Shifting is Optimal under Gap-ETH: A Lower Bound Framework for Geometric Approximation Schemes

2026-07-07

AI Summary

Under the Gap-ETH assumption, this work establishes the conditional optimality of shifting-based PTAS algorithms for a range of d-dimensional unit ball graph problems—including Maximum Independent Set and Maximum Induced Forest—as well as the Unit Ball Piercing problem, all of which admit running times of $n^{O(1/\varepsilon^{d-1})}$. This result extends, for the first time, the known optimality of shifting techniques from two dimensions to arbitrary constant dimensions. To achieve this, the authors develop a unified maximization framework for geometric constraint satisfaction problems (CSPs), integrating De Berg’s cube wiring theorem with the reduction methodology of Marx and Sidiropoulos. This framework provides a common foundation for proving conditional hardness results across high-dimensional geometric optimization problems.

Abstract

The shifting technique of Hochbaum and Maass [J.ACM'85] produces PTASes with the fastest known running times $n^{O(1/\varepsilon^{d-1})}$ for several $d$-dimensional geometric problems. However, it is only known, due to Marx [FOCS'07], that these algorithms are indeed optimal for dimension $d=2$. We show that these running times are optimal under Gap-ETH for every constant dimension. More precisely, we develop a framework that enables us to prove the conditional optimality of the shifting algorithms for several problems on unit ball graphs, such as maximum independent set, maximum induced forest, and others, as well as for the problem of piercing unit balls. Our framework is built using the cube wiring theorem of De Berg et al. [SICOMP'20] and the reduction steps of Marx and Sidiropoulos [SoCG'14] to create a convenient maximization version of geometric CSP that can be used as a basis for reductions.

Problem

• shifting technique
• geometric approximation schemes
• Gap-ETH
• optimality
• running time

Innovation

• shifting technique
• Gap-ETH
• geometric approximation schemes
• conditional lower bounds
• geometric CSP