30 research outputs found
More Applications of the d-Neighbor Equivalence: Connectivity and Acyclicity 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. For all these problems, we obtain 2^O(k)* n^O(1), 2^O(k log(k))* n^O(1), 2^O(k^2) * n^O(1) and n^O(k) time algorithms parameterized respectively by clique-width, Q-rank-width, rank-width and maximum induced matching width. Our approach simplifies and unifies the known algorithms for each of the parameters and match asymptotically also the running time of the best algorithms for basic NP-hard problems such as Vertex Cover and Dominating Set. Our framework is based on the d-neighbor equivalence defined in [Bui-Xuan, Telle and Vatshelle, TCS 2013]. The results we obtain highlight the importance and the generalizing power of this equivalence relation on width measures. We also prove that this equivalence relation could be useful for Max Cut: a W[1]-hard problem parameterized by clique-width. For this latter problem, we obtain n^O(k), n^O(k) and n^(2^O(k)) time algorithm parameterized by clique-width, Q-rank-width and rank-width
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
Tight Lower Bounds for Problems Parameterized by Rank-Width
We show that there is no 2o(k2)nO(1) time algorithm for Independent Set on n-vertex graphs with rank-width k, unless the Exponential Time Hypothesis (ETH) fails. Our lower bound matches the 2O(k2)nO(1) time algorithm given by Bui-Xuan, Telle, and Vatshelle [Discret. Appl. Math., 2010] and it answers the open question of Bergougnoux and Kanté [SIAM J. Discret. Math., 2021]. We also show that the known 2O(k2)nO(1) time algorithms for Weighted Dominating Set, Maximum Induced Matching and Feedback Vertex Set parameterized by rank-width k are optimal assuming ETH. Our results are the first tight ETH lower bounds parameterized by rank-width that do not follow directly from lower bounds for n-vertex graphs
New Width Parameters for Independent Set: One-sided-mim-width and Neighbor-depth
We study the tractability of the maximum independent set problem from the
viewpoint of graph width parameters, with the goal of defining a width
parameter that is as general as possible and allows to solve independent set in
polynomial-time on graphs where the parameter is bounded. We introduce two new
graph width parameters: one-sided maximum induced matching-width (o-mim-width)
and neighbor-depth. O-mim-width is a graph parameter that is more general than
the known parameters mim-width and tree-independence number, and we show that
independent set and feedback vertex set can be solved in polynomial-time given
a decomposition with bounded o-mim-width. O-mim-width is the first width
parameter that gives a common generalization of chordal graphs and graphs of
bounded clique-width in terms of tractability of these problems.
The parameter o-mim-width, as well as the related parameters mim-width and
sim-width, have the limitation that no algorithms are known to compute
bounded-width decompositions in polynomial-time. To partially resolve this
limitation, we introduce the parameter neighbor-depth. We show that given a
graph of neighbor-depth , independent set can be solved in time
even without knowing a corresponding decomposition. We also show that
neighbor-depth is bounded by a polylogarithmic function on the number of
vertices on large classes of graphs, including graphs of bounded o-mim-width,
and more generally graphs of bounded sim-width, giving a quasipolynomial-time
algorithm for independent set on these graph classes. This resolves an open
problem asked by Kang, Kwon, Str{\o}mme, and Telle [TCS 2017].Comment: 26 pages, 1 figure. This is the full version of an extended abstract
that will appear in WG202
Towards a Polynomial Kernel for Directed Feedback Vertex Set
In the Directed Feedback Vertex Set (DFVS) problem, the input is
a directed graph D and an integer k. The objective is to determine
whether there exists a set of at most k vertices intersecting every
directed cycle of D. DFVS was shown to be fixed-parameter tractable when parameterized by solution size by Chen, Liu, Lu, O\u27Sullivan and
Razgon [JACM 2008]; since then, the existence of a polynomial kernel for this problem has become one of the largest open problems in the area of parameterized algorithmics.
In this paper, we study DFVS parameterized by the feedback vertex
set number of the underlying undirected graph. We provide two main contributions: a polynomial kernel for this problem on general instances, and a linear kernel for the case where the input digraph is embeddable on a surface of bounded genus
Kernelization for Finding Lineal Topologies (Depth-First Spanning Trees) with Many or Few Leaves
For a given graph , a depth-first search (DFS) tree of is an
-rooted spanning tree such that every edge of is either an edge of
or is between a \textit{descendant} and an \textit{ancestor} in . A graph
together with a DFS tree is called a \textit{lineal topology} . Sam et al. (2023) initiated study of the parameterized complexity
of the \textsc{Min-LLT} and \textsc{Max-LLT} problems which ask, given a graph
and an integer , whether has a DFS tree with at most and
at least leaves, respectively. Particularly, they showed that for the dual
parameterization, where the tasks are to find DFS trees with at least and
at most leaves, respectively, these problems are fixed-parameter
tractable when parameterized by . However, the proofs were based on
Courcelle's theorem, thereby making the running times a tower of exponentials.
We prove that both problems admit polynomial kernels with \Oh(k^3) vertices.
In particular, this implies FPT algorithms running in k^{\Oh(k)}\cdot
n^{O(1)} time. We achieve these results by making use of a \Oh(k)-sized
vertex cover structure associated with each problem. This also allows us to
demonstrate polynomial kernels for \textsc{Min-LLT} and \textsc{Max-LLT} for
the structural parameterization by the vertex cover number.Comment: 16 pages, accepted for presentation at FCT 202
On Dasgupta’s Hierarchical Clustering Objective and Its Relation to Other Graph Parameters
Postponed access: the file will be available after 2022-09-09The minimum height of vertex and edge partition trees are well-studied graph parameters known as, for instance, vertex and edge ranking number. While they are NP-hard to determine in general, linear-time algorithms exist for trees. Motivated by a correspondence with Dasgupta’s objective for hierarchical clustering we consider the total rather than maximum depth of vertices as an alternative objective for minimization. For vertex partition trees this leads to a new parameter with a natural interpretation as a measure of robustness against vertex removal. As tools for the study of this family of parameters we show that they have similar recursive expressions and prove a binary tree rotation lemma. The new parameter is related to trivially perfect graph completion and therefore intractable like the other three are known to be. We give polynomial-time algorithms for both total-depth variants on caterpillars and on trees with a bounded number of leaf neighbors. For general trees, we obtain a 2-approximation algorithm.acceptedVersio
Sédimentation de particules en écoulement tourbillonnaire : effets collectifs et anisotropie
Les écoulements de particules sont présents dans un grand nombre de processus naturels ou industriels. Dans bien des cas, la sédimentation des particules est un phénomène dominant, complexe et fascinant, qu'il est important de contrôler et de comprendre de façon fondamentale. De nombreuses avancées ont été réalisées à bas nombres de Reynolds (sans inertie) mais le régime inertiel et turbulent n'a reçu que peu d'attention. L'objet de cette communication est l'étude de la sédimentation de particules à travers un écoulement constitué d'un réseau bidimensionnel de tourbillons. Après avoir analysé la sédimentation d'une particule sphérique isolée, nous nous intéressons, d'une part, au cas d'une fibre isolée, et, d'autre part, au cas d'un nuage de sphères