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arxiv: 2605.00010 · v1 · submitted 2026-03-11 · 🧮 math.OC

The Keplerian Traveling Salesperson Problem

Max Bannach , Giacomo Acciarini , Dario Izzo This is my paper

Pith reviewed 2026-05-15 12:29 UTC · model grok-4.3

classification 🧮 math.OC
keywords Keplerian traveling salesperson problemtime-unfoldingtime-expanded networkinterplanetary trajectory optimizationinteger linear programmingdynamic discretization discoveryspace mission designbenchmark suite
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The pith

The continuous problem of visiting multiple moving orbital targets can be exactly recast as a discrete optimization task on a time-expanded network.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper formalizes the Keplerian Traveling Salesperson Problem in which a spacecraft must rendezvous with several bodies whose positions change under Keplerian dynamics, producing time-dependent and asymmetric transfer costs. This models practical tasks such as debris removal or asteroid-belt surveys. A time-unfolding construction converts the continuous trajectory problem into a discrete network problem whose nodes represent states at chosen times. Standard integer-linear-programming solvers can then be applied directly to the resulting instances. The authors also supply a benchmark set and supporting heuristics so that the same problems become accessible to researchers outside astrodynamics.

Core claim

The central claim is that the Keplerian TSP admits a rigorous time-unfolded formulation that turns the original continuous, time-dependent planning task into a discrete optimization problem on a time-expanded network; this network can be solved exactly by integer-linear-programming methods that incorporate Interval-based Dynamic Discretization Discovery, and the same representation yields a publicly released benchmark suite together with optimality-preserving preprocessing and heuristic routines.

What carries the argument

The time-unfolding technique that builds a time-expanded network whose discrete paths correspond to feasible continuous Keplerian trajectories.

If this is right

  • Small-to-medium multi-rendezvous missions become solvable to proven optimality with off-the-shelf ILP solvers.
  • The released benchmark set supplies a standard test bed for comparing exact and heuristic methods on astrodynamic TSP variants.
  • Preprocessing routines that preserve optimality can be used to reduce instance size before calling any solver.
  • The same time-expanded encoding supports both exact solution and the construction of fast improvement heuristics.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same unfolding idea could be tested on higher-fidelity dynamics that include perturbations once the discretization error is quantified.
  • Integration of the network model with existing mission-design tools would allow automatic generation of feasible itineraries for active-debris-removal campaigns.
  • The benchmark instances may stimulate development of specialized branch-and-cut routines that exploit orbital symmetry.

Load-bearing premise

The chosen time discretization and dynamic discovery procedure keep the continuous orbital dynamics close enough that the discrete solutions remain feasible and near-optimal for the mission sizes considered.

What would settle it

A concrete instance in which a discrete network solution, when simulated with the true continuous Keplerian dynamics, violates a rendezvous window or exceeds the reported cost bound by more than the solver tolerance.

Figures

Figures reproduced from arXiv: 2605.00010 by Dario Izzo, Giacomo Acciarini, Max Bannach.

