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

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× 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× n matrices A,B, and T, and the goal is to output a Boolean n× 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) = Ω(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, Künnemann, 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.

## Authors

10/10/2019

### Truly Subcubic Min-Plus Product for Less Structured Matrices, with Applications

The goal of this paper is to get truly subcubic algorithms for Min-Plus ...
08/27/2018

### Max-Min and Min-Max universally yield Gumbel

"A chain is only as strong as its weakest link" says the proverb. But wh...
04/29/2020

### Quantum and approximation algorithms for maximum witnesses of Boolean matrix products

The problem of finding maximum (or minimum) witnesses of the Boolean pro...
09/30/2020

### Monochromatic Triangles, Intermediate Matrix Products, and Convolutions

The most studied linear algebraic operation, matrix multiplication, has ...
07/17/2020

### A Hölderian backtracking method for min-max and min-min problems

We present a new algorithm to solve min-max or min-min problems out of t...
06/24/2013

### A Decomposition of the Max-min Fair Curriculum-based Course Timetabling Problem

We propose a decomposition of the max-min fair curriculum-based course t...
05/06/2021

### Faster Monotone Min-Plus Product, Range Mode, and Single Source Replacement Paths

One of the most basic graph problems, All-Pairs Shortest Paths (APSP) is...
##### This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.