TY - GEN
T1 - Structural parameterizations of dominating set variants
AU - Goyal, Dishant
AU - Jacob, Ashwin
AU - Kumar, Kaushtubh
AU - Majumdar, Diptapriyo
AU - Raman, Venkatesh
N1 - Publisher Copyright:
© 2018, Springer International Publishing AG, part of Springer Nature.
PY - 2018/1/1
Y1 - 2018/1/1
N2 - We consider structural parameterizations of the fundamental dominating set problem and its variants in the parameter ecology program. We give improved fixed-parameter tractable (FPT) algorithms and lower bounds under well-known conjectures for dominating set in graphs that are k vertices away from a cluster graph or a split graph. These are graphs in which there is a set of k vertices (called the modulator) whose deletion results in a cluster graph or a split graph. We also call k as the deletion distance (to the appropriate class of graphs). Specifically, we show the following results. When parameterized by the deletion distance k to cluster graphs, we can find a minimum dominating set in O*(3K) time (O* notation ignores polynomial factors of input). Within the same time, we can also find a minimum independent dominating set (IDS) or a minimum efficient dominating set (EDS) or a minimum total dominating set. These algorithms are obtained through a dynamic programming approach for an interesting generalization of set cover which may be of independent interest.We complement our upper bound results by showing that at least for dominating set and total dominating set, O*((2Ȣ∈)k) time algorithm is not possible for any ∈ > 0 under, what is known as, Set Cover Conjecture. We also show that most of these variants of dominating set do not have polynomial sized kernel. The standard dominating set and most of its variants are NP-hard or W[2]-hard in split graphs. For the two variants IDS and EDS that are polynomial time solvable in split graphs, we show that when parameterized by the deletion distance k to split graphs, IDS can be solved in O*(2k) time and we provide an Ω(2k) lower bound under the strong exponential time hypothesis (SETH);the 2k barrier can be broken for EDS by designing an O*(3k/2) algorithm. This is one of the very few problems with a runtime better than O*(2k) in the realm of structural parameterization. We also show that no 2o(k) algorithm is possible unless the exponential time hypothesis (ETH) is false.
AB - We consider structural parameterizations of the fundamental dominating set problem and its variants in the parameter ecology program. We give improved fixed-parameter tractable (FPT) algorithms and lower bounds under well-known conjectures for dominating set in graphs that are k vertices away from a cluster graph or a split graph. These are graphs in which there is a set of k vertices (called the modulator) whose deletion results in a cluster graph or a split graph. We also call k as the deletion distance (to the appropriate class of graphs). Specifically, we show the following results. When parameterized by the deletion distance k to cluster graphs, we can find a minimum dominating set in O*(3K) time (O* notation ignores polynomial factors of input). Within the same time, we can also find a minimum independent dominating set (IDS) or a minimum efficient dominating set (EDS) or a minimum total dominating set. These algorithms are obtained through a dynamic programming approach for an interesting generalization of set cover which may be of independent interest.We complement our upper bound results by showing that at least for dominating set and total dominating set, O*((2Ȣ∈)k) time algorithm is not possible for any ∈ > 0 under, what is known as, Set Cover Conjecture. We also show that most of these variants of dominating set do not have polynomial sized kernel. The standard dominating set and most of its variants are NP-hard or W[2]-hard in split graphs. For the two variants IDS and EDS that are polynomial time solvable in split graphs, we show that when parameterized by the deletion distance k to split graphs, IDS can be solved in O*(2k) time and we provide an Ω(2k) lower bound under the strong exponential time hypothesis (SETH);the 2k barrier can be broken for EDS by designing an O*(3k/2) algorithm. This is one of the very few problems with a runtime better than O*(2k) in the realm of structural parameterization. We also show that no 2o(k) algorithm is possible unless the exponential time hypothesis (ETH) is false.
UR - http://www.scopus.com/inward/record.url?scp=85048044146&partnerID=8YFLogxK
U2 - 10.1007/978-3-319-90530-3_14
DO - 10.1007/978-3-319-90530-3_14
M3 - Conference contribution
AN - SCOPUS:85048044146
SN - 9783319905297
T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
SP - 157
EP - 168
BT - Computer Science - Theory and Applications - 13th International Computer Science Symposium in Russia, CSR 2018, Proceedings
A2 - Podolskii, Vladimir V.
A2 - Fomin, Fedor V.
PB - Springer Verlag
T2 - 13th International Computer Science Symposium in Russia, CSR 2018
Y2 - 6 June 2018 through 10 June 2018
ER -