  Some of the most fundamental and well-studied graph parameters are the
Diameter (the largest shortest paths distance) and Radius (the smallest
distance for which a "center" node can reach all other nodes). The natural and
important $ST$-variant considers two subsets $S$ and $T$ of the vertex set and
lets the $ST$-diameter be the maximum distance between a node in $S$ and a node
in $T$, and the $ST$-radius be the minimum distance for a node of $S$ to reach
all nodes of $T$. The bichromatic variant is the special case in which $S$ and
$T$ partition the vertex set.
  In this paper we present a comprehensive study of the approximability of $ST$
and Bichromatic Diameter, Radius, and Eccentricities, and variants, in graphs
with and without directions and weights. We give the first nontrivial
approximation algorithms for most of these problems, including time/accuracy
trade-off upper and lower bounds. We show that nearly all of our obtained
bounds are tight under the Strong Exponential Time Hypothesis (SETH), or the
related Hitting Set Hypothesis.
  For instance, for Bichromatic Diameter in undirected weighted graphs with $m$
edges, we present an $\tilde{O}(m^{3/2})$ time $5/3$-approximation algorithm,
and show that under SETH, neither the running time, nor the approximation
factor can be significantly improved while keeping the other unchanged.
