represented differently depending on state.

The implication for TA-TSP is precise. A trigger mechanism can be modeled in at least three nonexclusive ways suggested by these papers. First, the feasible state space can be restricted so that only trigger-consistent partial tours exist. Second, the transition system can be augmented so that route history becomes part of the node label. Third, violations can be absorbed into the objective through penalties large enough to prevent infeasible routes from becoming minima. These are not presented in the cited papers as a unified TA-TSP doctrine, but they recur as the operative design choices for route-dependent TSP variants.

The same theme also clarifies what is not sufficient. Merely allowing revisits or asymmetric costs does not by itself create trigger-arc logic. Likewise, representing a route as a Hamiltonian cycle may be too rigid when route validity depends on intermediate conditions rather than only on endpoint incidence. The literature considered here therefore points toward TA-TSP as a problem of history-aware route encoding rather than only combinatorial tour enumeration (Ross et al., 2020, Luna et al., 7 Apr 2026, Morandi et al., 2022).

3. Continuous-time functional formulations as a precursor

The most general precursor among the cited works is "An Optimal Control Theory for the Traveling Salesman Problem and Its Variants" (Ross et al., 2020). It does not define TA-TSP explicitly, but it replaces the ordinary discrete graph model with an $\mathcal{F}$-graph in which arcs and vertices are functionals. An $\mathcal{F}$-control is a sequence $\langle \psi_0,\psi_1,\ldots,\psi_n\rangle$ of measurable functions, and a control walk alternates consistently between vertex-functionals and arc-functionals. The specialization to a $\mathcal{T}$-graph introduces a label space $L$, partitioned into measurable subsets associated with vertices and arcs, so that a continuous label-space trajectory generates a control walk.

This formulation is relevant to TA-TSP because visitation and feasibility become path-dependent. The time-on-task functional

$$T^i[(\cdot), t_0, t_f] := \int_{t_0}^{t_f} \mathcal{I}((t), L^i(t))\, dt$$

defines whether a vertex $i$ is visited through the condition $T^i[(\cdot), t_0, t_f]\ne 0$. The derivative-based functional

$c'=\frac{c}{1-\Delta y/2}$0

then yields a Hamiltonicity criterion: the trajectory is Hamiltonian if and only if $c'=\frac{c}{1-\Delta y/2}$1 for all $c'=\frac{c}{1-\Delta y/2}$2. The paper emphasizes that, unlike discrete formulations, explicit subtour elimination constraints are unnecessary because continuity enforces a single closed control walk.

Several variant formulations display trigger-like behavior even though they are not named TA-TSP. In orienteering with neighborhoods, reward is activated only if a thresholded return condition is met: $c'=\frac{c}{1-\Delta y/2}$3 In the fast/minimum-time TSP,

$c'=\frac{c}{1-\Delta y/2}$4

so a visit becomes valid only after sufficient dwell time. Time windows are incorporated by augmenting the label space, $c'=\frac{c}{1-\Delta y/2}$5. At the most general level, the $c'=\frac{c}{1-\Delta y/2}$6 optimal-control problem allows constraint functionals $c'=\frac{c}{1-\Delta y/2}$7 to encode time-on-task, walk indicators, degree constraints, capacity constraints, and others. For TA-TSP interpretation, this is the broadest available template: a trigger can be represented as a trajectory-activated functional constraint or reward rather than as a static edge annotation.

4. Augmented-state graph formulations and exact shortest-path equivalence

"Augmented Graphs of Convex Sets and the Traveling Salesman Problem" (Luna et al., 7 Apr 2026) provides the closest discrete structural analogue to TA-TSP among the cited works. The paper studies a robot in $c'=\frac{c}{1-\Delta y/2}$8 with target sets $c'=\frac{c}{1-\Delta y/2}$9, formulates the TSP requirement in STL as eventual visitation of all targets, and then constructs an augmented graph of convex sets (AGCS) in which nodes are annotated by the subset of targets already visited. The TSP is thereby transformed into a shortest-path problem in the augmented state graph.

The key TA-TSP-relevant feature is that admissible transitions depend on state. The AGCS consists of copies of the original graph, each copy encoding a specific visited set. These copies are arranged in layers by visited-set cardinality: the base layer contains the initial target, intermediate layers correspond to subsets of increasing size, and the final layer represents the full visited set. Directed edges connect only compatible subset states, i.e., from a graph copy for $\mathcal{F}$0 to one for $\mathcal{F}$1. The paper states that these directed edges between subgraphs prevent backtracking in a dynamic-programming style.

This is not an explicit trigger-arc rule of the form “if arc $\mathcal{F}$2 is selected, then arc $\mathcal{F}$3 must be selected,” but it is very close structurally. The current visited-set state determines which transition copies are available. A plausible implication is that many TA-TSP constraints can be re-expressed as state augmentation, with triggers encoded in the lifted graph rather than in the base graph. The paper’s exactness claim is correspondingly strong: every path in the AGCS-TSPS from the initial target set to the terminal target set satisfies the traveling salesman specifications, and a shortest such path corresponds to an optimal solution of the original problem.

The connection to Bellman-Held-Karp is explicit. Both the AGCS-TSPS and Bellman-Held-Karp use binary subsets, sort them by size, and reuse optimal partial paths when extending to larger subsets. In the AGCS construction this reuse is embedded directly as a layered graph. The paper also notes that the network flow constraints of the shortest-path formulation replace the degree constraints of each target set, while the directed edges between and within subgraphs replace subtour constraints. For TA-TSP, the conceptual significance is that trigger dependence can be enforced by reachable-state design rather than only by separate logical constraints.

