SIGEST

Abstract

The SIGEST article in this issue is “From Circuit Complexity to Faster All-Pairs Shortest Paths,” by R. Ryan Williams. The original paper appeared in SIAM Journal on Computing (SICOMP) in 2018 and, as indicated by the high citation count, ideas from this work have inspired a broad range of future activity. For a given $n$-node graph, the problem tackled here is to compute all shortest paths between any pair of nodes, the so-called all-pairs shortest path (APSP) problem. As explained by the author (see Corollary 1.2) a solution to the APSP problem leads to solutions for many other problems in discrete mathematics. The key result in this work is a new algorithm that improves on the classical $O(n^3)$ complexity for the first time by a factor that is superpolylogarithmic. This is done by introducing a new technique, called the polynomial method. Key ingredients of the work are the use of tropical, or max-plus, algebra, where $\min$ plays the role of addition and $+$ is promoted to multiplication, and a new reduction to rectangular matrix multiplication over the field of two elements. In addition to creating an algorithm that runs faster than $n^3/\log^k n$ for every constant $k$, the author discusses the potential for achieving a truly subcubic speed of $n^{3-\epsilon}$ for some $\epsilon >0$. The original (SICOMP) article has been given an extended introduction to increase accessibility to the SIAM readership. Thanks to the wide applicability of the APSP problem, and the potential for further developments in the field, this paper should be of interest both to researchers working on algorithms and to those working on complexity.