Subcubic Equivalences between Path, Matrix and Triangle Problems

We say an algorithm on <i>n</i> × <i>n</i> matrices with integer entries in [−<i>M</i>,<i>M</i>] (or <i>n</i>-node graphs with edge weights from [−<i>M</i>,<i>M</i>]) is <i>truly subcubic</i> if it runs in <i>O</i>(<i>n</i>^{3 − Δ} ċ poly(log <i>M</i>)) time for some Δ > 0. We define a notion of <i>subcubic reducibility</i> and show that many important problems on graphs and matrices solvable in <i>O</i>(<i>n</i>³) time are <i>equivalent</i> under subcubic reductions. Namely, the following weighted problems either <i>all</i> have truly subcubic algorithms, or none of them do: •The all-pairs shortest paths problem on weighted digraphs (APSP). •Detecting if a weighted graph has a triangle of negative total edge weight. •Listing up to <i>n</i>^2.99 negative triangles in an edge-weighted graph. •Finding a minimum weight cycle in a graph of non-negative edge weights. •The replacement paths problem on weighted digraphs. •Finding the second shortest simple path between two nodes in a weighted digraph. •Checking whether a given matrix defines a metric. •Verifying the correctness of a matrix product over the (min, +)-semiring. •Finding a maximum subarray in a given matrix. Therefore, if APSP cannot be solved in <i>n</i>^3−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 improved BMM algorithms: a derandomization of the combinatorial BMM algorithm of Bansal and Williams (FOCS’09), and an improved quantum algorithm for BMM.