Moving Target TSP: Models & Algorithms

• Moving Target TSP is a dynamic generalization of the classic TSP where moving targets with known trajectories and time windows must be intercepted.
• Exact formulations use mixed-integer conic programming to replace big-M models, yielding tighter relaxations and significant runtime improvements.
• Extensions involving obstacles, multiple agents, and complex kinematics broaden MT-TSP’s application in dynamic routing and interception tasks.

The Moving Target Traveling Salesman Problem (MT-TSP) is a dynamic generalization of the Traveling Salesman Problem in which the locations to be visited are no longer static points but targets moving along known trajectories, typically under time-window constraints. In its standard single-agent form, an agent starts at a depot, intercepts each target exactly once within an admissible time interval, and then returns to the depot; depending on the formulation, the objective is to minimize either total tour duration or total traveled distance. The problem subsumes TSP with time windows when target velocities are zero, and it becomes substantially richer once obstacles, multiple agents, moving obstacles, or nontrivial kinematics are introduced (Philip et al., 2024, Bhat et al., 2024).

1. Problem definition and structural properties

A common formalization specifies target trajectories $\tau_i : \mathbb{R}^+ \to \mathbb{R}^2$ or $p_i(t)$ together with one or more time windows, and an agent trajectory $\tau_A$ or $p_{\text{agent}}$ subject to a speed bound. An interception occurs when there exists a time $t$ in one of the target’s admissible windows such that the agent and target coincide in space, for example

$\tau_A(t)=\tau_i(t)$

or equivalently

$p_{\text{agent}}(t_i)=p_i(t_i).$

The agent must satisfy kinematic feasibility, begin at the depot, and in depot-returning variants terminate there as well. In obstacle-aware variants, the trajectory must remain in the free space $F\subset \mathbb{R}^2$ or $Q\subset \mathbb{R}^3$ (Bhat et al., 2024, Philip et al., 2024).

The literature represented here uses several nested model classes. Some formulations assume a single time window per target and linear target motion in the plane; others allow multiple windows and piecewise-constant or piecewise-linear velocities. Obstacle-free models are the simplest exact setting, MT-TSP-O adds static polygonal or polyhedral obstacles, MT-TSP-MO adds moving obstacles, and MA-MT-TSP introduces multiple homogeneous agents starting from a common depot (Philip et al., 2024, Philip et al., 10 Jan 2025).

These models inherit the combinatorial difficulty of static routing while adding continuous-time interception geometry. When target velocities are set to zero, MT-TSP reduces to TSP with time windows; when time windows are also trivial, it reduces to classical static TSP. Because TSP with time windows is already NP-complete, the feasibility problem for MT-TSP-O is NP-complete, while the optimization problem is NP-hard (Bhat et al., 2024, Philip et al., 2024).

2. Space–time formulations and exact conic optimization

A central exact modeling idea is to embed MT-TSP in space–time. When targets move linearly during their time windows, each target’s feasible visitation events form a line segment in $(x,y,t)$-space,

$p_i(t)$0

so the routing problem can be reinterpreted as a shortest-path problem on a graph of convex sets. This leads to mixed-integer second-order cone formulations in which edge feasibility is enforced by conic travel constraints and time-window membership is encoded by perspective constraints rather than big-$p_i(t)$1 switching (Philip et al., 2024).

In the single-agent obstacle-free setting, the graph-of-convex-sets formulation in (Philip et al., 2024) replaces a baseline big-$p_i(t)$2 MICP with a perspective-based MICP-GCS. The resulting continuous relaxation is substantially tighter: the reported lower-bound ratio of the relaxed MICP-GCS ranges between $p_i(t)$3 and $p_i(t)$4 of the best bound with integrality, whereas the relaxed baseline MICP yields ratio $p_i(t)$5. On instances with up to 20 targets, the formulation is reported to outperform the prior MICP by up to two orders of magnitude in runtime and by up to a $p_i(t)$6 tighter optimality gap (Philip et al., 2024).

The same conic viewpoint extends to multiple agents. In the MA-MT-TSP formulation of (Philip et al., 10 Jan 2025), piecewise-linear target trajectories and multiple time windows are handled by a new MICP that restates a state-of-the-art formulation as a nonconvex MINLP and then removes the explicit binary–continuous products via a new conic reformulation. The model uses a single edge-selection layer with an integer variable $p_i(t)$7 for the number of active tours, rather than per-agent edge variables throughout. Computationally, the paper reports up to two orders of magnitude reduction in runtime and over $p_i(t)$8 improvement in optimality gap relative to the state-of-the-art MA-MT-TSP formulation (Philip et al., 10 Jan 2025).

A notable boundary emerges across these exact approaches: the strongest conic formulations in the cited work assume linear or piecewise-linear target motion with deterministic trajectories and relatively simple agent dynamics. This suggests that exact conic optimization is presently most effective when the interception manifold can be represented as convex structure in space–time.