5. Quantum route-cost encoding and why the published solver is not TA-TSP

The paper "Quantum iterative approach to the Traveling Salesman Problem" (Rodríguez-Almazán et al., 10 Jun 2026) proposes a quantum iterative framework that combines Quantum Phase Estimation (QPE) with Grover-Long amplitude amplification. A route register $\mathcal{F}$4 uses $\mathcal{F}$5 qubits, and the cost register $\mathcal{F}$6 uses $\mathcal{F}$7 qubits with $\mathcal{F}$8. Route costs are normalized into $\mathcal{F}$9, encoded as phases, and then read through QPE. The route-cost operator is assembled from diagonal operators $\mathcal{F}$0 into

$\mathcal{F}$1

so that applying $\mathcal{F}$2 to a route basis state accumulates the route cost in the eigenphase. After QPE, Grover-Long exact amplitude amplification iteratively marks routes below a threshold and updates that threshold in a Dürr–Høyer-style minimum-finding loop.

The expected complexity claimed in the paper is

$\mathcal{F}$3

with the explicit caveat that the initialization term $\mathcal{F}$4 is omitted and is expected to become subdominant only after future refinements. The proof-of-concept is toy-scale: a 4-city Salamanca example is used for both symmetric and asymmetric instances, and finite precision causes measured costs to fluctuate around tabulated values without, according to the authors, affecting route separation.

For TA-TSP, however, the paper’s direct relevance lies in its limitations. It explicitly does not address trigger-arc, precedence, or conditional-edge constraints. It does state how the framework could be adapted in principle. Four modifications are identified. Feasible-route encoding would need to restrict the superposition to TA-TSP-feasible tours or penalize infeasible tours heavily. Constraint-aware oracle design would require the Grover-Long marking oracle to identify only routes satisfying trigger or predecessor constraints. Modified cost operator $\mathcal{F}$5 would have to encode penalties or conditional costs so that infeasible routes are not selected as minima. State preparation $\mathcal{F}$6 would need to generate only feasible tours, or at least be strongly biased toward them. The paper therefore provides a quantum search-and-evaluation skeleton for ordinary TSP, not a TA-TSP solver.

6. Arc retraversal, terminological pitfalls, and adjacent variants

A separate but important terminological issue is raised by "The TSP with drones: The benefits of retraversing the arcs" (Morandi et al., 2022). This paper is not about trigger arcs in the sense of conditional activation; instead, it studies whether optimal truck routes in a truck-and-drone setting may need to retraverse the same arc multiple times. The truck route is modeled as a closed walk rather than a Hamiltonian cycle, and the paper proves that revisiting nodes and retraversing arcs are distinct phenomena.

This distinction matters for TA-TSP because “arc-triggered” and “arc-retraversing” can be conflated, but they refer to different structures. TA-TSP-style formulations concern dependence of cost or feasibility on route history; the truck-drone paper concerns multiplicity of arc usage in a richer routing model. The latter shows that route structure may need to exceed simple cycles. Under restrictive conditions—single drone, invertible sorties, and either one-customer sorties or the original FSTSP-like assumptions—at least one optimal solution is not arc-retraversing. Outside those conditions, the paper gives Euclidean counterexamples in which all optimal solutions are arc-retraversing, including cases where forbidding retraversal increases the optimum from $\mathcal{F}$7 to $\mathcal{F}$8 and from $\mathcal{F}$9 to $\langle \psi_0,\psi_1,\ldots,\psi_n\rangle$0.

The methodological lesson for TA-TSP is cautious rather than direct. One should not assume that a Hamiltonian-cycle representation remains adequate once route feasibility depends on auxiliary operations, state changes, or conditional logic. This does not imply that TA-TSP requires retraversal, because that claim is not made in the cited work. It does suggest that trigger-dependent routing models may require richer path spaces than those used in classical TSP.

7. Synthesis and open modeling directions

Taken together, the cited arXiv papers position TA-TSP as an emerging interpretive category rather than a fully canonical problem statement. The functional-optimal-control framework of (Ross et al., 2020) shows how arc and vertex behavior can depend on the trajectory itself, how visitation can be defined by measurable-function integrals, and how continuity can subsume subtour elimination. The augmented-state construction of (Luna et al., 7 Apr 2026) shows that history-dependent admissibility can be encoded exactly by visited-set state augmentation, reducing TSP to shortest path in a layered graph. The quantum paper (Rodríguez-Almazán et al., 10 Jun 2026) shows that, for ordinary TSP, route cost can be phase-encoded and searched iteratively, while also specifying the extra machinery TA-TSP would require. The truck-drone work (Morandi et al., 2022) clarifies that arc multiplicity is a separate modeling axis and that route-history effects may force a departure from simple cycle-based representations.

A plausible synthesis is that TA-TSP can be formulated along either of two principal lines. One line is state augmentation: triggers are compiled into augmented nodes or subgraphs so that only trigger-consistent transitions exist. The other is functional constraint encoding: triggers are represented by path-dependent costs, rewards, dwell conditions, or feasibility functionals. A hybrid of these two approaches is also suggested by the literature, though not explicitly developed there.

What remains absent from the cited arXiv record is an explicit discrete formulation with named trigger-arc variables and constraints of the form “if a route segment is taken, a dependent segment becomes available, mandatory, or penalized.” The literature reviewed here therefore supports TA-TSP most strongly as a conceptual and structural generalization of TSP, rather than as a settled standalone model with a universally adopted formal definition.