Figure 1
Figure 1. Figure 1: Illustration of the progression from classical Euclidean traveling salesperson problem (left), [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: illustrates the introduced concepts and the resulting graph G [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: Left: Intervals λα, λβ, and λγ corresponding to bodies α, β, γ. The x-axis illustrates time, and black dots within the intervals indicate the time points used for the transfer between the intervals. A solution of ΓI can use both edges, while a solution of Γktsp cannot. Right: A subtour in a time-interval network starting and ending at λγ. The Flow Constraint does not exclude solutions that select this cycl… view at source ↗
Figure 5
Figure 5. Figure 5: The DDD algorithm for ktsp as schematic flowchart. Red boxes describe processes, while green boxes define decisions. The concrete realization of the initial interval par￾tition and the refinement step is not relevant, as long as the refinement partitions the interval that is partici￾pating in the conflict. (M, A, t0, tmax, αs) • compute T • compute initial interval partition refine interval partition solve… view at source ↗
Figure 6
Figure 6. Figure 6: Illustration of (a) the Vee Rule, (b) the Shortcut Rule, and (c) the Far Away Rule. 4.2 A Constructive Heuristic to Find an Initial Solution In order to quickly generate an initial solution, we use an adaptation of the insertion heuristics for the asymmetric traveling salesman problem with time windows [1]. For our setting, we introduce the permutation-schedule representation of trajectories. In this repre… view at source ↗
Figure 7
Figure 7. Figure 7: Sample solutions from the benchmark set for both the asteroid belt and Jovian moons cases. [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Average number of arcs removed from time [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Comparison of the direct encoding of Section 2.2 with the dynamic discretization encoding [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The init heuristic that generates initial feasible trajectories by successively inserting elements to a growing trajectory. 8.3 Details of the Improving Heuristic As in the previous section, we will assume for simplicity that all transfers are feasible. Hence, every pair (π, σ) represents a feasible trajectory P and we can define val(π, σ) := val(P). Let us further call a trajectory 2-opt if the swap heur… view at source ↗
Figure 11
Figure 11. Figure 11: The details of the b-swan heuristic. As input it obtains a time-expanded network G, a set S of feasible trajectories in permutation-schedule representation, a beamwidth w ≥ n, a shrink-factor f ≤ 1, and a value k > 2 indicating that perturbations are performed using k-swaps. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_11.png] view at source ↗
read the original abstract

We address a fundamental challenge in space mission design and space logistics: planning interplanetary trajectories for missions that must rendezvous with multiple bodies. Such mission occur, for instance, in active debris removal, in-orbit servicing, or asteroid belt exploration. We model these problems as a variant of the Traveling salesperson problem (TSP), which we term the Keplerian TSP (KTSP). Unlike the well-studied TSP, the KTSP accounts for the motion of orbital targets, leading to time-dependent and asymmetric transfer costs that capture key real-world effects in astrodynamics. We provide a rigorous formalization of the KTSP and release a benchmark suite to support its study. Central to our approach is a time-unfolding technique that reformulates the continuous problem as a discrete optimization task in a time-expanded network. This representation makes the benchmark accessible to researchers in discrete optimization even without prior knowledge of celestial mechanics. We also develop an alternative encoding as an integer linear program using Interval-based Dynamic Discretization Discovery to handle the time-dependent nature of transfers. We leverage state-of-the-art ILP solvers to solve the KTSP instances, accompanied by a detailed computational study that highlights their strengths and limitations. We complement these exact methods with an initial solution heuristic, an improvement heuristic, and preprocessing routines that preserve optimality.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper formalizes the Keplerian Traveling Salesperson Problem (KTSP) as a time-dependent, asymmetric TSP variant for rendezvous missions with moving orbital targets. It introduces a time-unfolding reformulation that converts the continuous problem into a discrete optimization task over a time-expanded network, develops an integer linear programming encoding via Interval-based Dynamic Discretization Discovery (IDDD), releases a benchmark suite, and reports computational results from state-of-the-art ILP solvers together with initial and improvement heuristics.

Significance. If the discretization scheme produces solutions whose extracted continuous trajectories remain feasible under true Keplerian dynamics, the work supplies a concrete bridge between combinatorial optimization and astrodynamics. The benchmark release and the explicit time-expanded network representation lower the barrier for discrete-optimization researchers to engage with realistic space-logistics instances.

major comments (3)
  1. [§3] §3 (Time-unfolding construction): the manuscript asserts that discrete optima map to feasible continuous Keplerian trajectories, yet provides neither a priori error bounds on the chosen time intervals nor a convergence argument as the discretization is refined. Without such guarantees, it is unclear whether the reported ILP solutions satisfy rendezvous conditions to within typical astrodynamic tolerances (position < 10 km, velocity < 0.1 m/s).
  2. [§4.2] §4.2 (IDDD formulation): the linear cost matrix used inside the ILP is obtained from Lambert or Keplerian transfer approximations evaluated at interval midpoints; the paper does not quantify the deviation of these costs from the true nonlinear transfer time and Δv when the optimal discrete tour is re-propagated in continuous time.
  3. [Table 2] Table 2 (computational results): the reported optimality gaps and runtimes are given only for the discretized ILP; no post-processing validation column shows the position/velocity residuals of the recovered continuous trajectories, leaving open the possibility that some “optimal” tours are infeasible once the continuous dynamics are restored.
minor comments (2)
  1. [§3] Notation for the time-expanded graph (nodes, arcs, and interval indices) is introduced without a compact summary table; a single table collecting all symbols would improve readability.
  2. [§5] The benchmark suite is stated to be released, but the manuscript does not specify the exact file formats, orbital-element conventions, or how the continuous reference trajectories are stored for validation.

