Monochromatic Triangles, Triangle Listing and APSP

Virginia Vassilevska Williams, Yinzhan Xu

Abstract: All-Pairs Shortest Paths (APSP) is one of the most basic problems in graph algorithms. Given an n-node directed or undirected graph with integer weights in {-nc, ..., nc} and no negative cycles, APSP asks to compute the shortest paths distance between every pair of vertices. The fastest known algorithm for APSP runs in n3/2Θ(√{logn}) time [Williams'14], and no truly subcubic time algorithms are known. One of the main hypotheses in fine-grained complexity is that APSP requires n3-o(1) time. Another famous hypothesis in fine-grained complexity is that the 3SUM problem for n integers (which can be solved in O(n2) time) requires n2-o(1) time. Although there are no direct reductions between 3SUM and APSP, it is known that they are related: the (min, +)-convolution problem reduces in a fine-grained way to both, and both fine-grained reduce to the Exact Triangle problem. 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 m4/3-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(n1.5), and can be solved in O(m1.41) time [Alon et al.'97] with the current matrix multiplication bounds, and in Õ(m4/3) 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 m4/3-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 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(n2.5-ε) time algorithm for , then both the APSP and 3SUM hypotheses are false. The fastest algorithm for All-Edges Monochromatic Triangle runs in Õ(n(3+ω)/2) time [Vassilevska et al.'06], and our new reduction shows that if ω = 2, this algorithm is best possible, unless 3SUM or APSP can be solved faster. Besides 3SUM, previously the only problems known to be fine-grained reducible to All-Edges Monochromatic Triangle were the seemingly easier problems directed unweighted APSP and Min-Witness Product [Lincoln et al.'20]. Our reduction shows that this problem is much harder. We also connect the problem to other “intermediate” problems, whose runtimes are between O(nω) and O(n3), such as the Max-Min product problem.