Hardness for Triangle Problems under Even More Believable Hypotheses: Reductions from Real APSP, Real 3SUM, and OV

The $3$SUM hypothesis, the APSP hypothesis and SETH are the three main hypotheses in fine-grained complexity. So far, within the area, the first two hypotheses have mainly been about integer inputs in the Word RAM model of computation. The "Real APSP" and "Real $3$SUM" hypotheses, which assert that the APSP and $3$SUM hypotheses hold for real-valued inputs in a reasonable version of the Real RAM model, are even more believable than their integer counterparts. Under the very believable hypothesis that at least one of the Integer $3$SUM hypothesis, Integer APSP hypothesis or SETH is true, Abboud, Vassilevska W. and Yu [STOC 2015] showed that a problem called Triangle Collection requires $n^{3-o(1)}$ time on an $n$-node graph. Our main result is a nontrivial lower bound for a slight generalization of Triangle Collection, called All-Color-Pairs Triangle Collection, under the even more believable hypothesis that at least one of the Real $3$SUM, the Real APSP, and the OV hypotheses is true. Combined with slight modifications of prior reductions, we obtain polynomial conditional lower bounds for problems such as the (static) ST-Max Flow problem and dynamic Max Flow, now under the new weaker hypothesis. Our main result is built on the following two lines of reductions. * Real APSP and Real $3$SUM hardness for the All-Edges Sparse Triangle problem. Prior reductions only worked from the integer variants of these problems. * Real APSP and OV hardness for a variant of the Boolean Matrix Multiplication problem. Along the way we show that Triangle Collection is equivalent to a simpler restricted version of the problem, simplifying prior work. Our techniques also have other interesting implications, such as a super-linear lower bound of Integer All-Numbers $3$SUM based on the Real $3$SUM hypothesis, and a tight lower bound for a string matching problem based on the OV hypothesis.