Optimality program in segment and string graphs

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.Comment: 19 pages, 15 figure

Algorithms and Complexity for Geodetic Sets on Planar and Chordal Graphs

We study the complexity of finding the geodetic number on subclasses of planar graphs and chordal graphs. A set S of vertices of a graph G is a geodetic set if every vertex of G lies in a shortest path between some pair of vertices of S. The Minimum Geodetic Set (MGS) problem is to find a geodetic set with minimum cardinality of a given graph. The problem is known to remain NP-hard on bipartite graphs, chordal graphs, planar graphs and subcubic graphs. We first study MGS on restricted classes of planar graphs: we design a linear-time algorithm for MGS on solid grids, improving on a 3-approximation algorithm by Chakraborty et al. (CALDAM, 2020) and show that MGS remains NP-hard even for subcubic partial grids of arbitrary girth. This unifies some results in the literature. We then turn our attention to chordal graphs, showing that MGS is fixed parameter tractable for inputs of this class when parameterized by their treewidth (which equals the clique number minus one). This implies a linear-time algorithm for k-trees, for fixed k. Then, we show that MGS is NP-hard on interval graphs, thereby answering a question of Ekim et al. (LATIN, 2012). As interval graphs are very constrained, to prove the latter result we design a rather sophisticated reduction technique to work around their inherent linear structure

On dominating set of some subclasses of string graphs

We provide constant factor approximation algorithms for the Minimum Dominating Set (MDS) problem on several subclasses of string graphs i.e. intersection graphs of simple curves on the plane. For k ≥ 0, unit Bk-VPG graphs are intersection graphs of simple rectilinear curves having at most k cusps (bends) and each segment of the curve being unit length. We give an 18-approximation algorithm for the MDS problem on unit B0-VPG graphs. This partially addresses a question of Katz et al. (2005) [24]. We also give an O(k4)- approximation algorithm for the MDS problem on unit Bk-VPG graphs. We show that there is an 8-approximation algorithm for the MDS problem on vertically-stabbed L-graphs. We also give a 656-approximation algorithm for the MDS problem on stabbed rectangle overlap graphs. This is the first constant-factor approximation algorithm for the MDS problem on stabbed rectangle overlap graphs and extends a result of Bandyapadhyay et al. (2019) [31]. We prove some hardness results to complement the above results

Optimality Program in Segment and String Graphs

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