  Planar graphs are known to allow subexponential algorithms running in time
$2^{O(\sqrt n)}$ or $2^{O(\sqrt n \log n)}$ for most of the paradigmatic
problems, while the brute-force time $2^{\Theta(n)}$ is very likely to be
asymptotically best on general graphs. Intrigued by an algorithm packing curves
in $2^{O(n^{2/3}\log n)}$ by Fox and Pach [SODA'11], we investigate which
problems have subexponential algorithms on the intersection graphs of curves
(string graphs) or segments (segment intersection graphs) and which problems
have no such algorithms under the ETH (Exponential Time Hypothesis). Among our
results, we show that, quite surprisingly, 3-Coloring can also be solved in
time $2^{O(n^{2/3}\log^{O(1)}n)}$ on string graphs while an algorithm running
in time $2^{o(n)}$ for 4-Coloring even on axis-parallel segments (of unbounded
length) would disprove the ETH. For 4-Coloring of unit segments, we show a
weaker ETH lower bound of $2^{o(n^{2/3})}$ which exploits the celebrated
Erd\H{o}s-Szekeres theorem. The subexponential running time also carries over
to Min Feedback Vertex Set but not to Min Dominating Set and Min Independent
Dominating Set.
