# Approximating APSP without Scaling: Equivalence of Approximate Min-Plus and Exact Min-Max

## TL;DR

This paper develops strongly polynomial approximation algorithms for APSP and related graph problems that avoid the dependence on weight scale W, providing faster solutions especially for directed graphs, and establishes an equivalence between approximating APSP and Min-Max Product.

## Contribution

It introduces strongly polynomial approximation schemes for APSP and related problems, removing the dependence on weight scale W, and proves the equivalence between approximating directed APSP and Min-Max Product.

## Key findings

- Strongly polynomial algorithms for undirected APSP and graph characteristics.
- Faster strongly polynomial approximation scheme for directed APSP.
- Proved the equivalence between approximating directed APSP and Min-Max Product.

## Abstract

Zwick's $(1+\varepsilon)$-approximation algorithm for the All Pairs Shortest Path (APSP) problem runs in time $\widetilde{O}(\frac{n^\omega}{\varepsilon} \log{W})$, where $\omega \le 2.373$ is the exponent of matrix multiplication and $W$ denotes the largest weight. This can be used to approximate several graph characteristics including the diameter, radius, median, minimum-weight triangle, and minimum-weight cycle in the same time bound. Since Zwick's algorithm uses the scaling technique, it has a factor $\log W$ in the running time. In this paper, we study whether APSP and related problems admit approximation schemes avoiding the scaling technique. That is, the number of arithmetic operations should be independent of $W$; this is called strongly polynomial. Our main results are as follows. - We design approximation schemes in strongly polynomial time $O(\frac{n^\omega}{\varepsilon} \text{polylog}(\frac{n}{\varepsilon}))$ for APSP on undirected graphs as well as for the graph characteristics diameter, radius, median, minimum-weight triangle, and minimum-weight cycle on directed or undirected graphs. - For APSP on directed graphs we design an approximation scheme in strongly polynomial time $O(n^{\frac{\omega + 3}{2}} \varepsilon^{-1} \text{polylog}(\frac{n}{\varepsilon}))$. This is significantly faster than the best exact algorithm. - We explain why our approximation scheme for APSP on directed graphs has a worse exponent than $\omega$: Any improvement over our exponent $\frac{\omega + 3}{2}$ would improve the best known algorithm for Min-Max Product In fact, we prove that approximating directed APSP and exactly computing the Min-Max Product are equivalent.