Approximation Algorithms for Maximum Independent Set of Pseudo-Disks (1103.1431v1)

Abstract: We present approximation algorithms for maximum independent set of pseudo-disks in the plane, both in the weighted and unweighted cases. For the unweighted case, we prove that a local search algorithm yields a \PTAS. For the weighted case, we suggest a novel rounding scheme based on an \LP relaxation of the problem, which leads to a constant-factor approximation. Most previous algorithms for maximum independent set (in geometric settings) relied on packing arguments that are not applicable in this case. As such, the analysis of both algorithms requires some new combinatorial ideas, which we believe to be of independent interest.

• The paper introduces a Polynomial Time Approximation Scheme (PTAS) for the unweighted maximum independent set problem on pseudo-disks using local search and a constant-factor approximation for the weighted case via LP relaxation.
• It employs novel techniques like planar graph properties, new combinatorial lemmas for pseudo-disk intersections, and LP contention schemes to achieve these approximations.
• The results advance the understanding of geometric intersection graphs and offer frameworks potentially applicable to broader problems and for future practical implementations or efficiency enhancements.

Approximation Algorithms for Maximum Independent Set of Pseudo-Disks

In this paper, the authors introduce approximation algorithms for addressing the problem of finding the maximum independent set of pseudo-disks in the plane, providing solutions for both weighted and unweighted cases. In geometric graphs, identifying the maximum independent set is classically tied to computational complexity, with the problem known to be \textsf{NP-Complete}. The work focuses on pseudo-disks, which present unique computational challenges due to intersecting boundary behaviors that differ from fat objects like disks or squares.

Main Contributions

1. Unweighted Pseudo-Disks: PTAS via Local Search

The authors propose a Polynomial Time Approximation Scheme (PTAS) for the unweighted variant of the problem by employing a local search strategy. In previous geometric formulations, packing arguments were utilized, but these are insufficient for pseudo-disks. The local search approach used here is shown to circumvent these limitations. It involves iteratively constructing a locally optimal solution where subsets of pseudo-disks are swapped to increase the size of the independent set.

1. Weighted Pseudo-Disks: Constant-Factor Approximation via LP Relaxation

For the weighted instance, a novel algorithm based on Linear Programming (LP) relaxation is developed. The authors use a rounding scheme to convert the fractional solutions into an integral solution while maintaining a constant proximity to the optimal solution. This is particularly potent when applied to objects having linear union complexity, including pseudo-disks.

1. Theoretical Implications and Novel Techniques

• Planar Graph Techniques: By exploiting properties of planar graphs and integrating combinatorial geometric ideas, the authors offer structural insights into solving independent set problems for pseudo-disks.
• Combinatorial Lemmas: New combinatorial ideas that address intersections and resistances for pseudo-disks, conducive to both the theoretical understanding and algorithmic efficiency.
• LP Contention Schemes: Algorithms that operate within a contention resolution framework, drawing from fractional LP solutions to construct approximate solutions efficiently.

Implications and Future Directions

The results furnish a deeper understanding of geometric intersection graphs, illuminating insights that could be extrapolated to broader classes than pseudo-disks alone. The PTAS for unweighted pseudo-disks revitalizes connections between planar separators and geometric intersections. Meanwhile, the weighted case's reliance on LP showcases a promising path for tackling more general combinatorial optimization problems, such as covering problems or maximizing submodular functions where weights and intersections vary.

Future explorations could extend these approximation frameworks to higher dimensions or increase focus on practical implementations for computational geometry applications. Moreover, enhancing the efficiency of these algorithms or refining the techniques for specific subclasses of pseudo-disks poses an engaging challenge for subsequent studies.

Ultimately, this paper offers substantial contributions to combinatorial optimization and geometric approximation algorithms, presenting significant theoretical advancements alongside robust algorithmic frameworks for tackling independent set problems in pseudo-disks.