Results 1  10
of
40
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.
Parameterizing above or below guaranteed values
 J. of Computer and System Sciences
"... Abstract We consider new parameterizations of NPoptimization problems that have nontrivial lower and/or upper bounds on their optimum solution size. The natural parameter, we argue, is the quantity above the lower bound or below the upper bound. We show that for every problem in MAX SNP, the optim ..."
Abstract

Cited by 29 (3 self)
 Add to MetaCart
(Show Context)
Abstract We consider new parameterizations of NPoptimization problems that have nontrivial lower and/or upper bounds on their optimum solution size. The natural parameter, we argue, is the quantity above the lower bound or below the upper bound. We show that for every problem in MAX SNP, the optimum value is bounded below by an unbounded function of the inputsize, and that the aboveguarantee parameterization with respect to this lower bound is fixedparameter tractable. We also observe that approximation algorithms give nontrivial lower or upper bounds on the solution size and that the above or below guarantee question with respect to these bounds is fixedparameter tractable for a subclass of NPoptimization problems. We then introduce the notion of 'tight' lower and upper bounds and exhibit a number of problems for which the aboveguarantee and belowguarantee parameterizations with respect to a tight bound is fixedparameter tractable or Whard. We show that if we parameterize "sufficiently" above or below the tight bounds, then these parameterized versions are not fixedparameter tractable unless P = NP, for a subclass of NPoptimization problems. We also list several directions to explore in this paradigm.
Compression via Matroids: A Randomized Polynomial Kernel for Odd Cycle Transversal
"... The Odd Cycle Transversal problem (OCT) asks whether a given graph can be made bipartite by deleting at most k of its vertices. In a breakthrough result Reed, Smith, and Vetta (Operations Research Letters, 2004) gave a O(4 k kmn) time algorithm for it, the first algorithm with polynomial runtime of ..."
Abstract

Cited by 19 (4 self)
 Add to MetaCart
(Show Context)
The Odd Cycle Transversal problem (OCT) asks whether a given graph can be made bipartite by deleting at most k of its vertices. In a breakthrough result Reed, Smith, and Vetta (Operations Research Letters, 2004) gave a O(4 k kmn) time algorithm for it, the first algorithm with polynomial runtime of uniform degree for every fixed k. It is known that this implies a polynomialtime compression algorithm that turns OCT instances into equivalent instances of size at most O(4 k), a socalled kernelization. Since then the existence of a polynomial kernel for OCT, i.e., a kernelization with size bounded polynomially in k, has turned into one of the main open questions in the study of kernelization. Despite the impressive progress in the area, including the recent development of lower bound techniques (Bodlaender
Finding small separators in linear time via treewidth reduction
"... We present a method for reducing the treewidth of a graph while preserving all of its minimal s−t separators up to a certain fixed size k. This technique allows us to solve s−t Cut and Multicut problems with various additional restrictions (e.g., the vertices being removed from the graph form an ind ..."
Abstract

Cited by 17 (1 self)
 Add to MetaCart
We present a method for reducing the treewidth of a graph while preserving all of its minimal s−t separators up to a certain fixed size k. This technique allows us to solve s−t Cut and Multicut problems with various additional restrictions (e.g., the vertices being removed from the graph form an independent set or induce a connected graph) in linear time for every fixed number k of removed vertices. Our results have applications for problems that are not directly defined by separators, but the known solution methods depend on some variant of separation. For example, we can solve similarly restricted generalizations of Bipartization (delete at most k vertices from G to make it bipartite) in almost linear time for every fixed number k of removed vertices. These results answer a number of open questions in the area of parameterized complexity. Furthermore, our technique turns out to be relevant for (H,C,K)and (H,C,≤K)coloring problems as well, which are cardinality constrained variants of the classical Hcoloring problem. We make progress in the classification of the parameterized complexity of these problems by identifying new cases that can be solved in almost linear time for every fixed cardinality bound.
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
Backdoors to satisfaction
 The Multivariate Algorithmic Revolution and Beyond  Essays Dedicated to Michael R. Fellows on the Occasion of His 60th Birthday, volume 7370 of Lecture
"... ar ..."
(Show Context)
Iterative compression for exactly solving nphard minimization problems
 in Algorithmics of Large and Complex Networks, Lecture Notes in Computer Science
"... Abstract. We survey the conceptual framework and several applications of the iterative compression technique introduced in 2004 by Reed, Smith, and Vetta. This technique has proven very useful for achieving a number of recent breakthroughs in the development of fixedparameter algorithms for NPhard ..."
Abstract

Cited by 12 (6 self)
 Add to MetaCart
(Show Context)
Abstract. We survey the conceptual framework and several applications of the iterative compression technique introduced in 2004 by Reed, Smith, and Vetta. This technique has proven very useful for achieving a number of recent breakthroughs in the development of fixedparameter algorithms for NPhard minimization problems. There is a clear potential for further applications as well as a further development of the technique itself. We describe several algorithmic results based on iterative compression and point out some challenges for future research. 1
Clustering with Local Restrictions
"... Abstract. We study a family of graph clustering problems where each cluster has to satisfy a certain local requirement. Formally, let µ be a function on the subsets of vertices of a graph G. In the (µ, p,q)PARTITION problem, the task is to find a partition of the vertices where each cluster C satis ..."
Abstract

Cited by 11 (2 self)
 Add to MetaCart
(Show Context)
Abstract. We study a family of graph clustering problems where each cluster has to satisfy a certain local requirement. Formally, let µ be a function on the subsets of vertices of a graph G. In the (µ, p,q)PARTITION problem, the task is to find a partition of the vertices where each cluster C satisfies the requirements that (1) at most q edges leave C and (2) µ(C) ≤ p. Our first result shows that if µ is an arbitrary polynomialtime computable monotone function, then (µ, p, q)PARTITION can be solved in time n O(q) , i.e., it is polynomialtime solvable for every fixed q. We study in detail three concrete functions µ (number of nonedges in the cluster, maximum degree of nonedges in the cluster, number of vertices in the cluster), which correspond to natural clustering problems. For these functions, we show that (µ, p,q)PARTITION can be solved in time 2 O(p) · n O(1) and in randomized time 2 O(q) · n O(1) , i.e., the problem is fixedparameter tractable parameterized by p or by q. 1
New races in parameterized algorithmics
 IN: PROCEEDINGS OF THE 37TH INTERNATIONAL SYMPOSIUM ON MATHEMATICAL FOUNDATIONS OF COMPUTER SCIENCE (MFCS ’12), LNCS
"... Once having classified an NPhard problem fixedparameter tractable with respect to a certain parameter, the race for the most efficient fixedparameter algorithm starts. Herein, the attention usually focuses on improving the running time factor exponential in the considered parameter, and, in case ..."
Abstract

Cited by 10 (7 self)
 Add to MetaCart
(Show Context)
Once having classified an NPhard problem fixedparameter tractable with respect to a certain parameter, the race for the most efficient fixedparameter algorithm starts. Herein, the attention usually focuses on improving the running time factor exponential in the considered parameter, and, in case of kernelization algorithms, to improve the bound on the kernel size. Both from a practical as well as a theoretical point of view, however, there are further aspects of efficiency that deserve attention. We discuss several of these aspects and particularly focus on the search for “stronger parameterizations” in developing fixedparameter algorithms.