TY - GEN

T1 - Hardness and Approximation for the Geodetic Set Problem in Some Graph Classes

AU - Chakraborty, Dibyayan

AU - Foucaud, Florent

AU - Gahlawat, Harmender

AU - Ghosh, Subir Kumar

AU - Roy, Bodhayan

N1 - Publisher Copyright:
© 2020, Springer Nature Switzerland AG.

PY - 2020/1/1

Y1 - 2020/1/1

N2 - In this paper, we study the computational complexity of finding the geodetic number of graphs. A set of vertices S of a graph G is a geodetic set if any vertex of G lies in some shortest path between some pair of vertices from S. The Minimum Geodetic Set (MGS) problem is to find a geodetic set with minimum cardinality. In this paper, we prove that solving MGS is NP-hard on planar graphs with a maximum degree six and line graphs. We also show that unless there is no polynomial time algorithm to solve MGS with sublogarithmic approximation factor (in terms of the number of vertices) even on graphs with diameter 2. On the positive side, we give an 3 \of {n}\log n\right) -approximation algorithm for MGS on general graphs of order n. We also give a 3-approximation algorithm for MGS on solid grid graphs which are planar.

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

UR - http://www.scopus.com/inward/record.url?scp=85079554051&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-39219-2_9

DO - 10.1007/978-3-030-39219-2_9

M3 - Conference contribution

AN - SCOPUS:85079554051

SN - 9783030392185

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 102

EP - 115

BT - Algorithms and Discrete Applied Mathematics - 6th International Conference, CALDAM 2020, Proceedings

A2 - Changat, Manoj

A2 - Das, Sandip

PB - Springer

T2 - 6th International Conference on Algorithms and Discrete Applied Mathematics, CALDAM 2020

Y2 - 13 February 2020 through 15 February 2020

ER -