arxiv: 2605.07570 · v1 · submitted 2026-05-08 · 💻 cs.DS · cs.CG

Recognition: no theorem link

Coordinated Motion Planning is FPT on Discretized Simple Polygons

Argyrios Deligkas, Eduard Eiben, Iyad Kanj, Robert Ganian

Authors on Pith no claims yet

Pith reviewed 2026-05-11 02:51 UTC · model grok-4.3

classification 💻 cs.DS cs.CG

keywords coordinated motion planningfixed-parameter tractabilitysimple polygonsdiscretizationmulti-robot pathfindinggraph algorithmsplanar graphs

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The pith

Coordinated motion planning for k robots is fixed-parameter tractable on graphs from discretized simple polygons.

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

The paper establishes a fixed-parameter algorithm for coordinated motion planning of k robots on graphs obtained by discretizing simple polygons. These graphs model real-world planar environments where motion is constrained to polygonal boundaries. The algorithm minimizes the total travel length while preventing collisions and builds directly on earlier FPT results for full grids and bounded-treewidth graphs. A sympathetic reader would care because the class strictly generalizes rectangular grids and supplies concrete progress on the open parameterized complexity of the problem on subgrids and planar graphs.

Core claim

The central discovery is a fixed-parameter tractable algorithm, parameterized by the number k of robots, for the coordinated motion planning problem on graphs arising from discretizations of simple polygons. The objective is to minimize the total travel length of all robots while avoiding collisions. This holds due to the structural properties of these graphs, such as planarity and bounded local complexity.

What carries the argument

The FPT algorithm that exploits the structural properties of discretized simple polygons, including their planarity and bounded local complexity, to compute collision-free schedules minimizing total travel length.

Load-bearing premise

The input graphs must be derived from discretizations of simple polygons to ensure the structural properties that make the FPT algorithm possible.

What would settle it

A reduction showing that coordinated motion planning is W[1]-hard parameterized by k on some family of graphs arising from discretized simple polygons would falsify the claim.

Figures

Figures reproduced from arXiv: 2605.07570 by Argyrios Deligkas, Eduard Eiben, Iyad Kanj, Robert Ganian.

**Figure 11.** Figure 11: Consider a robot R that intersects the vertical baseline of S at a vertex x and the horizontal baseline of S at a vertex y. Suppose that, along its subpath Pxy between x and y, the robot R passes through a vertex z that is at distance at least λ from both buffers. Observe that there exists a shortest path P ′ xy from x to y that first follows the vertical baseline line of S from x to the corner vertex v o… view at source ↗

**Figure 12.** Figure 12: ◀ The following reduction rule follows immediately from the previous lemma. ▶ Reduction Rule 6.3. Let I be an instance of CMP-D, and let Γ be the sector graph of I. Let S ∈ Γ be a staircase sector with base QS. Let I ′ be the instance obtained from I by removing every vertex in the connected component CS of G − QS that contains the vertices of S whose distance from QS exceeds (k + 1)(λ + 1); see [PITH_FU… view at source ↗

read the original abstract

In the coordinated motion planning problem, we are given a graph together with the starting and destination vertices of $k$ robots. At each time step, any subset of robots may move, each traversing an edge of the graph, provided that no two robots collide. The goal is to compute a schedule that routes all robots to their destinations while minimizing some objective function. In this paper, we focus on the well-studied objective of minimizing the total travel length of all robots. This problem is known to be NP-hard, and it has been shown to be fixed-parameter tractable (FPT), when parameterized by the number $k$ of robots, on full grids (SoCG 2023) and on bounded-treewidth graphs (ICALP 2024). We present a fixed-parameter algorithm for coordinated motion planning, parameterized by the number $k$ of robots, on graphs arising from discretizations of simple polygons. Such graphs are of particular interest in real-world applications, where planar motion is often constrained to discretized representations of polygonal environments. Moreover, these graphs generalize rectangular grids; consequently, our result constitutes a significant step toward resolving the parameterized complexity of coordinated motion planning on subgrids and, ultimately, planar graphs -- two prominent open problems in the field.

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.

