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arxiv: 2509.20903 · v3 · submitted 2025-09-25 · 💻 cs.CG

Further Results on Rendering Geometric Intersection Graphs Sparse by Dispersion

Nicol\'as Honorato-Droguett , Kazuhiro Kurita , Tesshu Hanaka , Hirotaka Ono , Alexander Wolff This is my paper

Pith reviewed 2026-05-18 14:37 UTC · model grok-4.3

classification 💻 cs.CG
keywords geometric graph edit distanceintersection graphsunit circular arcsdispersionNP-hardnessinterval graphspoints spreadingXP algorithm
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The pith

An O(n log n) algorithm disperses unit circular arcs so their intersection graph becomes edgeless and k-clique-free.

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

The paper continues the study of Geometric Graph Edit Distance, where the goal is to move geometric objects by the smallest total distance so that their intersection graph satisfies a sparsity condition such as having no edges. For unit circular arcs the authors give a deterministic O(n log n) procedure that achieves an edgeless and k-clique-free outcome. The same procedure also settles an open case of the points-spreading problem on cyclic domains. The paper further proves that the same minimization task is strongly NP-hard already for unweighted interval graphs and for d-dimensional balls or cubes when d is at least two, and supplies an XP algorithm for weighted unit intervals when the number of maximal cliques is the parameter.

Core claim

We present an algorithm for rendering the intersection graph of a set of unit circular arcs edgeless and k-clique-free in O(n log n) time, where n is the number of arcs. The algorithm can be also used to solve an open case of the points-spreading problem on cyclic domains. We also show that GGED remains strongly NP-hard on unweighted interval graphs. We complement this result by showing that GGED is strongly NP-hard on sets of d-balls and d-cubes, for any d >= 2. Finally, we present an XP algorithm (parameterised by the number of maximal cliques) that removes all edges from the intersection graph of a set of weighted unit intervals.

What carries the argument

Dispersion of unit circular arcs by continuous movement that minimises the sum of Euclidean distances while eliminating all pairwise intersections.

If this is right

  • The points-spreading problem on cyclic domains admits an O(n log n) solution.
  • No polynomial-time algorithm exists for GGED on unweighted interval graphs unless P equals NP.
  • GGED is strongly NP-hard for any collection of d-balls or d-cubes when d is at least two.
  • When the number of maximal cliques is treated as a parameter, all edges can be removed from weighted unit-interval intersection graphs in time f(k) * n^O(1).

Where Pith is reading between the lines

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

  • The same sorting-based dispersion technique may extend to other circular or periodic geometric objects.
  • Practical map-labelling systems could adopt the O(n log n) procedure for overlap removal when objects are approximately unit size.
  • The hardness results motivate the search for constant-factor approximation algorithms on interval and higher-dimensional instances.
  • Scheduling applications that model tasks as arcs on a circle may now solve overlap-free assignments in near-linear time.

Load-bearing premise

The only allowed edit operation is continuous movement of the geometric objects and the cost equals the total Euclidean distance traveled, with no additional constraints on feasible final positions.

What would settle it

A set of unit circular arcs for which every sequence of moves that eliminates all intersections requires more than linearithmic time, or a polynomial-time algorithm that solves GGED exactly on unweighted interval graphs.

Figures

Figures reproduced from arXiv: 2509.20903 by Alexander Wolff, Hirotaka Ono, Kazuhiro Kurita, Nicol\'as Honorato-Droguett, Tesshu Hanaka.

Figure 1
Figure 1. Figure 1: Reduction overview: Skeleton of the reduction IA = I ∪ B. for all I ∈ I, we have that r(I i j ) = d i j for all i ∈ [m] and j ∈ [3]. Then, r(I i j+1) − r(I i j ) = d i j+1 − d i j = a i j+1 = len(I i j+1). Hence, the intervals in I + DI are disjoint. Similarly, for all i ∈ [m], it holds that r(I i 1 +d i 1 )−r(Bi−1) = len(I i 1 ) and ℓ(Bi) − ℓ(I i 3 ) = len(I i 3 ). In other words, the tuple I + DI does no… view at source ↗
Figure 2
Figure 2. Figure 2: Skeleton of the reduction of Corollary 2 (squares). 3.2 Extending the Reduction to Higher Dimensions In this section, we show that Theorem 1 can be extended to d-dimensional ob￾jects. Let (A, L) be an instance of 3-Partition. We assume that d = 2 and show a reduction that works for squares and disks. We give a multiset of squares and disks based on the structure of IA. To maintain simplicity, we give the d… view at source ↗
Figure 3
Figure 3. Figure 3: Further restriction of free spaces: (a) Restricted space where squares of S can be placed (left) and an example of three squares placed into the free space where ai + aj + ak = L (right); (b) Restricted space where disks of S can be placed (left) and an example of three disks placed into the free space where ai + aj + ak = L (right). The restricted space for disks is defined by disks of diameter L/2 + ∆. S… view at source ↗
Figure 4
Figure 4. Figure 4: Movement of squares of the barrier: At most T · (L/4) squares (bounded by the black box) are moved with moving distance of at most one unit. The rest (L/4 + ∆)T 2 − T L/4 of squares is moved by one unit to the i + 1th free space. space in a valid solution. Moreover, the area of free spaces is bounded by poly(n), consequently, adding these objects maintains the polynomial size of the instance. If S ∈ S is a… view at source ↗
read the original abstract

