4 research outputs found
Fast exact algorithms for some connectivity problems parametrized by clique-width
Given a clique-width -expression of a graph , we provide time algorithms for connectivity constraints on locally checkable properties
such as Node-Weighted Steiner Tree, Connected Dominating Set, or Connected
Vertex Cover. We also propose a time algorithm for Feedback
Vertex Set. The best running times for all the considered cases were either
or worse
Fast exact algorithms for some connectivity problems parametrized by clique-width
Given a clique-width -expression of a graph , we provide time algorithms for connectivity constraints on locally checkable properties such as Node-Weighted Steiner Tree, Connected Dominating Set, or Connected Vertex Cover. We also propose a time algorithm for Feedback Vertex Set. The best running times for all the considered cases were either or worse
More applications of the d-neighbor equivalence: acyclicity and connectivity constraints
In this paper, we design a framework to obtain efficient algorithms for
several problems with a global constraint (acyclicity or connectivity) such as
Connected Dominating Set, Node Weighted Steiner Tree, Maximum Induced Tree,
Longest Induced Path, and Feedback Vertex Set. We design a meta-algorithm that
solves all these problems and whose running time is upper bounded by
, , and where is respectively the clique-width,
-rank-width, rank-width and maximum induced matching width of a
given decomposition. Our meta-algorithm simplifies and unifies the known
algorithms for each of the parameters and its running time matches
asymptotically also the running times of the best known algorithms for basic
NP-hard problems such as Vertex Cover and Dominating Set. Our framework is
based on the -neighbor equivalence defined in [Bui-Xuan, Telle and
Vatshelle, TCS 2013]. The results we obtain highlight the importance of this
equivalence relation on the algorithmic applications of width measures.
We also prove that our framework could be useful for -hard problems
parameterized by clique-width such as Max Cut and Maximum Minimal Cut. For
these latter problems, we obtain , and time
algorithms where is respectively the clique-width, the
-rank-width and the rank-width of the input graph