Simulated Author's Rebuttal

3 responses · 0 unresolved

Thank you for the thorough and constructive review of our manuscript on the Keplerian Traveling Salesperson Problem. We appreciate the emphasis on validating the continuous feasibility of the discrete solutions and agree that additional analysis will strengthen the paper. We address each major comment below, outlining the revisions we plan to incorporate.

read point-by-point responses

  1. Referee: [§3] §3 (Time-unfolding construction): the manuscript asserts that discrete optima map to feasible continuous Keplerian trajectories, yet provides neither a priori error bounds on the chosen time intervals nor a convergence argument as the discretization is refined. Without such guarantees, it is unclear whether the reported ILP solutions satisfy rendezvous conditions to within typical astrodynamic tolerances (position < 10 km, velocity < 0.1 m/s).

    Authors: We acknowledge that the manuscript does not provide explicit a priori error bounds or a formal convergence argument for the time-unfolding discretization. In the revised version, we will add a dedicated subsection on discretization error that includes empirical validation: we will re-propagate all reported optimal discrete tours under continuous Keplerian dynamics and tabulate the resulting position and velocity residuals. While deriving tight theoretical bounds is beyond the current scope, the numerical evidence will demonstrate that residuals remain within typical astrodynamic tolerances for the benchmark instances, thereby confirming practical feasibility. revision: yes

  2. Referee: [§4.2] §4.2 (IDDD formulation): the linear cost matrix used inside the ILP is obtained from Lambert or Keplerian transfer approximations evaluated at interval midpoints; the paper does not quantify the deviation of these costs from the true nonlinear transfer time and Δv when the optimal discrete tour is re-propagated in continuous time.

    Authors: We agree that quantifying the approximation error introduced by midpoint evaluations is necessary. We will revise §4.2 to include a quantitative comparison: for each optimal tour, we will solve the continuous boundary-value problem to obtain exact transfer times and Δv values, then report the absolute and relative deviations from the midpoint-based costs used in the ILP. This analysis will be supported by additional figures or tables showing the distribution of these deviations across the benchmark set. revision: yes

  3. Referee: [Table 2] Table 2 (computational results): the reported optimality gaps and runtimes are given only for the discretized ILP; no post-processing validation column shows the position/velocity residuals of the recovered continuous trajectories, leaving open the possibility that some “optimal” tours are infeasible once the continuous dynamics are restored.

    Authors: This observation is correct. We will extend Table 2 (or add a companion table) with columns reporting the maximum position and velocity residuals obtained after continuous propagation of each recovered trajectory. A short paragraph will describe the post-processing procedure used to extract and validate the continuous solutions from the discrete optima. These additions will directly address concerns about feasibility under true Keplerian dynamics. revision: yes

Circularity Check

0 steps flagged

No significant circularity in KTSP formalization or time-unfolding reformulation

full rationale

The paper introduces a new formalization of the Keplerian TSP as a time-dependent asymmetric TSP variant, reformulated via time-unfolding into a discrete optimization problem on a time-expanded network and solved via ILP with Interval-based Dynamic Discretization Discovery. No derivation step reduces by construction to fitted parameters, self-definitions, or load-bearing self-citations; the central claims rest on standard TSP encodings and orbital-mechanics transfer models without tautological equivalence to inputs. The benchmark suite and heuristics are presented as independent contributions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard two-body Keplerian assumptions and classical TSP encodings; no new free parameters, invented entities, or ad-hoc axioms are described in the abstract.

axioms (1)
  • domain assumption Two-body Keplerian orbital motion with point-mass gravity and no perturbations
    Used to define time-dependent transfer costs between targets.

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