Algorithms for Optimally Shifting Intervals under Intersection Graph Models

TL;DR

The paper proposes a new graph-edit model for intersection graphs in which vertices are geometric objects that can be moved, and the objective is to minimize the total moving distance to achieve a target property $\Pi$. It provides a linear-time algorithm for interval graphs to transform into a complete graph by locating an optimal gathering point on the line, and extends this approach to unit square graphs under $L_1$ distance for the same goal. It also gives an $O(n \log n)$-time method to realize a $k$-clique on unit interval graphs when the moving-distance function is unique, and develops LP formulations to realize several other properties (edgeless, acyclic, not containing a $k$-clique, and $k$-connectivity) for unit interval graphs. Together, these results establish an efficient framework for editing geometric intersection graphs via object movement with potential applications in pattern recognition and spatial network design.

Abstract

We propose a new model for graph editing problems on intersection graphs. In well-studied graph editing problems, adding and deleting vertices and edges are used as graph editing operations. As a graph editing operation on intersection graphs, we propose moving objects corresponding to vertices. In this paper, we focus on interval graphs as an intersection graph. We give a linear-time algorithm to find the total moving distance for transforming an interval graph into a complete graph. The concept of this algorithm can be applied for (i) transforming a unit square graph into a complete graph over $L_1$ distance and (ii) attaining the existence of a $k$-clique on unit interval graphs. In addition, we provide LP-formulations to achieve several properties in the associated graph of unit intervals.