Monochromatic Triangles, Triangle Listing and APSP
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One of the main hypotheses in fine-grained complexity is that All-Pairs Shortest Paths (APSP) for n-node graphs requires n^3-o(1) time. Another famous hypothesis is that the 3SUM problem for n integers requires n^2-o(1) time. Although there are no direct reductions between 3SUM and APSP, it is known that they are related: there is a problem, (min,+)-convolution that reduces in a fine-grained way to both, and a problem Exact Triangle that both fine-grained reduce to. In this paper we find more relationships between these two problems and other basic problems. Pătraşcu had shown that under the 3SUM hypothesis the All-Edges Sparse Triangle problem in m-edge graphs requires m^4/3-o(1) time. The latter problem asks to determine for every edge e, whether e is in a triangle. It is equivalent to the problem of listing m triangles in an m-edge graph where m=Õ(n^1.5), and can be solved in O(m^1.41) time [Alon et al.'97] with the current matrix multiplication bounds, and in Õ(m^4/3) time if ω=2. We show that one can reduce Exact Triangle to All-Edges Sparse Triangle, showing that All-Edges Sparse Triangle (and hence Triangle Listing) requires m^4/3-o(1) time also assuming the APSP hypothesis. This allows us to provide APSP-hardness for many dynamic problems that were previously known to be hard under the 3SUM hypothesis. We also consider the previously studied All-Edges Monochromatic Triangle problem. Via work of [Lincoln et al.'20], our result on All-Edges Sparse Triangle implies that if the All-Edges Monochromatic Triangle problem has an O(n^2.5-ϵ) time algorithm for ϵ>0, then both the APSP and 3SUM hypotheses are false. We also connect the problem to other “intermediate” problems, whose runtimes are between O(n^ω) and O(n^3), such as the Max-Min product problem.
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