Removing overlaps is a central task in domains such as scheduling, visibility, and map labelling. This can be modelled using graphs, where overlap removals correspond to enforcing a certain sparsity constraint on the graph structure. We continue the study of the problem Geometric Graph Edit Distance (GGED), where the aim is to minimise the total cost of editing a geometric intersection graph to obtain a graph contained in a specific graph class. For us, the edit operation is the movement of objects, and the cost is the movement distance. We present an algorithm for rendering the intersection graph of a set of unit circular arcs edgeless and $k$-clique-free in $O(n\log n)$ time, where $n$ is the number of arcs. The algorithm can be also used to solve an open case of the points-spreading problem on cyclic domains [Li \& Wang, CGT 2025]. We also show that GGED remains strongly NP-hard on unweighted interval graphs, solving an open problem of Honorato-Droguett et al. [WADS 2025]. We complement this result by showing that GGED is strongly NP-hard on sets of $d$-balls and $d$-cubes, for any $d\ge 2$. Finally, we present an XP algorithm (parameterised by the number of maximal cliques) that removes all edges from the intersection graph of a set of weighted unit intervals.

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

0 major / 3 minor

Summary. The manuscript studies the Geometric Graph Edit Distance (GGED) problem of moving geometric objects by the minimum total Euclidean distance to render their intersection graph edgeless or k-clique-free. It gives an O(n log n) algorithm for unit circular arcs that sorts endpoints and uses a linear-time dispersion pass exploiting unit length and circular ordering (including cyclic wrap-around via the maximum gap). The paper proves strong NP-hardness of GGED on unweighted interval graphs (resolving an open question) and on d-balls/d-cubes for any d ≥ 2, and supplies an XP algorithm for weighted unit intervals parameterized by the number of maximal cliques. It also settles an open case of the points-spreading problem on cyclic domains.

Significance. If the claims hold, the work is a solid contribution to geometric dispersion and graph editing problems. The O(n log n) algorithm for unit circular arcs is efficient and leverages natural structure (unit length, circular order) to obtain optimality; the resolution of the open CGT 2025 case and the WADS 2025 open problem on interval graphs are clear advances. The hardness results for higher-dimensional objects and the parameterized XP algorithm broaden the complexity picture. Machine-checked proofs or reproducible code are not mentioned, but the self-contained reductions and explicit algorithmic construction are strengths.

minor comments (3)
  1. [Abstract] Abstract: the claim that the algorithm 'can be also used to solve an open case' of the points-spreading problem would benefit from a one-sentence parenthetical identifying the precise open question from Li & Wang (CGT 2025).
  2. [Algorithm section (presumably §3 or §4)] The description of the cyclic wrap-around handling (maximum gap after sorting) is referenced in the skeptic note but should appear with a short optimality argument in the main text of the algorithm section to make the O(n log n) claim fully self-contained for readers.
  3. [Hardness sections] Ensure that the strong NP-hardness reductions for interval graphs and d-balls explicitly state the source problem (e.g., 3-Partition) and confirm that the constructed instances remain polynomial in size, as required for strong NP-hardness.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the positive assessment, accurate summary of our contributions, and recommendation for minor revision. We are pleased that the O(n log n) algorithm, hardness results, and resolution of open problems are viewed as advances.

read point-by-point responses

  1. Referee: The manuscript studies the Geometric Graph Edit Distance (GGED) problem of moving geometric objects by the minimum total Euclidean distance to render their intersection graph edgeless or k-clique-free. It gives an O(n log n) algorithm for unit circular arcs that sorts endpoints and uses a linear-time dispersion pass exploiting unit length and circular ordering (including cyclic wrap-around via the maximum gap).

    Authors: We thank the referee for this accurate description. The algorithm indeed relies on sorting endpoints in O(n log n) time followed by a linear-time dispersion pass that exploits unit length and the circular ordering, with wrap-around handled by identifying the maximum gap. No changes are required. revision: no

  2. Referee: The paper proves strong NP-hardness of GGED on unweighted interval graphs (resolving an open question) and on d-balls/d-cubes for any d ≥ 2, and supplies an XP algorithm for weighted unit intervals parameterized by the number of maximal cliques. It also settles an open case of the points-spreading problem on cyclic domains.

    Authors: We appreciate the referee highlighting these results. The strong NP-hardness on unweighted interval graphs resolves the open question from WADS 2025, while the d-dimensional hardness and XP algorithm complement the picture. The cyclic points-spreading case is settled as a byproduct of the arc algorithm. No revisions needed. revision: no

  3. Referee: Machine-checked proofs or reproducible code are not mentioned, but the self-contained reductions and explicit algorithmic construction are strengths.

    Authors: We note the referee's observation. The reductions are fully self-contained with explicit constructions, and the algorithm is described in sufficient detail for implementation. If the editor prefers, we can add a brief note on reproducibility in the final version. revision: partial

Circularity Check

0 steps flagged

No significant circularity in algorithmic constructions or reductions

full rationale

The paper's central contributions consist of explicit algorithmic constructions (sorting endpoints followed by a linear-time dispersion pass exploiting unit length and circular ordering) and direct NP-hardness reductions on interval graphs and d-balls. These steps are presented as self-contained procedures with verifiable time bounds and do not reduce to fitted parameters, self-definitions, or unverified self-citations. References to prior work (including by overlapping authors) serve only to identify open cases solved by the new results; the load-bearing arguments remain independent and externally falsifiable via the described sorting and assignment methods.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on standard assumptions from computational geometry and graph theory; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)
  • domain assumption Intersection graphs of geometric objects correctly model overlap relations under Euclidean movement costs.
    Central modeling premise stated in the abstract's problem definition.

pith-pipeline@v0.9.0 · 5802 in / 1114 out tokens · 43752 ms · 2026-05-18T14:37:02.314187+00:00 · methodology

discussion (0)

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