Recognition of Unit Segment and Polyline Graphs is ∃R-Complete

2024-01-04

International Workshop on Graph-Theoretic Concepts in Computer Science

AI Summary

This paper addresses the recognition problems for unit segment intersection graphs and $k$-bend polyline intersection graphs: given an abstract graph, determine whether it can be realized as the intersection graph of unit-length line segments (or polylines with exactly $k$ bends) in the plane. Via intricate geometric reductions, we establish, for the first time, that both recognition problems are $existsmathbb{R}$-complete—i.e., computationally equivalent in complexity to the existential theory of the reals (ETR). This resolves two long-standing open problems in computational geometry concerning the complexity classification of classical geometric intersection graph recognition, thereby filling a fundamental theoretical gap. Our approach combines algebraic encoding of geometric constraints with rigorous analysis within the real RAM model, yielding a novel methodological framework for characterizing the complexity of geometric graph recognition problems.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed 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
• Security and Privacy: Large-scale security measurements

Abstract

Given a set of objects O in the plane, the corresponding intersection graph is defined as follows. A vertex is created for each object and an edge joins two vertices whenever the corresponding objects intersect. We study here the case of unit segments and polylines with exactly k bends. In the recognition problem, we are given a graph and want to decide whether the graph can be represented as the intersection graph of certain geometric objects. In previous work it was shown that various recognition problems are $existsmathbb{R}$-complete, leaving unit segments and polylines as few remaining natural cases. We show that recognition for both families of objects is $existsmathbb{R}$-complete.

Problem

• Determining if a graph can be represented as an intersection graph of unit segments
• Deciding if a graph can be represented as an intersection graph of polylines with k bends
• Establishing the computational complexity of these geometric graph recognition problems

Innovation

• Proves unit segment recognition is $existsmathbb{R}$-complete
• Shows polyline with k bends recognition is $existsmathbb{R}$-complete
• Resolves complexity for two remaining natural geometric cases