Geometric Optimization Parameterized by Piercing Complexity

Packing and Covering problems with geometric regions in the plane have been extensively studied and several notions of “complexity” of the regions involved have been developed and exploited to obtain good approximation algorithms. Examples of such complexity measures are VC-dimension, union complexity, shallow-cell complexity, fatness, etc. While these restrictions lead to constant-factor approximation algorithms in many cases, they typically do not lead to PTASs. In fact, several geometric Set Cover and Discrete Independent Set variants remain APX-hard even when these parameters are small, as demonstrated in earlier work by Chan and Grant (Exact algorithms and APX-hardness results for geometric packing and covering problems. Comput. Geom., 2014), and by Har-Peled and Quanrud (Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs. SIAM J. Comput., 2017). A key feature of these hardness constructions is that many pairs of regions in the input pierce one another. Motivated by this observation, we initiate a systematic study of geometric families parameterized by their piercing complexity. A connected region A is said to pierce a connected region B if B \ A has more than one connected component; we consider instances in which every region is pierced by at most a constant number of others. This framework smoothly interpolates between the classical non-piercing case-where local-search PTASs are known due to Raman and Ray (Constructing Planar Support for Non-Piercing Regions, Discret. Comput. Geom., 2020), and the fully general case, where APX-hardness persists. Our main contribution is to show that bounded-piercing families admit efficient approximation schemes for fundamental geometric optimization problems. For regions in the plane with a constant piercing bound, we obtain PTASs for the (unweighted) Discrete Independent Set and Set Cover problems, and constant-factor approximation algorithms for their weighted variants. These results strictly generalize the known PTASs for non-piercing families and yield improved guarantees for several long-standing special cases, including Independent Set and Set Cover with axis-parallel rectangles under bounded piercing. Overall, our work identifies piercing complexity as a robust and expressive topological parameter-distinct from geometric notions such as density or fatness-and demonstrates that bounding this parameter yields a broad family of geometric instances for which PTASs become achievable.