TY - GEN

T1 - Mixed dominating set

T2 - 43rd International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2017

AU - Jain, Pallavi

AU - Jayakrishnan, M.

AU - Panolan, Fahad

AU - Sahu, Abhishek

N1 - Funding Information:
The research leading to these results have received funding from the Research Council via ERC Advanced Investigator Grant 267959.
Publisher Copyright:
© 2017, Springer International Publishing AG.

PY - 2017/1/1

Y1 - 2017/1/1

N2 - In the mixed dominating set (mds) problem, we are given an n-vertex graph G and a positive integer k, and the objective is to decide whether there exists a set S⊆ V(G) ∪ E(G) of cardinality at most k such that every element x∈ (V(G) ∪ E(G)) \ S is either adjacent to or incident with an element of S. We show that mds can be solved in time 7. 465 knO ( 1 ) on general graphs, and in time 2O(k)nO(1) on planar graphs. We complement this result by showing that mds does not admit an algorithm with running time 2 o ( k )nO ( 1 ) unless the Exponential Time Hypothesis (ETH) fails, and that it does not admit a polynomial kernel unless coNP ⊆ NP/ poly. In addition, we provide an algorithm which, given a graph G together with a tree decomposition of width tw, solves mds in time 6 twnO ( 1 ). We finally show that unless the Set Cover Conjecture (SeCoCo) fails, mds does not admit an algorithm with running time O((2 - ϵ) tw ( G )nO ( 1 )) for any ϵ> 0, where tw(G) is the tree-width of G.

AB - In the mixed dominating set (mds) problem, we are given an n-vertex graph G and a positive integer k, and the objective is to decide whether there exists a set S⊆ V(G) ∪ E(G) of cardinality at most k such that every element x∈ (V(G) ∪ E(G)) \ S is either adjacent to or incident with an element of S. We show that mds can be solved in time 7. 465 knO ( 1 ) on general graphs, and in time 2O(k)nO(1) on planar graphs. We complement this result by showing that mds does not admit an algorithm with running time 2 o ( k )nO ( 1 ) unless the Exponential Time Hypothesis (ETH) fails, and that it does not admit a polynomial kernel unless coNP ⊆ NP/ poly. In addition, we provide an algorithm which, given a graph G together with a tree decomposition of width tw, solves mds in time 6 twnO ( 1 ). We finally show that unless the Set Cover Conjecture (SeCoCo) fails, mds does not admit an algorithm with running time O((2 - ϵ) tw ( G )nO ( 1 )) for any ϵ> 0, where tw(G) is the tree-width of G.

UR - http://www.scopus.com/inward/record.url?scp=85034050071&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-68705-6_25

DO - 10.1007/978-3-319-68705-6_25

M3 - Conference contribution

AN - SCOPUS:85034050071

SN - 9783319687049

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

SP - 330

EP - 343

BT - Graph-Theoretic Concepts in Computer Science - 43rd International Workshop, WG 2017, Revised Selected Papers

A2 - Woeginger, Gerhard J.

A2 - Bodlaender, Hans L.

PB - Springer Verlag

Y2 - 21 June 2017 through 23 June 2017

ER -