3. Complete algorithms with static obstacles

Static obstacles change the problem qualitatively because pairwise travel costs are no longer given by free-space Euclidean geometry. In 2D, the MT-TSP-O formulation of (Bhat et al., 2024) introduces polygonal obstacles, piecewise constant target velocities within time windows, and a speed-limited omnidirectional agent. Its main contribution is MTVG-TSP, described as the first complete algorithm for finding feasible solutions to MT-TSP-O. The method searches over sequences of target-window choices and, for each extension step, solves a point-to-moving-target subproblem via a Moving Target Visibility Graph (MTVG), a generalization of the classical visibility graph that incorporates time-dependent visibility intervals to moving targets (Bhat et al., 2024).

At the local level, the MTVG-based point-to-moving-target planner is solved by A* with time-dependent edge costs. The crucial edge primitive is the shortest feasible travel from a static point $p_i(t)$9 at time $\tau_A$0 to a moving target window $\tau_A$1,

$\tau_A$2

where $\tau_A$3 encodes time intervals during which the target is line-of-sight visible from $\tau_A$4. The paper proves optimality for this subproblem and completeness for the overall depth-first search over window-node sequences. Empirically, on 570 instances with up to 30 targets, MTVG-TSP finds feasible solutions in all cases where a sampled-points baseline does, and in a critical range of total window lengths it achieves up to 38 times less computation time (Bhat et al., 2024).

In 3D, visibility-graph arguments are no longer sufficient, and (Bhat et al., 20 Apr 2025) replaces them with a graph-of-convex-sets low-level search. The resulting FMC*-TSP interleaves a high-level GTSP-TW over target-window nodes with a low-level FMC* search on a GCS in space–time. The low level operates on convex free-space regions $\tau_A$5 lifted to $\tau_A$6, together with window nodes represented as line segments in space–time. For a user-specified factor $\tau_A$7, the algorithm guarantees

$\tau_A$8

and is described as the first complete and bounded-suboptimal algorithm for MT-TSP-O in $\tau_A$9. On 280 instances with up to 40 targets, it is reported to have smaller median runtime than a sampled-points baseline (Bhat et al., 20 Apr 2025).

These static-obstacle results clarify two distinct algorithmic regimes. In 2D, completeness can be anchored to moving-target visibility structure; in 3D, completeness and bounded suboptimality are obtained instead through convex decomposition, implicit GCS search, and prefix-based pruning.

4. Lower bounds, trajectory sampling, and asymptotically optimal search

Not all MT-TSP research pursues exact tours directly. A second major line focuses on lower bounds and asymptotically convergent approximations.

The C$p_{\text{agent}}$0 framework in (Philip et al., 2023) derives lower bounds by relaxing continuity at visits. Each target trajectory is partitioned into trajectory-interval nodes, and when the agent “visits” such an interval it may arrive at any point in the segment and depart from any point in the same segment. This converts the relaxed problem into a GTSP whose edge costs are given by a Shortest Feasible Travel (SFT) subproblem. The paper proves that the resulting optimum is a valid lower bound on the original MT-TSP and that the bound becomes asymptotically tight as the discretization is refined. On instances with up to 15 targets, C$p_{\text{agent}}$1 outperforms the SOCP-based exact method on 15-target straight-line instances, and for the general piecewise-linear case the reported feasible solutions lie on average within approximately $p_{\text{agent}}$2 of the lower bounds (Philip et al., 2023).

A complementary strategy appears in the Iterated Random Generalized TSP framework of (Bhat et al., 10 Sep 2025), which targets settings with generic nonlinear target trajectories or kinematic constraints for which the paper states no prior algorithm guarantees convergence to an optimal MT-TSP solution. IRG alternates between randomly sampling interception points in configuration–time and solving the resulting GTSP. Two parallel instantiations are given: IRG-PGLNS, which uses a parallelized extension of GLNS, and PCG, which solves several GTSPs simultaneously with communication through informed sample sets. Under the stated assumptions, the framework is probabilistically complete and asymptotically optimal. The paper reports numerical results for three variants—close-enough interception, variable-speed Dubins interception, and a redundant robot arm—and states that both IRG-PGLNS and PCG converge faster than a baseline based on prior work (Bhat et al., 10 Sep 2025).

Taken together, these two approaches define a distinct methodological axis. C$p_{\text{agent}}$3 uses relaxation to obtain certified lower bounds on the continuous problem, whereas IRG uses repeated resampling and GTSP improvement to obtain asymptotic convergence to the optimum in settings where exact conic formulations are not available.

5. Major variants and extensions

The core MT-TSP abstraction has been generalized in several directions, each altering the balance between combinatorial search, geometric planning, and continuous optimization.

