Subcubic Equivalences Between Path, Matrix, and Triangle Problems

• Published in: Journal of the ACM Vol. 65; no. 5; pp. 1 - 38
• Main Authors: Williams, Virginia Vassilevska, Williams, R. Ryan
• Format: Journal Article
• Language: English
• Published: New York, NY, USA ACM 29.08.2018; Association for Computing Machinery
• Subjects: Algorithms; Boolean algebra; Combinatorial analysis; Computational complexity and cryptography; Design and analysis of algorithms; Equivalence; Graph algorithms; Graph algorithms analysis; Graph theory; Graphs; Integer programming; Matrix; Minimum weight; Problems, reductions and completeness; Shortest paths; Theory of computation; all-pairs shortest paths; subcubic time; Fine-grained complexity; fine-grained reductions; equivalences
• ISSN: 0004-5411; 1557-735X
• DOI: 10.1145/3186893

We say an algorithm on n × n matrices with integer entries in [−M,M] (or n-node graphs with edge weights from [−M,M]) is truly subcubic if it runs in O(n3 − δ poly(log M)) 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 on weighted digraphs (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 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 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 improved BMM algorithms: a derandomization of the combinatorial BMM algorithm of Bansal and Williams (FOCS’09), and an improved quantum algorithm for BMM.