Exploring the Approximability Landscape of 3SUM
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Exploring the Approximability Landscape of 3SUM

Authors Karl Bringmann , Ahmed Ghazy , Marvin Künnemann

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Author Details

Karl Bringmann
  • Saarland University, Saarland Informatics Campus, Saarbrücken, Germany
  • Max Planck Institute for Informatics, Saarland Informatics Campus, Saarbrücken, Germany
Ahmed Ghazy
  • CISPA Helmholtz Center for Information Security, Saarbrücken, Germany
Marvin Künnemann
  • Karlsruhe Institute of Technology, Germany

Funding

  • Bringmann, Karl: This work is part of the project TIPEA that has received funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement No. 850979).
  • Künnemann, Marvin: Research partially supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – 462679611.

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Karl Bringmann, Ahmed Ghazy, and Marvin Künnemann. Exploring the Approximability Landscape of 3SUM. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 34:1-34:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ESA.2024.34

Abstract

Since an increasing number of problems in P have conditional lower bounds against exact algorithms, it is natural to study which of these problems can be efficiently approximated. Often, however, there are many potential ways to formulate an approximate version of a problem. We ask: How sensitive is the (in-)approximability of a problem in P to its precise formulation?
To this end, we perform a case study using the popular 3SUM problem. Its many equivalent formulations give rise to a wide range of potential approximate relaxations. Specifically, to obtain an approximate relaxation in our framework, one can choose among the options: (a) 3SUM or Convolution 3SUM, (b) monochromatic or trichromatic, (c) allowing under-approximation, over-approximation, or both, (d) approximate decision or approximate optimization, (e) single output or multiple outputs and (f) implicit or explicit target (given as input). 
We show general reduction principles between some variants and find that we can classify the remaining problems (over polynomially bounded positive integers) into three regimes:  
1) (1+ε)-approximable in near-linear time Õ(n + 1/ε), 
2) (1+ε)-approximable in near-quadratic time Õ(n/ε) or Õ(n+1/ε²), or 
3) non-approximable, i.e., requiring time n^{2± o(1)} even for any approximation factor.  In each of these three regimes, we provide matching upper and conditional lower bounds.
To prove our results, we establish two results that may be of independent interest: Over polynomially bounded integers, we show subquadratic equivalence of (min,+)-convolution and polyhedral 3SUM, and we prove equivalence of the Strong 3SUM conjecture and the Strong Convolution 3SUM conjecture.

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
Keywords
  • Fine-grained Complexity
  • Conditional Lower Bounds
  • Approximation Schemes
  • Min-Plus Convolution

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