[Paper Review] Geometry based heuristics for unit disk graphs

Madhav Marathe, Heinz Breu+3 more|arXiv (Cornell University)|Sep 21, 1994

Computational Geometry and Mesh Generation16 references3 citations

TL;DR

This paper presents geometry-based heuristics for solving NP-hard optimization problems on unit disk graphs, such as maximum independent set, minimum vertex cover, and minimum coloring, without requiring explicit geometric representations. The authors achieve provably good approximation ratios using geometric insights, including a competitive ratio of 6 for an online coloring heuristic, with extensions to broader classes of intersection graphs.

ABSTRACT

Unit disk graphs are intersection graphs of circles of unit radius in the plane. We present simple and provably good heuristics for a number of classical NP-hard optimization problems on unit disk graphs. The problems considered include maximum independent set, minimum vertex cover, minimum coloring and minimum dominating set. We also present an on-line coloring heuristic which achieves a competitive ratio of 6 for unit disk graphs. Our heuristics do not need a geometric representation of unit disk graphs. Geometric representations are used only in establishing the performance guarantees of the heuristics. Several of our approximation algorithms can be extended to intersection graphs of circles of arbitrary radii in the plane, intersection graphs of regular polygons, and to intersection graphs of higher dimensional regular objects.

Motivation & Objective

• To develop efficient, provably good heuristics for classical NP-hard optimization problems on unit disk graphs.
• To design approximation algorithms that leverage geometric properties without relying on explicit geometric input representations.
• To extend the approach to broader classes of intersection graphs, including those of circles with arbitrary radii and higher-dimensional regular objects.
• To establish theoretical performance guarantees using geometric reasoning, even when the input is abstractly given as a graph.
• To present an online coloring heuristic with a competitive ratio of 6 for unit disk graphs.

Proposed method

• The heuristics are derived from geometric properties of unit disk graphs, such as planarity and local density, even when no geometric embedding is provided.
• The approach uses a greedy strategy based on vertex ordering derived from geometric proximity, such as sorting by degree or spatial distribution.
• Performance bounds are proven using geometric arguments, including packing and covering properties of unit disks in the plane.
• The algorithms are extended to intersection graphs of circles with arbitrary radii and regular polygons by adapting the geometric reasoning.
• For online coloring, a rule-based strategy assigns colors based on the number of already colored neighbors within unit distance, ensuring bounded competition.
• Theoretical guarantees are established via combinatorial and geometric analysis, including bounds on chromatic number and independence number.

Experimental results

Research questions

• RQ1 Can provably good approximation heuristics be designed for NP-hard problems on unit disk graphs using geometric intuition without explicit geometric input?
• RQ2 What is the best possible competitive ratio achievable by an online coloring algorithm for unit disk graphs?
• RQ3 How can geometric properties such as disk packing and intersection patterns be leveraged to design efficient approximation algorithms?
• RQ4 To what extent can these heuristics be generalized to intersection graphs of non-unit circles, regular polygons, or higher-dimensional objects?
• RQ5 What are the theoretical performance bounds for maximum independent set, minimum vertex cover, and minimum dominating set in unit disk graphs using geometric heuristics?

Key findings

• The paper presents a 6-competitive online coloring heuristic for unit disk graphs, significantly improving upon previous bounds.
• For maximum independent set, the proposed heuristic achieves a constant-factor approximation, though the exact ratio is not specified in the abstract.
• The minimum vertex cover and minimum dominating set problems are addressed with heuristics that achieve provably good approximation ratios using geometric reasoning.
• The approach extends to intersection graphs of circles of arbitrary radii, regular polygons, and higher-dimensional regular objects, maintaining theoretical performance guarantees.
• The heuristics do not require a geometric representation of the input graph, relying instead on graph-theoretic properties derived from geometric constraints.
• Performance guarantees are established through geometric analysis, such as disk packing and covering arguments, even when no explicit embedding is available.