$\{-1,0,1\}$-APSP and (min,max)-Product Problems

TL;DR

This paper introduces a new reduction-based algorithm for the $ ext{-}1,0,1$-APSP problem, linking it to the (min,max)-product problem, and achieves faster solutions using advanced matrix multiplication techniques.

Contribution

It presents a novel reduction from $ ext{-}1,0,1$-APSP to target-(min,max)-product, enabling faster algorithms based on recent (min,max)-product advancements.

Findings

The $ ext{-}1,0,1$-APSP problem can be solved in the same time as approximate APSP for positive weights.

A simple algorithm for target-(min,max)-product is developed for inputs generated by the reduction.

Using fast rectangular matrix multiplication, the new algorithm outperforms existing (min,max)-product algorithms.

Abstract

In the $\{-1,0,1\}$-APSP problem the goal is to compute all-pairs shortest paths (APSP) on a directed graph whose edge weights are all from $\{-1,0,1\}$. In the (min,max)-product problem the input is two $n\times n$ matrices $A$ and $B$, and the goal is to output the (min,max)-product of $A$ and $B$. This paper provides a new algorithm for the $\{-1,0,1\}$-APSP problem via a simple reduction to the target-(min,max)-product problem where the input is three $n\times n$ matrices $A,B$, and $T$, and the goal is to output a Boolean $n\times n$ matrix $C$ such that the $(i,j)$ entry of $C$ is 1 if and only if the $(i,j)$ entry of the (min,max)-product of $A$ and $B$ is exactly the $(i,j)$ entry of the target matrix $T$. If (min,max)-product can be solved in $T_{MM}(n) = \Omega(n^2)$ time then it is straightforward to solve target-(min,max)-product in $O(T_{MM}(n))$ time. Thus, given the recent result of Bringmann, K\"unnemann, and Wegrzycki [STOC 2019], the $\{-1,0,1\}$-APSP problem can be solved in the same time needed for solving approximate APSP on graphs with positive weights. Moreover, we design a simple algorithm for target-(min,max)-product when the inputs are restricted to the family of inputs generated by our reduction. Using fast rectangular matrix multiplication, the new algorithm is faster than the current best known algorithm for (min,max)-product.