Hardness for triangle problems under even more believable hypotheses: reductions from real APSP, real 3SUM, and OV

The 3SUM hypothesis, the All-Pairs Shortest Paths (APSP) hypothesis and the Strong Exponential Time Hypothesis are the three main hypotheses in the area of 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 3SUM" hypotheses, which assert that the APSP and 3SUM 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 3SUM 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. The main result of this paper 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 3SUM, the Real APSP, and the Orthogonal Vector (OV) hypotheses is true. Combined with slight modifications of prior reductions from Triangle Collection, we obtain polynomial conditional lower bounds for problems such as the (static) ST-Max Flow problem and dynamic versions of Max Flow, Single-Source Reachability Count, and Counting Strongly Connected Components, now under the new weaker hypothesis. Our main result is built on the following two lines of reductions. • Real APSP and Real 3SUM 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 3SUM based on the Real 3SUM hypothesis, and a tight lower bound for a string matching problem based on the OV hypothesis.