Exploring the Approximability Landscape of 3SUM

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),
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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.