Optimality program in segment and string graphs

TL;DR

This paper investigates the existence of subexponential algorithms for coloring and other problems on string and segment intersection graphs, revealing surprising results like 3-Coloring being solvable in subexponential time.

Contribution

It demonstrates that certain problems like 3-Coloring have subexponential algorithms on string graphs, and establishes ETH-based lower bounds for problems like 4-Coloring.

Findings

3-Coloring solvable in $2^{O(n^{2/3}\log^{O(1)}n)}$ time on string graphs

ETH lower bound for 4-Coloring on axis-parallel segments

Subexponential algorithms extend to Min Feedback Vertex Set but not to Min Dominating Set or Min Independent Dominating Set

Abstract

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.