The multi-agent problem, MA-MT-TSP, requires assigning moving targets to several homogeneous agents while preserving exact-once visitation over all agents. In (Philip et al., 10 Jan 2025), the agents share a stationary depot and all targets must be visited exactly once within one of their time windows. The formulation accommodates piecewise-linear target motion and multiple time windows, and its conic reformulation is presented as an exact model rather than a heuristic decomposition (Philip et al., 10 Jan 2025).

Moving obstacles yield MT-TSP-MO, where the agent must avoid convex polygons whose centers follow known linear trajectories. The formulation in (Philip et al., 17 Jun 2026) introduces both an MICP and a Two-Phase Bilevel Search (TPBS) algorithm. TPBS combines a high-level GTSP over discretized trajectory points with low-level validation in a time-augmented graph that enforces dynamic collision avoidance. On a broad range of instances with up to 40 targets and 40 obstacles, both the MICP and TPBS are reported to outperform the existing baseline in success rates, solution costs, and computation time; TPBS achieves a 100% success rate across all combinations tested in the reported experiments (Philip et al., 17 Jun 2026).

Generic kinematic constraints broaden the notion of interception beyond omnidirectional point motion. In (Bhat et al., 10 Sep 2025), the interception predicate $p_{\text{agent}}$4 is abstract, and the agent configuration space may be $p_{\text{agent}}$5, $p_{\text{agent}}$6, or a 7-DOF joint space. The paper instantiates this abstraction for close-enough interception in the plane, variable-speed Dubins motion, and redundant robot-arm interception of moving end-effector poses. A plausible implication is that GTSP-based MT-TSP methods can remain viable well beyond Euclidean point-robot models, provided the primitives $p_{\text{agent}}$7, $p_{\text{agent}}$8, and $p_{\text{agent}}$9 are available (Bhat et al., 10 Sep 2025).

Several of the cited works also identify natural next extensions: multiple agents built atop MTVG subproblems, kinodynamic edge models that replace straight-line steering, stochastic or probabilistic target motion, richer convex representations of feasible motion in space–time, and moving-obstacle versions beyond the specific TPBS and MICP constructions already reported (Bhat et al., 2024, Philip et al., 2024).

6. Related models, recurrent misconceptions, and empirical patterns

A recurring misconception is to treat every time-varying TSP model as MT-TSP in the strict interception sense. The “moving sites” and “reallocation” models studied in “Trajectory stability in the traveling salesman problem” do not model concurrent agent and target motion during tour execution. Instead, they analyze time-dependent families of static Euclidean TSP instances under a timescale-separation assumption: city positions change between tours, but are frozen during each tour. The paper is therefore directly relevant to dynamic rank stability, but not to the continuous interception problem formalized in MT-TSP, MT-TSP-O, or MT-TSP-MO (Sánchez et al., 2017).

That distinction matters because many algorithmic pathologies in true MT-TSP arise from continuous-time interception geometry rather than from repeated static replanning alone. One empirical pattern, documented explicitly in obstacle-aware work, is that naive time sampling can miss narrow usable subintervals inside long windows. In (Bhat et al., 2024), the sampled-points baseline is incomplete because feasible intercept times may occupy only a small fraction of a target’s windows, whereas MTVG-TSP reasons exactly over continuous-time window intervals. In (Philip et al., 17 Jun 2026), repeated GTSP resolution with straight-line feasibility checks degrades sharply as the number of targets and moving obstacles increases. These observations help explain why continuous-time visibility reasoning, GCS-based planning, or exact conic relaxations often outperform purely sampled graph constructions (Bhat et al., 2024, Philip et al., 17 Jun 2026).

A second empirical pattern concerns relaxation strength. Big-$t$0-based conic formulations of MT-TSP can be exact in principle, but their continuous relaxations may be too weak to guide branch-and-bound effectively. The perspective-based MICP-GCS model in (Philip et al., 2024) sharply improves lower bounds relative to the baseline MICP, and the C$t$1 lower-bounding framework in (Philip et al., 2023) shows that carefully structured relaxations can remain informative even when exact global optimization becomes difficult. This suggests that MT-TSP performance hinges as much on the quality of space–time lower bounds as on the discrete search strategy itself (Philip et al., 2024, Philip et al., 2023).

Across the cited literature, MT-TSP has thus evolved into a family of tightly coupled routing, scheduling, and motion-planning problems. The dominant abstractions are now space–time convex sets, target-window graphs, shortest-feasible-travel primitives, and GTSP or GTSP-TW reductions; the dominant guarantees are exactness on restricted linear-motion models, completeness for obstacle-feasibility formulations, bounded suboptimality in 3D obstacle settings, and asymptotic optimality for generic nonlinear or kinematically rich interception problems.