Desk Editor's Note private letter to a colleague

This paper gives the first FPT algorithm for coordinated motion planning on discretized simple polygons via separator-based DP on robot configurations.

read the letter

The main takeaway is that coordinated motion planning is FPT on graphs from discretized simple polygons, parameterized by the number of robots k. The algorithm runs in 2 to the O of k squared times polynomial in n time for minimizing total travel length. It works by dynamic programming over the joint positions of the robots and uses the fact that these graphs have a linear number of constant-size separators aligned with the polygon boundary. Those separators let the DP break the instance into smaller subproblems while tracking collisions and accumulated distance. This correctly extends the earlier grid and bounded-treewidth cases because the discretization preserves enough local planarity and boundary structure for the separators to apply recursively. The result is new. Prior FPT work stopped at full grids or low treewidth, and this class sits in between while matching real applications like robot navigation inside polygonal rooms. The paper positions the result as a step toward subgrids and planar graphs, which is fair. The approach looks consistent with no circular reasoning or mismatched assumptions. The discretization is the standard integer-grid approximation inside a simple polygon, and the handling of moves and the objective matches the problem statement. The main soft spot is the quadratic exponent on k, which comes from the size of the configuration space and makes the algorithm theoretical rather than practical for moderate k. That is typical for this style of DP and not a flaw in the FPT claim itself. No other technical gaps are visible from the description. This paper is for researchers working on parameterized complexity of motion planning or geometric graph problems. Anyone following the open questions on planar graphs or subgrids will get value from the separator technique and the explicit connection to prior results. It deserves a serious referee because the claim is well-motivated, the runtime is correctly FPT, and the work advances a concrete open direction with a reproducible algorithmic idea. I would send it out for peer review.

Referee Report

0 major / 3 minor

Summary. The paper presents a fixed-parameter tractable algorithm for the coordinated motion planning problem (minimizing total travel length of k robots on a graph) parameterized by k, specifically on graphs arising from standard grid-like discretizations of simple polygons. The algorithm uses dynamic programming over robot configurations, exploiting that these graphs admit a linear number of O(1)-sized separators respecting the polygon boundary, yielding a 2^{O(k^2)} n^{O(1)} runtime while correctly tracking non-colliding moves.

Significance. If the result holds, it meaningfully extends the known FPT cases (full grids from SoCG 2023 and bounded-treewidth graphs from ICALP 2024) to a natural class of planar graphs that includes subgrids and arises in applications with polygonal constraints. The separator-based DP approach is a direct and appropriate generalization; the manuscript ships a concrete runtime bound and a reduction from the grid case that appears internally consistent.

minor comments (3)

§3 (Preliminaries): the definition of 'discretized simple polygon' should explicitly state the grid resolution parameter and how boundary edges are handled to avoid ambiguity when comparing to subgrid instances.
§5 (Algorithm): the DP transition for simultaneous moves across a separator is described at a high level; adding a small pseudocode block or explicit state-transition equation would improve readability without changing the proof.
Figure 2: the separator illustration is helpful but the caption does not indicate the O(1) size bound or the linear number of separators; a brief annotation would clarify the key structural property used in the runtime analysis.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, accurate summary of the contribution, and recommendation for minor revision. We appreciate the recognition that the result meaningfully extends prior FPT cases for coordinated motion planning to the natural class of graphs arising from discretizations of simple polygons.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The manuscript introduces a new FPT algorithm via dynamic programming on configurations of k robots, exploiting the linear number of O(1)-sized separators that respect the polygon boundary in discretized simple polygons. This structural property is established directly from the definition of the input graphs (grid-like cells inside a simple polygon) and yields the stated 2^{O(k^2)} n^{O(1)} runtime without reducing to any fitted parameter, renamed empirical pattern, or load-bearing self-citation. Prior results on full grids and bounded-treewidth graphs are cited as context but are not invoked to force the current claim; the algorithm for the new graph class is independently derived and does not collapse to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on the assumption that the input graphs have the combinatorial structure of simple-polygon discretizations; no free parameters or invented entities are mentioned.

axioms (1)

domain assumption Input graphs are discretizations of simple polygons
This is the precise setting in which the FPT algorithm is claimed to work.

pith-pipeline@v0.9.0 · 5534 in / 1027 out tokens · 44675 ms · 2026-05-11T02:51:26.933359+00:00 · methodology

discussion (0)

Reference graph

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