Optimality Program in Segment and String Graphs

Planar graphs are known to allow subexponential algorithms running in time 2O(<) or 2O(<logn) for most of the paradigmatic problems, while the brute-force time 2(n) is very likely to be asymptotically best on general graphs. Intrigued by an algorithm packing curves in 2O(n2/3logn) 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 Exponential Time Hypothesis (ETH). Among our results, we show that, quite surprisingly, 3-Coloring can also be solved in time 2O(n2/3logO(1)n) on string graphs while an algorithm running in time 2o(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 lower bound, excluding a 2o(n2/3) algorithm (under the ETH). The construction exploits the celebrated Erds-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.