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تسجيل مستخدم جديدClose relatives of Feedback Vertex Set without single-exponential algorithms parameterized by treewidth
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The Cut & Count technique and the rank-based approach have lead to single-exponential FPT algorithms parameterized by treewidth, that is, running in time $2^{O(tw)}n^{O(1)}$, for Feedback Vertex Set and connect
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It has long been known that Feedback Vertex Set can be solved in time $2^{mathcal{O}(wlog w)}n^{mathcal{O}(1)}$ on $n$-vertex graphs of treewidth $w$, but it was only recently that this running time was improved to $2^{mathcal{O}(w)}n^{mathcal{O}(1)}
$, that is, to single-exponential parameterized by treewidth. We investigate which generalizations of Feedback Vertex Set can be solved in a similar running time. Formally, for a class $mathcal{P}$ of graphs, the Bounded $mathcal{P}$-Block Vertex Deletion problem asks, given a graph~$G$ on $n$ vertices and positive integers~$k$ and~$d$, whether $G$ contains a set~$S$ of at most $k$ vertices such that each block of $G-S$ has at most $d$ vertices and is in $mathcal{P}$. Assuming that $mathcal{P}$ is recognizable in polynomial time and satisfies a certain natural hereditary condition, we give a sharp characterization of when single-exponential parameterized algorithms are possible for fixed values of $d$: if $mathcal{P}$ consists only of chordal graphs, then the problem can be solved in time $2^{mathcal{O}(wd^2)} n^{mathcal{O}(1)}$, and if $mathcal{P}$ contains a graph with an induced cycle of length $ellge 4$, then the problem is not solvable in time $2^{o(wlog w)} n^{mathcal{O}(1)}$ even for fixed $d=ell$, unless the ETH fails. We also study a similar problem, called Bounded $mathcal{P}$-Component Vertex Deletion, where the target graphs have connected components of small size rather than blocks of small size, and we present analogous results. For this problem, we also show that if $d$ is part of the input and $mathcal{P}$ contains all chordal graphs, then it cannot be solved in time $f(w)n^{o(w)}$ for some function $f$, unless the ETH fails.
After the number of vertices, Vertex Cover is the largest of the classical graph parameters and has more and more frequently been used as a separate parameter in parameterized problems, including problems that are not directly related to the Vertex C
over. Here we consider the TREEWIDTH and PATHWIDTH problems parameterized by k, the size of a minimum vertex cover of the input graph. We show that the PATHWIDTH and TREEWIDTH can be computed in O*(3^k) time. This complements recent polynomial kernel results for TREEWIDTH and PATHWIDTH parameterized by the Vertex Cover.
We study the recently introduced Connected Feedback Vertex Set (CFVS) problem from the view-point of parameterized algorithms. CFVS is the connected variant of the classical Feedback Vertex Set problem and is defined as follows: given a graph G=(V,E)
and an integer k, decide whether there exists a subset F of V, of size at most k, such that G[V F] is a forest and G[F] is connected. We show that Connected Feedback Vertex Set can be solved in time $O(2^{O(k)}n^{O(1)})$ on general graphs and in time $O(2^{O(sqrt{k}log k)}n^{O(1)})$ on graphs excluding a fixed graph H as a minor. Our result on general undirected graphs uses as subroutine, a parameterized algorithm for Group Steiner Tree, a well studied variant of Steiner Tree. We find the algorithm for Group Steiner Tree of independent interest and believe that it will be useful for obtaining parameterized algorithms for other connectivity problems.
In the Directed Feedback Vertex Set (DFVS) problem, the input is a directed graph $D$ on $n$ vertices and $m$ edges, and an integer $k$. The objective is to determine whether there exists a set of at most $k$ vertices intersecting every directed cycl
e of $D$. Whether or not DFVS admits a fixed parameter tractable (FPT) algorithm was considered the most important open problem in parameterized complexity until Chen, Liu, Lu, OSullivan and Razgon [JACM 2008] answered the question in the affirmative. They gave an algorithm for the problem with running time $O(k!4^kk^4nm)$. Since then, no faster algorithm for the problem has been found. In this paper, we give an algorithm for DFVS with running time $O(k!4^kk^5(n+m))$. Our algorithm is the first algorithm for DFVS with linear dependence on input size. Furthermore, the asymptotic dependence of the running time of our algorithm on the parameter $k$ matches up to a factor $k$ the algorithm of Chen, Liu, Lu, OSullivan and Razgon. On the way to designing our algorithm for DFVS, we give a general methodology to shave off a factor of $n$ from iterative-compression based algorithms for a few other well-studied covering problems in parameterized complexity. We demonstrate the applicability of this technique by speeding up by a factor of $n$, the current best FPT algorithms for Multicut [STOC 2011, SICOMP 2014] and Directed Subset Feedback Vertex Set [ICALP 2012, TALG 2014].
We introduce and study two natural generalizations of the Connected VertexCover (VC) problem: the $p$-Edge-Connected and $p$-Vertex-Connected VC problem (where $p geq 2$ is a fixed integer). Like Connected VC, both new VC problems are FPT, but do not
admit a polynomial kernel unless $NP subseteq coNP/poly$, which is highly unlikely. We prove however that both problems admit time efficient polynomial sized approximate kernelization schemes. We obtain an $O(2^{O(pk)}n^{O(1)})$-time algorithm for the $p$-Edge-Connected VC and an $O(2^{O(k^2)}n^{O(1)})$-time algorithm for the $p$-Vertex-Connected VC. Finally, we describe a $2(p+1)$-approximation algorithm for the $p$-Edge-Connected VC. The proofs for the new VC problems require more sophisticated arguments than for Connected VC. In particular, for the approximation algorithm we use Gomory-Hu trees and for the approximate kernels a result on small-size spanning $p$-vertex/edge-connected subgraph of a $p$-vertex/edge-connected graph obtained independently by Nishizeki and Poljak (1994) and Nagamochi and Ibaraki (1992).
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