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Zwick's $(1+\varepsilon)$-approximation algorithm for the All Pairs ShortestPath (APSP) problem runs in time $\widetilde{O}(\frac{n^\omega}{\varepsilon}\log{W})$, where $\omega \le 2.373$ is the exponent of matrix multiplicationand $W$ denotes the largest weight. This can be used to approximate severalgraph characteristics including the diameter, radius, median, minimum-weighttriangle, 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 problemsadmit approximation schemes avoiding the scaling technique. That is, the numberof arithmetic operations should be independent of $W$; this is called stronglypolynomial. Our main results are as follows.  - We design approximation schemes in strongly polynomial time$O(\frac{n^\omega}{\varepsilon} \text{polylog}(\frac{n}{\varepsilon}))$ forAPSP on undirected graphs as well as for the graph characteristics diameter,radius, median, minimum-weight triangle, and minimum-weight cycle on directedor undirected graphs.  - For APSP on directed graphs we design an approximation scheme in stronglypolynomial time $O(n^{\frac{\omega + 3}{2}} \varepsilon^{-1}\text{polylog}(\frac{n}{\varepsilon}))$. This is significantly faster than thebest exact algorithm.  - We explain why our approximation scheme for APSP on directed graphs has aworse exponent than $\omega$: Any improvement over our exponent $\frac{\omega +3}{2}$ would improve the best known algorithm for Min-Max Product In fact, weprove that approximating directed APSP and exactly computing the Min-MaxProduct are equivalent.

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