Results 1 
9 of
9
Fixedparameter tractability of multicut parameterized by the size of the cutset
, 2011
"... Given an undirected graph G, a collection {(s1, t1),...,(sk, tk)} of pairs of vertices, and an integer p, the EDGE MULTICUT problem ask if there is a set S of at most p edges such that the removal of S disconnects every si from the corresponding ti. VERTEX MULTICUT is the analogous problem where S i ..."
Abstract

Cited by 32 (6 self)
 Add to MetaCart
Given an undirected graph G, a collection {(s1, t1),...,(sk, tk)} of pairs of vertices, and an integer p, the EDGE MULTICUT problem ask if there is a set S of at most p edges such that the removal of S disconnects every si from the corresponding ti. VERTEX MULTICUT is the analogous problem where S is a set of at most p vertices. Our main result is that both problems can be solved in time 2O(p3) · nO(1), i.e., fixedparameter tractable parameterized by the size p of the cutset in the solution. By contrast, it is unlikely that an algorithm with running time of the form f (p) · nO(1) exists for the directed version of the problem, as we show it to be W[1]hard parameterized by the size of the cutset.
FixedParameter Tractability of Directed Multiway Cut Parameterized by the Size of the Cutset
"... Given a directed graph G, a set of k terminals and an integer p, the DIRECTED VERTEX MULTIWAY CUT problem asks if there is a set S of at most p (nonterminal) vertices whose removal disconnects each terminal from all other terminals. DIRECTED EDGE MULTIWAY CUT is the analogous problem where S is a se ..."
Abstract

Cited by 14 (5 self)
 Add to MetaCart
(Show Context)
Given a directed graph G, a set of k terminals and an integer p, the DIRECTED VERTEX MULTIWAY CUT problem asks if there is a set S of at most p (nonterminal) vertices whose removal disconnects each terminal from all other terminals. DIRECTED EDGE MULTIWAY CUT is the analogous problem where S is a set of at most p edges. These two problems indeed are known to be equivalent. A natural generalization of the multiway cut is the multicut problem, in which we want to disconnect only a set of k given pairs instead of all pairs. Marx (Theor. Comp. Sci. 2006) showed that in undirected graphs multiway cut is fixedparameter tractable (FPT) parameterized by p. Marx and Razgon (STOC 2011) showed that undirected multicut is FPT and directed multicut is W[1]hard parameterized by p. We complete the picture here by our main result which is that both DIRECTED VERTEX MULTIWAY CUT and DIRECTED EDGE MULTIWAY CUT can be solved in time 22O(p) nO(1) , i.e., FPT parameterized by size p of the cutset of the solution. This answers an open question raised by Marx (Theor. Comp. Sci. 2006) and Marx and Razgon (STOC 2011). It follows from our result that DIRECTED MULTICUT is FPT for the case of k = 2 terminal pairs, which answers another open problem raised in Marx and Razgon (STOC 2011). 1
Fixedparameter tractability of multicut in directed acyclic graphs
, 2015
"... The Multicut problem, given a graph G, a set of terminal pairs T = {(si, ti)  1 ≤ i ≤ r}, and an integer p, asks whether one can find a cutset consisting of at most p nonterminal vertices that separates all the terminal pairs, i.e., after removing the cutset, ti is not reachable from si for each ..."
Abstract

