19 research outputs found
Dominator Coloring and CD Coloring in Almost Cluster Graphs
In this paper, we study two popular variants of Graph Coloring -- Dominator
Coloring and CD Coloring. In both problems, we are given a graph and a
natural number as input and the goal is to properly color the vertices
with at most colors with specific constraints. In Dominator Coloring, we
require for each , a color such that dominates all vertices
colored . In CD Coloring, we require for each color , a
which dominates all vertices colored . These problems, defined due to their
applications in social and genetic networks, have been studied extensively in
the last 15 years. While it is known that both problems are fixed-parameter
tractable (FPT) when parameterized by where is the treewidth of
, we consider strictly structural parameterizations which naturally arise
out of the problems' applications.
We prove that Dominator Coloring is FPT when parameterized by the size of a
graph's cluster vertex deletion (CVD) set and that CD Coloring is FPT
parameterized by CVD set size plus the number of remaining cliques. En route,
we design a simpler and faster FPT algorithms when the problems are
parameterized by the size of a graph's twin cover, a special CVD set. When the
parameter is the size of a graph's clique modulator, we design a randomized
single-exponential time algorithm for the problems. These algorithms use an
inclusion-exclusion based polynomial sieving technique and add to the growing
number of applications using this powerful algebraic technique.Comment: 29 pages, 3 figure
LIPIcs, Volume 244, ESA 2022, Complete Volume
LIPIcs, Volume 244, ESA 2022, Complete Volum
Algorithmes exponentiels pour l'Ć©tiquetage, la domination et l'ordonnancement
This manuscript of Habilitation aĢ Diriger des Recherches enlights some results obtained since my PhD, I defended in 2007. The presented results have been mainly published in international conferences and journals. Exponential-time algorithms are given to solve various decision, optimization and enumeration problems. First, we are interested in solving the L(2,1)-labeling problem for which several algorithms are described (based on branching, divide-and-conquer and dynamic programming). Some combinatorial bounds are also established to analyze those algorithms. Then we solve domination-like problems. We develop algorithms to solve a generalization of the dominating set problem and we give algorithms to enumerate minimal dominating sets in some graph classes. As a consequence, the analysis of these algorithms implies combinatorial bounds. Finally, we extend our field of applications of moderately exponential-time algorithms to scheduling problems. By using dynamic programming paradigm and by extending the sort-and-search approach, we are able to solve a family of scheduling problems.Ce manuscrit dāHabilitation aĢ Diriger des Recherches met en lumieĢre quelques reĢsultats obtenus depuis ma theĢse de doctorat soutenue en 2007. Ces reĢsultats ont eĢteĢ, pour lāessentiel, publieĢs dans des confeĢrences et des journaux internationaux. Des algorithmes exponentiels sont donneĢs pour reĢsoudre des probleĢmes de deĢcision, dāoptimisation et dāeĢnumeĢration. On sāinteĢresse tout dāabord au probleĢme dāeĢtiquetage L(2,1) dāun graphe, pour lequel diffeĢrents algorithmes sont deĢcrits (baseĢs sur du branchement, le paradigme diviser-pour-reĢgner, ou la programmation dynamique). Des bornes combinatoires, neĢcessaires aĢ lāanalyse de ces algorithmes, sont eĢgalement eĢtablies. Dans un second temps, nous reĢsolvons des probleĢmes autour de la domination. Nous deĢveloppons des algorithmes pour reĢsoudre une geĢneĢralisation de la domination et nous donnons des algorithmes pour eĢnumeĢrer les ensembles dominants minimaux dans des classes de graphes. Lāanalyse de ces algorithmes implique des bornes combinatoires. Finalement, nous eĢtendons notre champ dāapplications de lāalgorithmique modeĢreĢment exponentielle aĢ des probleĢmes dāordonnancement. Par le deĢveloppement dāapproches de type programmation dynamique et la geĢneĢralisation de la meĢthode trier-et-chercher, nous proposons la reĢsolution de toute une famille de probleĢmes dāordonnancement
LIPIcs, Volume 274, ESA 2023, Complete Volume
LIPIcs, Volume 274, ESA 2023, Complete Volum
LIPIcs, Volume 248, ISAAC 2022, Complete Volume
LIPIcs, Volume 248, ISAAC 2022, Complete Volum
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum