On the Complexity of Minimising the Moving Distance for Dispersing Objects

TL;DR

This work formalizes Geometric Graph Edit Distance (GGED), where moving geometric objects to modify intersection graphs incurs a total moving distance cost. It gives an $O(n\log n)$ algorithm for unit interval graphs to disperse objects and satisfy sparse properties such as edgeless, acyclic, or $k$-clique-free states, while proving strong NP-hardness for weighted interval graphs under the same targets. It further shows that the minimax variant on unit disk graphs is strongly NP-hard under both $L_1$ and $L_2$, via a Planar 3-SAT gadget-based reduction. The results delineate a sharp contrast between tractable unit-interval scenarios and intractable weighted/disk settings, and they introduce a detailed gadget-based reduction framework that could inform future geometric modification problems in higher dimensions or with different edit operations.

Abstract

We study Geometric Graph Edit Distance (GGED), a graph-editing model to compute the minimum edit distance of intersection graphs that uses moving objects as an edit operation. We first show an $O(n\log n)$-time algorithm that minimises the total moving distance to disperse unit intervals. This algorithm is applied to render a given unit interval graph (i) edgeless, (ii) acyclic and (iii) $k$-clique-free. We next show that GGED becomes strongly NP-hard when rendering a weighted interval graph (i) edgeless, (ii) acyclic and (iii) $k$-clique-free. Lastly, we prove that minimising the maximum moving distance for rendering a unit disk graph edgeless is strongly NP-hard over the $L_1$ and $L_2$ distances.

Figures (19)

• Figure 1: Reduction Overview
• Figure 2: Reduction Overview: An arbitrary instance $\Phi$ of Planar 3-SAT with its rectilinear embedding $G_\Phi$.
• Figure 3: Reduction Overview: (a) The skeleton given by the instance $(\Phi, G_\Phi)$ of \ref{['fig:reduction_overview_a']}; (b) The intersection of the gadget for $c = (x_1 \lor \overline{x_2} \lor x_4)$ is removed by moving disks in a way that a free slot of the gadget for $x_2$ is used. Since $c = \mathit{true}$ when $x_2 = \mathit{false}$, the free slots for the other two gadgets become blocked, being unable to remove their intersection using the variable gadget for $x_2$.
• Figure 4: Disks used in the reduction: Transition disk $D$ and $k$-heavy disks, $k \in\{1,2,6\}$, with their corresponding moving distance function.
• Figure 5: Range of movement and feasible area: (a) illustration of a collection of disks $\mathcal{D} = \{D,D_1,D_2,D_3\} \subseteq \mathcal{D}^{(i)}_\Phi$ and the range of movement $\mathcal{A}_D$; (b) the feasible area of $D$, $\mathcal{F}_D$. In particular, $\mathcal{F}_D = \mathcal{A}_D \setminus \{\mathcal{B}_{D_1} \cup \mathcal{B}_{D_2} \cup \mathcal{B}_{D_3}\}$ is the region marked with a bold dotted line. The disk $D$ can be moved to an arbitrary point $p$ contained in $\mathcal{F}_D$. On the other hand, $D$ cannot be moved to the point $q \in \mathcal{A}_D$ without exceeding the minimum moving distance $K$ even if $d_D(q) \le K$ holds.

Theorems & Definitions (39)

• definition 1: Equispace function
• proof
• proof
• definition 2: Optimally Equispaceable Collections
• lemma 1
• proof
• proof
• corollary 1
• proof
• proof