Cited by 9 (0 self)
 Add to MetaCart
(Show Context)
The Multicut problem, given a graph G, a set of terminal pairs T = {(si, ti)  1 ≤ i ≤ r}, and an integer p, asks whether one can find a cutset consisting of at most p nonterminal vertices that separates all the terminal pairs, i.e., after removing the cutset, ti is not reachable from si for each 1 ≤ i ≤ r. The fixedparameter tractability of Multicut in undirected graphs, parameterized by the size of the cutset only, has been recently proved by Marx and Razgon [SIAM J. Comput., 43 (2014), pp. 355–388] and, independently, by Bousquet, Daligault, and Thomassé [Proceedings of STOC, ACM, 2011, pp. 459–468], after resisting attacks as a longstanding open problem. In this paper we prove that Multicut is fixedparameter tractable on directed acyclic graphs when parameterized both by the size of the cutset and the number of terminal pairs. We complement this result by showing that this is implausible for parameterization by the size of the cutset only, as this version of the problem remains W [1]hard.
Linear Time Parameterized Algorithms via SkewSymmetric Multicuts
, 2013
"... A skewsymmetric graph (D = (V,A), σ) is a directed graph D with an involution σ on the set of vertices and arcs. Flows on skewsymmetric graphs have been used to generalize maximum flow and maximum matching problems on graphs, initially by Tutte [1967], and later by Goldberg and Karzanov [1994, 199 ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
A skewsymmetric graph (D = (V,A), σ) is a directed graph D with an involution σ on the set of vertices and arcs. Flows on skewsymmetric graphs have been used to generalize maximum flow and maximum matching problems on graphs, initially by Tutte [1967], and later by Goldberg and Karzanov [1994, 1995]. In this paper, we introduce a separation problem, dSkewSymmetric Multicut, where we are given a skewsymmetric graph D, a family of T of dsized subsets of vertices and an integer k. The objective is to decide if there is a set X ⊆ A of k arcs such that every set J in the family has a vertex v such that v and σ(v) are in different connected components of D ′ = (V,A \ (X ∪ σ(X)). In this paper, we give an algorithm for dSkewSymmetric Multicut which runs in time O((4d)k(m+ n+ `)), where m is the number of arcs in the graph, n the number of vertices and ` the length of the family given in the input. This problem, apart from being independently interesting, also abstracts out and captures the main combinatorial obstacles towards solving numerous classical problems. Our algorithm for dSkewSymmetric Multicut paves the way for the first linear time
Communicated by:
, 2013
"... We give a 5 k n O(1) time fixedparameter algorithm for determining whether a given undirected graph on n vertices has a subset of at most k vertices whose deletion results in a tree. Such a subset is a restricted form of a feedback vertex set. While parameterized complexity of feedback vertex set p ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
(Show Context)
We give a 5 k n O(1) time fixedparameter algorithm for determining whether a given undirected graph on n vertices has a subset of at most k vertices whose deletion results in a tree. Such a subset is a restricted form of a feedback vertex set. While parameterized complexity of feedback vertex set problem and several of its variations have been well studied, to the best of our knowledge, this is the first fixedparameter algorithm for this version of feedback vertex set. Submitted:
List Hcoloring a graph by removing few vertices
, 2013
"... In the deletion version of the list homomorphism problem, we are given graphs G and H, a list L(v) ⊆ V (H) for each vertex v ∈ V (G), and an integer k. The task is to decide whether there exists a set W ⊆ V (G) of size at most k such that there is a homomorphism from G \W to H respecting the lists ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
In the deletion version of the list homomorphism problem, we are given graphs G and H, a list L(v) ⊆ V (H) for each vertex v ∈ V (G), and an integer k. The task is to decide whether there exists a set W ⊆ V (G) of size at most k such that there is a homomorphism from G \W to H respecting the lists. We show that DLHom(H), parameterized by k and H, is fixedparameter tractable for any (P6, C6)free bipartite graph H; already for this restricted class of graphs, the problem generalizes Vertex Cover, Odd Cycle Transversal, and Vertex Multiway Cut parameterized by the size of the cutset and the number of terminals. We conjecture that DLHom(H) is fixedparameter tractable for the class of graphs H for which the list homomorphism problem (without deletions) is polynomialtime solvable; by a result of Feder et al. [9], a graph H belongs to this class precisely if it is a bipartite graph whose complement is a circular arc graph. We show that this conjecture is equivalent to the fixedparameter tractability of a single fairly natural satisfiability problem, Clause Deletion ChainSAT.
Halfintegrality, LPbranching and FPT Algorithms
, 2014
"... A recent trend in parameterized algorithms is the application of polytope tools (specifically, LPbranching) to FPT algorithms (e.g., Cygan et al., 2011; Narayanaswamy et al., 2012). Though the list of work in this direction is short, the results are already interesting, yielding significant speedup ..."
Abstract
 Add to MetaCart
A recent trend in parameterized algorithms is the application of polytope tools (specifically, LPbranching) to FPT algorithms (e.g., Cygan et al., 2011; Narayanaswamy et al., 2012). Though the list of work in this direction is short, the results are already interesting, yielding significant speedups for a range of important problems. However, the existing approaches require the underlying polytope to have very restrictive properties, including halfintegrality and NemhauserTrotterstyle persistence properties. To date, these properties are essentially known to hold only for two classes of polytopes, covering the cases of Vertex Cover (Nemhauser and Trotter, 1975) and Node Multiway Cut (Garg et al., 1994). Taking a slightly different approach, we view halfintegrality as a discrete relaxation of a problem, e.g., a relaxation of the search space from {0, 1}V to {0, 1/2, 1}V such that the new problem admits a polynomialtime exact solution. Using tools from CSP (in particular Thapper and Živný, 2012) to study the existence of such relaxations, we are able to provide a much broader class of halfintegral polytopes with the required properties. Our results unify and significantly extend the previously known cases. In addition to the new insight into problems with halfintegral relaxations, our results yield a range of new and improved FPT algo
Linear Time Parameterized Algorithms for Subset Feedback Vertex Set
"... In the Subset Feedback Vertex Set (Subset FVS) problem, the input is a graph G on n vertices and m edges, a subset of vertices T, referred to as terminals, and an integer k. The objective is to determine whether there exists a set of at most k vertices intersecting every cycle that contains a termin ..."
Abstract
 Add to MetaCart
In the Subset Feedback Vertex Set (Subset FVS) problem, the input is a graph G on n vertices and m edges, a subset of vertices T, referred to as terminals, and an integer k. The objective is to determine whether there exists a set of at most k vertices intersecting every cycle that contains a terminal. The study of parameterized algorithms for this generalization of the Feedback Vertex Set problem has received significant attention over the last few years. In fact the parameterized complexity of this problem was open until 2011, when two groups independently showed that the problem is fixed parameter tractable (FPT). Using tools from graph minors Kawarabayashi and Kobayashi obtained an algorithm for Subset FVS running in time O(f(k) · n2m) [SODA 2012, JCTB 2012]. Independently, Cygan et al. [ICALP 2011, SIDMA 2013] designed an algorithm for Subset FVS running in time 2O(k log k) · nO(1). More recently, Wahlström obtained the first single exponential time algorithm for Subset FVS, running in time 4k · nO(1) [SODA 2014]. While the 2O(k) dependence on the parameter k is optimal under the Exponential Time Hypothesis (ETH), the dependence of this algorithm as well as those preceding it, on the input size is far from linear. In this paper we design the first linear time parameterized algorithms for Subset FVS. More precisely, we obtain the following new algorithms for Subset FVS. • A randomized algorithm for Subset FVS running in time O(25.6k · (n+m)). • A deterministic algorithm for Subset FVS running in time 2O(k log k) · (n+m). In particular, the first algorithm obtains the best possible dependence on both the parameter as well as the input size, up to the constant in the exponent. Both of our algorithms are based on “cut centrality”, in the sense that solution vertices are likely to show up in minimum size cuts between vertices sampled from carefully chosen distributions.