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(Multivariate) k-SUM as Barrier to Succinct Computation

Authors Geri Gokaj, Marvin Künnemann, Sabine Storandt , Carina Truschel

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ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Fine-grained complexity theory
  • sumsets
  • additive combinatorics
  • succinct inputs
  • computational geometry

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Abstract

How does the time complexity of a problem change when the input is given succinctly rather than explicitly? We study this question for several geometric problems defined on a set X of N points in ℤ^d. As succinct representation, we choose a sumset (or Minkowski sum) representation: Instead of receiving X explicitly, we are given sets A,B of n points that define X as A+B = {a+b∣ a ∈ A,b ∈ B}.
We investigate the fine-grained complexity of this succinct version for several Õ(N)-time computable geometric primitives. Remarkably, we can tie their complexity tightly to the complexity of corresponding k-SUM problems. Specifically, we introduce as All-ints 3-SUM(n,n,k) the following multivariate, multi-output variant of 3-SUM: given sets A,B of size n and set C of size k, determine for all c ∈ C whether there are a ∈ A and b ∈ B with a+b = c. We obtain the following results:  
1) Succinct closest L_∞-pair requires time N^{1-o(1)} under the 3-SUM hypothesis, while succinct furthest L_∞-pair can be solved in time Õ(n). 
2) Succinct bichromatic closest L_∞-Pair requires time N^{1-o(1)} iff the 4-SUM hypothesis holds. 
3) The following problems are fine-grained equivalent to All-ints 3-SUM(n,n,k): succinct skyline computation in 2D with output size k and succinct batched orthogonal range search with k given ranges. This establishes conditionally tight Õ(min{nk, N})-time algorithms for these problems. We obtain further connections with All-ints 3-SUM(n,n,k) for succinctly computing independent sets in unit interval graphs.  Thus, (Multivariate) k-SUM problems precisely capture the barrier for enabling sumset-succinct computation for various geometric primitives.

Cite As Get BibTex

Geri Gokaj, Marvin Künnemann, Sabine Storandt, and Carina Truschel. (Multivariate) k-SUM as Barrier to Succinct Computation. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 42:1-42:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ESA.2025.42

Author Details

Geri Gokaj
  • Karlsruhe Institute of Technology, Germany
Marvin Künnemann
  • Karlsruhe Institute of Technology, Germany
Sabine Storandt
  • University of Konstanz, Germany
Carina Truschel
  • University of Konstanz, Germany

Funding

  • Gokaj, Geri: Research supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 462679611.
  • Truschel, Carina: Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 251654672 – TRR 161.

Acknowledgements

We thank all reviewers for their in-depth review and feedback.

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