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Completeness Theorems for k-SUM and Geometric Friends: Deciding Fragments of Linear Integer Arithmetic

Authors Geri Gokaj , Marvin Künnemann

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Author Details

Geri Gokaj
  • Karlsruhe Institute of Technology, Germany
Marvin Künnemann
  • Karlsruhe Institute of Technology, Germany

Funding

This research is supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 462679611.

Acknowledgements

The authors like to thank the reviewers for constructive feedback as well as Karl Bringmann and Nick Fischer for helpful discussions.

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Geri Gokaj and Marvin Künnemann. Completeness Theorems for k-SUM and Geometric Friends: Deciding Fragments of Linear Integer Arithmetic. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 55:1-55:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ITCS.2025.55

Abstract

In the last three decades, the k-SUM hypothesis has emerged as a satisfying explanation of long-standing time barriers for a variety of algorithmic problems. Yet to this day, the literature knows of only few proven consequences of a refutation of this hypothesis. Taking a descriptive complexity viewpoint, we ask: What is the largest logically defined class of problems captured by the k-SUM problem?
To this end, we introduce a class FOP_ℤ of problems corresponding to deciding sentences in Presburger arithmetic/linear integer arithmetic over finite subsets of integers. We establish two large fragments for which the k-SUM problem is complete under fine-grained reductions:  
1) The k-SUM problem is complete for deciding the sentences with k existential quantifiers. 
2) The 3-SUM problem is complete for all 3-quantifier sentences of FOP_ℤ expressible using at most 3 linear inequalities.  Specifically, a faster-than-n^{⌈k/2⌉ ± o(1)} algorithm for k-SUM (or faster-than-n^{2 ± o(1)} algorithm for 3-SUM, respectively) directly translate to polynomial speedups of a general algorithm for all sentences in the respective fragment.
Observing a barrier for proving completeness of 3-SUM for the entire class FOP_ℤ, we turn to the question which other - seemingly more general - problems are complete for FOP_ℤ. In this direction, we establish FOP_ℤ-completeness of the problem pair of Pareto Sum Verification and Hausdorff Distance under n Translations under the L_∞/L₁ norm in ℤ^d. In particular, our results invite to investigate Pareto Sum Verification as a high-dimensional generalization of 3-SUM.

Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity theory and logic
Keywords
  • fine-grained complexity theory
  • descriptive complexity
  • presburger arithmetic
  • completeness results
  • k-SUM

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