[Paper Review] Connecting a Set of Circles with Minimum Sum of Radii

Erin Wolf Chambers, Sándor P. Fekete+6 more|arXiv (Cornell University)|May 4, 2011

Computational Geometry and Mesh Generation21 references14 citations

TL;DR

This paper studies the Connected Range Assignment (CRA) problem: assigning radii to a set of points in the plane to minimize the sum of radii while ensuring the resulting disk intersection graph is connected. It proves the problem is NP-hard when radii are bounded, provides a polynomial-time approximation scheme (PTAS) for unbounded radii, and gives constant-factor approximation guarantees for solutions using a bounded number of disks, with tight lower bounds and experimental validation of heuristic performance.

ABSTRACT

We consider the problem of assigning radii to a given set of points in the plane, such that the resulting set of circles is connected, and the sum of radii is minimized. We show that the problem is polynomially solvable if a connectivity tree is given. If the connectivity tree is unknown, the problem is NP-hard if there are upper bounds on the radii and open otherwise. We give approximation guarantees for a variety of polynomial-time algorithms, describe upper and lower bounds (which are matching in some of the cases), provide polynomial-time approximation schemes, and conclude with experimental results and open problems.

Motivation & Objective

• To minimize the sum of radii assigned to a set of points such that the resulting disks form a connected intersection graph.
• To determine the computational complexity of the problem under different constraints, especially with and without upper bounds on radii.
• To develop approximation algorithms with provable performance guarantees for cases with a bounded number of disks.
• To evaluate the effectiveness of heuristic solutions, such as one-disk or two-disk configurations, through experimental analysis.
• To explore whether the problem is polynomially solvable in the absence of radius bounds, based on empirical and theoretical evidence.

Proposed method

• Proves NP-hardness of the CRA problem in planar weighted graphs when radii are upper-bounded, using a reduction from a known NP-hard problem.
• Develops a polynomial-time approximation scheme (PTAS) for the unbounded radii case, leveraging geometric and graph-theoretic techniques.
• Introduces a linear programming-based approach to compute tight lower bounds on the optimal solution, particularly for 2-disk configurations.
• Uses a spanning tree enumeration method to compute the optimal solution for small instances, enabling comparison with heuristics.
• Employs experimental evaluation on uniformly distributed point sets in a 2D circular region, comparing optimal solutions with one-disk and two-disk heuristics.
• Applies techniques from computational geometry and graph connectivity to model and analyze the structure of feasible solutions.

Experimental results

Research questions

• RQ1 Is the Connected Range Assignment problem with unbounded radii NP-hard or polynomial-time solvable?
• RQ2 Can a constant-factor approximation be achieved for solutions using a bounded number of disks, and what are the tightest possible approximation ratios?
• RQ3 How effective is the one-disk solution as a heuristic, and does its performance remain stable as the number of points increases?
• RQ4 Can a PTAS be extended to the case where radii are bounded, and what assumptions are required for such an extension?
• RQ5 Does the worst-case approximation ratio of k-disk solutions converge to 1 as k increases, indicating asymptotic optimality?

Key findings

• The CRA problem is NP-hard when radii are upper-bounded, even in planar weighted graphs.
• A polynomial-time approximation scheme (PTAS) exists for the unbounded radii case, providing arbitrarily close approximations to the optimal sum of radii.
• For solutions using at most two disks, a 4/3-approximation is guaranteed in general, and a 5/4-approximation on a line, with matching lower bounds.
• The one-disk solution performs remarkably well in practice, with the ratio of its cost to the optimal cost remaining nearly constant across varying numbers of points.
• Linear programming was used to derive tight lower bounds, particularly for 2-disk configurations, confirming theoretical approximation guarantees.
• Experimental results show that the optimal solution can be computed efficiently for small instances, and the one-disk heuristic is consistently close to optimal, even in worst-case scenarios.