From Circuit Complexity to Faster All-Pairs Shortest Paths

Abstract

We present a randomized method for computing the min-plus product (a.k.a. tropical product) of two $n \times n$ matrices, yielding a faster algorithm for solving the all-pairs shortest path problem (APSP) in dense $n$-node directed graphs with arbitrary edge weights. In the real random-access machine model, where additions and comparisons of reals are unit cost (but all other operations have logarithmic cost), the algorithm runs in time $\frac{n^3}{2^{\Omega(\log n)^{1/2}}}$ and is correct with high probability. On the word random-access machine which permits constant-time operations on $\log(n)$-bit words, the algorithm runs in $n^3/2^{\Omega(\log n)^{1/2}} + n^{2+o(1)}\log(nM)$ time on graphs with edge weights in $([0,M] \cap \mathbb{Z})\cup\{\infty\}$. Prior algorithms needed either $\Theta(n^3/\log^c n)$ time for various $c \leq 2$, or $\Theta(M^{\alpha}n^{\beta})$ time for various $\alpha > 0$ and $\beta > 2$. Our algorithm applies a tool from circuit complexity, namely, the Razborov--Smolensky polynomials for approximately representing ${AC}^0[p]$ circuits, to efficiently reduce a matrix product over the min-plus algebra to a relatively small number of rectangular matrix products over $\mathbb{F}_2$. Each rectangular matrix product can be computed using a particularly efficient method due to Coppersmith. We also give a deterministic version of the algorithm running in $n^3/2^{\log^{\delta} n}$ time for some $\delta > 0$, which utilizes the Yao--Beigel--Tarui translation of ${AC}^0[m]$ circuits into “nice” depth-two circuits.