Monochromatic Triangles, Triangle Listing and APSP
Abstract
One of the main hypotheses in finegrained complexity is that AllPairs Shortest Paths (APSP) for $n$node graphs requires $n^{3o(1)}$ time. Another famous hypothesis is that the $3$SUM problem for $n$ integers requires $n^{2o(1)}$ time. Although there are no direct reductions between $3$SUM and APSP, it is known that they are related: there is a problem, $(\min,+)$convolution that reduces in a finegrained way to both, and a problem Exact Triangle that both finegrained reduce to. In this paper we find more relationships between these two problems and other basic problems. Pătrașcu had shown that under the $3$SUM hypothesis the AllEdges Sparse Triangle problem in $m$edge graphs requires $m^{4/3o(1)}$ 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 $m=\tilde{O}(n^{1.5})$, and can be solved in $O(m^{1.41})$ time [Alon et al.'97] with the current matrix multiplication bounds, and in $\tilde{O}(m^{4/3})$ time if $\omega=2$. We show that one can reduce Exact Triangle to AllEdges Sparse Triangle, showing that AllEdges Sparse Triangle (and hence Triangle Listing) requires $m^{4/3o(1)}$ time also assuming the APSP hypothesis. This allows us to provide APSPhardness for many dynamic problems that were previously known to be hard under the $3$SUM hypothesis. We also consider the previously studied AllEdges Monochromatic Triangle problem. Via work of [Lincoln et al.'20], our result on AllEdges Sparse Triangle implies that if the AllEdges Monochromatic Triangle problem has an $O(n^{2.5\epsilon})$ time algorithm for $\epsilon>0$, then both the APSP and $3$SUM hypotheses are false. We also connect the problem to other ``intermediate'' problems, whose runtimes are between $O(n^\omega)$ and $O(n^3)$, such as the MaxMin product problem.
 Publication:

arXiv eprints
 Pub Date:
 July 2020
 arXiv:
 arXiv:2007.09318
 Bibcode:
 2020arXiv200709318V
 Keywords:

 Computer Science  Computational Complexity
 EPrint:
 To appear in FOCS'20