Hardness and approximation for the geodetic set problem in some graph classes

TL;DR

This paper investigates the computational difficulty of finding the geodetic number in graphs, proving NP-hardness in certain classes and providing approximation algorithms for general and specific graph families.

Contribution

It establishes NP-hardness of the Minimum Geodetic Set problem on planar and line graphs, and introduces approximation algorithms for general and grid graphs.

Findings

NP-hardness on planar graphs with max degree six

No polynomial-time sublogarithmic approximation unless P=NP

O(√[3]{n} log n)-approximation for general graphs

Abstract

In this paper, we study the computational complexity of finding the \emph{geodetic number} of graphs. A set of vertices $S$ of a graph $G$ is a \emph{geodetic set} if any vertex of $G$ lies in some shortest path between some pair of vertices from $S$. The \textsc{Minimum Geodetic Set (MGS)} problem is to find a geodetic set with minimum cardinality. In this paper, we prove that solving the \textsc{MGS} problem is NP-hard on planar graphs with a maximum degree six and line graphs. We also show that unless $P=NP$, there is no polynomial time algorithm to solve the \textsc{MGS} problem with sublogarithmic approximation factor (in terms of the number of vertices) even on graphs with diameter $2$. On the positive side, we give an $O\left(\sqrt[3]{n}\log n\right)$-approximation algorithm for the \textsc{MGS} problem on general graphs of order $n$. We also give a $3$-approximation algorithm for the \textsc{MGS} problem on the family of solid grid graphs which is a subclass of planar graphs.