Geometric Optimization Parameterized by Piercing Complexity

2026-09-25

AI Summary

This study addresses the challenge that traditional complexity measures fail to guarantee polynomial-time approximation schemes (PTAS) for geometric packing and covering problems. To overcome this, it introduces piercing number as a novel parameter, integrating piercing graph analysis, recursive local search, balanced separator construction, and shallow cell complexity theory. The core contributions demonstrate that local search algorithms yield PTAS for discrete independent set and set cover problems. Furthermore, sublinear balanced separators depending solely on the piercing number are constructed, surpassing non-piercing family limitations and extending to weighted variants. Ultimately, this work achieves a PTAS for instances with fixed piercing numbers, an O(r)-approximation for weighted set cover, and a deterministic O(r+1)-approximation for independent sets.

Abstract

Packing and covering problems for geometric regions have been studied under many notions of complexity, including VC-dimension, union complexity, shallow-cell complexity, and fatness. Although these restrictions often yield constant-factor approximation algorithms, they do not by themselves generally lead to PTASs. A recurring feature of known hardness constructions is that one region may be \emph{pierced} by many others: a region $B$ pierces $A$ when $A\setminus B$ is disconnected. We study geometric instances through the \emph{piercing degree}. Since piercing is symmetric for Jordan regions, this is the maximum degree of the corresponding piercing graph. Our main result is that, for every fixed piercing degree, the standard local-search algorithms give PTASs for the unweighted \emph{Discrete Independent Set} and \emph{Set Cover} problems. The proof constructs a sublinear balanced separator for an appropriate locality graph and applies it adaptively throughout the recursive local-search analysis. This guarantee depends only on the piercing degree; in particular, it places no bound on the number of components created by an individual piercing pair. We also prove a polynomial shallow-trace bound depending only on the piercing degree. As consequences, for every fixed piercing degree, weighted Set Cover admits a deterministic $C_r$-approximation and weighted Discrete Independent Set admits a deterministic $O(r+1)$-approximation. These results extend the known guarantees for non-piercing families and apply, for example, to axis-parallel rectangles when every rectangle is pierced by only a bounded number of other rectangles.

Problem

• Piercing complexity
• Geometric optimization
• Discrete Independent Set
• Set Cover
• PTAS

Innovation

• Piercing Degree
• PTAS
• Local Search
• Balanced Separator
• Shallow-Trace Bound