# Subcubic equivalences between path, matrix, and triangle problems

## Abstract

We say an algorithm on n x n matrices with entries in [-M,M] (or n-node graphs with edge weights from [-M,M]) is truly subcubic if it runs in O(n 3-δ·poly(logM)) time for some δ > 0. We define a notion of subcubic reducibility, and show that many important problems on graphs and matrices solvable in O(n3) time are equivalent under subcubic reductions. Namely, the following weighted problems either all have truly subcubic algorithms, or none of them do: • The all-pairs shortest paths problem (APSP). • Detecting if a weighted graph has a triangle of negative total edge weight. • Listing up to n2.99 negative triangles in an edge-weighted graph. • Finding a minimum weight cycle in a graph of non-negative edge weights. • The replacement paths problem in an edge-weighted digraph. • Finding the second shortest simple path between two nodes in an edge-weighted digraph. • Checking whether a given matrix defines a metric. • Verifying the correctness of a matrix product over the (min, +)-semiring. Therefore, if APSP cannot be solved in n3-ε time for any ε > 0, then many other problems also need essentially cubic time. In fact we show generic equivalences between matrix products over a large class of algebraic structures used in optimization, verifying a matrix product over the same structure, and corresponding triangle detection problems over the structure. These equivalences simplify prior work on subcubic algorithms for all-pairs path problems, since it now suffices to give appropriate subcubic triangle detection algorithms. Other consequences of our work are new combinatorial approaches to Boolean matrix multiplication over the (OR,AND)-semiring (abbreviated as BMM). We show that practical advances in triangle detection would imply practical BMM algorithms, among other results. Building on our techniques, we give two new BMM algorithms: a derandomization of the recent combinatorial BMM algorithm of Bansal and Williams (FOCS'09), and an improved quantum algorithm for BMM. © 2010 IEEE.