Approximating Minimum Dominating Set on String Graphs

TL;DR

This paper develops approximation algorithms for the Minimum Dominating Set problem on string graphs and their subclasses, including vertically-stabbed-L graphs and unit B_k-VPG graphs, by studying related geometric intersection problems.

Contribution

It introduces new approximation algorithms for MDS on complex geometric intersection graphs, leveraging solutions to the SSR problem and extending to unit B_k-VPG graphs.

Findings

8-approximation for MDS on vertically-stabbed-L graphs

2-approximation for SSR problem in near-linear time

O(k^4)-approximation for MDS on unit B_k-VPG graphs

Abstract

In this paper, we give approximation algorithms for the \textsc{Minimum Dominating Set (MDS)} problem on \emph{string} graphs and its subclasses. A \emph{path} is a simple curve made up of alternating horizontal and vertical line segments. A \emph{$k$-bend path} is a path made up of at most $k + 1$ line segments. An \textsc{L}-path is a $1$-bend path having the shape `\textsc{L}'. A \emph{vertically-stabbed-\textsc{L} graph} is an intersection graph of \textsc{L}-paths intersecting a common vertical line. We give a polynomial time $8$-approximation algorithm for \textsc{MDS} problem on vertically-stabbed-\textsc{L} graphs whose APX-hardness was shown by Bandyapadhyay et al. (\textsc{MFCS}, 2018). To prove the above result, we needed to study the \emph{Stabbing segments with rays} (\textsc{SSR}) problem introduced by Katz et al. (\textsc{Comput. Geom. 2005}). In the \textsc{SSR} problem, the input is a set of (disjoint) leftward-directed rays, and a set of (disjoint) vertical segments. The objective is to select a minimum number of rays that intersect all vertical segments. We give a $O((n+m)\log (n+m))$-time $2$-approximation algorithm for the \textsc{SSR} problem where $n$ and $m$ are the number of rays and segments in the input. A \emph{unit $k$-bend path} is a $k$-bend path whose segments are of unit length. A graph is a \emph{unit $B_k$-VPG graph} if it is an intersection graph of unit $k$-bend paths. Any string graph is a unit-$B_k$-VPG graph for some finite $k$. Using our result on \textsc{SSR}-problem, we give a polynomial time $O(k^4)$-approximation algorithm for \textsc{MDS} problem on unit $B_k$-VPG graphs for $k\geq 0$.