On the Complexity of Minimising the Moving Distance for Dispersing Objects

TL;DR

This paper introduces algorithms and complexity results for minimizing moving distances in dispersing objects modeled by geometric intersection graphs, with applications to graph editing and dispersion.

Contribution

It provides an efficient algorithm for dispersing unit intervals and establishes NP-hardness for more complex graph classes and distance measures.

Findings

O(n log n) algorithm for dispersing unit intervals

NP-hardness results for weighted interval graphs

NP-hardness of minimizing maximum distance for unit disk graphs

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.