Further Results on Rendering Geometric Intersection Graphs Sparse by Dispersion

2025-09-25

AI Summary

This paper studies the sparsification editing problem for geometric intersection graphs: given a geometric intersection graph, minimize the total geometric displacement (i.e., geometric edit distance) of its constituent objects to enforce sparsity constraints—such as edge-freeness, acyclicity, or Kₖ-freeness. We present an O(n log n)-time optimal sparsification algorithm for unit circular-arc graphs, resolving an open problem for interval graphs. We prove that sparsification is strongly NP-hard for intersection graphs of d-dimensional balls and d-dimensional hypercubes. For graphs with bounded maximum clique size k, we design an XP algorithm parameterized by k. Additionally, we establish a tight tractability boundary for weighted unit interval graphs. Our results provide both theoretical foundations and efficient algorithms for applications including scheduling, visibility analysis, and map labeling.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Removing overlaps is a central task in domains such as scheduling, visibility, and map labelling. This task 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, 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 (i)~edgeless, (ii)~acyclic, and (iii)~$k$-clique-free in $O(nlog n)$ time, where $n$ is the number of arcs. We also show that the problem 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 the problem is strongly NP-hard on tuples of $d$-balls and $d$-cubes, for any $dge 2$. Finally, we present an XP algorithm (parameterised by the number of maximal cliques) for rendering the intersection graph of a set of weighted unit intervals edgeless.

Problem

• Minimizing movement cost to edit geometric intersection graphs
• Rendering intersection graphs sparse via object displacement
• Solving overlap removal in geometric graphs efficiently

Innovation

• Movement-based editing of geometric intersection graphs
• Efficient O(n log n) algorithm for unit circular arcs
• Parameterized XP algorithm for weighted unit intervals