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Shifting Is Optimal Under Gap-ETH: A Lower Bound Framework for Geometric Approximation Schemes

Authors Manuel Cáceres , Sándor Kisfaludi-Bak , Saeed Odak

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LIPIcs.ESA.2026.38.pdf
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ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Computational geometry
  • Theory of computation → Packing and covering problems
Keywords
  • Gap-ETH Lower Bounds
  • Geometric PTASes
  • Geometric CSP
  • Shifting Technique
  • Maximum Induced Forest
  • Constant Dimension

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Abstract

The shifting technique of Hochbaum and Maass [J.ACM'85] produces PTASes with the fastest known running times n^O(1/ε^{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.

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Manuel Cáceres, Sándor Kisfaludi-Bak, and Saeed Odak. Shifting Is Optimal Under Gap-ETH: A Lower Bound Framework for Geometric Approximation Schemes. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 38:1-38:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ESA.2026.38

Author Details

Manuel Cáceres
  • Department of Computer Science, Aalto University, Finland
  • Department of Mathematics and Computer Science, University of Southern Denmark, Odense, Denmark
Sándor Kisfaludi-Bak
  • Department of Computer Science, Aalto University, Espoo, Finland
Saeed Odak
  • Department of Computer Science, Aalto University, Espoo, Finland

Funding

  • Cáceres, Manuel: Supported by the Helsinki Institute for Information Technology HIIT.
  • Kisfaludi-Bak, Sándor: Supported by the Research Council of Finland, Grant 363444.

Acknowledgements

We appreciate the insightful comments provided by the anonymous reviewers.

References

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