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42
A 2approximation algorithm for the undirected feedback vertex set problem
 SIAM J. Discrete Math
, 1999
"... Abstract. A feedback vertex set of a graph is a subset of vertices that contains at least one vertex from every cycle in the graph. The problem considered is that of finding a minimum feedback vertex set given a weighted and undirected graph. We present a simple and efficient approximation algorithm ..."
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Cited by 65 (0 self)
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Abstract. A feedback vertex set of a graph is a subset of vertices that contains at least one vertex from every cycle in the graph. The problem considered is that of finding a minimum feedback vertex set given a weighted and undirected graph. We present a simple and efficient approximation algorithm with performance ratio of at most 2, improving previous best bounds for either weighted or unweighted cases of the problem. Any further improvement on this bound, matching the best constant factor known for the vertex cover problem, is deemed challenging. The approximation principle, underlying the algorithm, is based on a generalized form of the classical local ratio theorem, originally developed for approximation of the vertex cover problem, and a more flexible style of its application.
Complexity classification of some edge modification problems
, 2001
"... In an edge modification problem one has to change the edge set of a given graph as little as possible so as to satisfy a certain property. We prove the NPhardness of a variety of edge modification problems with respect to some wellstudied classes of graphs. These include perfect, chordal, chain, c ..."
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Cited by 40 (2 self)
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In an edge modification problem one has to change the edge set of a given graph as little as possible so as to satisfy a certain property. We prove the NPhardness of a variety of edge modification problems with respect to some wellstudied classes of graphs. These include perfect, chordal, chain, comparability, split and asteroidal triple free. We show that some of these problems become polynomial when the input graph has bounded degree. We also give a general constant factor approximation algorithm for deletion and editing problems on bounded degree graphs with respect to properties that can be characterized by a finite set of forbidden induced subgraphs.
Automated Generation of Search Tree Algorithms for Hard Graph Modification Problems
 Algorithmica
, 2004
"... We present a framework for an automated generation of exact search tree algorithms for NPhard problems. The purpose of our approach is twofoldrapid development and improved upper bounds. ..."
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Cited by 24 (10 self)
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We present a framework for an automated generation of exact search tree algorithms for NPhard problems. The purpose of our approach is twofoldrapid development and improved upper bounds.
FixedParameter Algorithms for Cluster Vertex Deletion
, 2008
"... We initiate the first systematic study of the NPhard Cluster Vertex Deletion (CVD) problem (unweighted and weighted) in terms of fixedparameter algorithmics. In the unweighted case, one searches for a minimum number of vertex deletions to transform a graph into a collection of disjoint cliques. Th ..."
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Cited by 23 (11 self)
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We initiate the first systematic study of the NPhard Cluster Vertex Deletion (CVD) problem (unweighted and weighted) in terms of fixedparameter algorithmics. In the unweighted case, one searches for a minimum number of vertex deletions to transform a graph into a collection of disjoint cliques. The parameter is the number of vertex deletions. We present efficient fixedparameter algorithms for CVD applying the fairly new iterative compression technique. Moreover, we study the variant of CVD where the maximum number of cliques to be generated is prespecified. Here, we exploit connections to fixedparameter algorithms for (weighted) Vertex Cover.
Constant Ratio Approximations of the Weighted Feedback . . .
"... We consider the weighted feedback vertex set problem for undirected graphs. It is shown that a generalized local ratio strategy leads to an efficient approximation with the performance guarantee of twice the optimal, improving the previous results for both weighted and unweighted cases. We further e ..."
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Cited by 21 (3 self)
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We consider the weighted feedback vertex set problem for undirected graphs. It is shown that a generalized local ratio strategy leads to an efficient approximation with the performance guarantee of twice the optimal, improving the previous results for both weighted and unweighted cases. We further elaborate our approach to treat the case when graphs are of bounded degree, and show that it achieves even better performance, 2 0 2 3102 , where 1 is the maximum degree of graphs.
Chordal deletion is fixedparameter tractable
 In 32nd International Workshop on GraphTheoretic Concepts in Computer Science, WG 2006, LNCS Proceedings
, 2004
"... Abstract. It is known to be NPhard to decide whether a graph can be made chordal by the deletion of k vertices. Here we present a uniformly polynomialtime algorithm for the problem: the running time is f(k) ·n α for some constant α not depending on k and some f depending only on k. For large value ..."
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Cited by 19 (1 self)
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Abstract. It is known to be NPhard to decide whether a graph can be made chordal by the deletion of k vertices. Here we present a uniformly polynomialtime algorithm for the problem: the running time is f(k) ·n α for some constant α not depending on k and some f depending only on k. For large values of n, such an algorithm is much better than trying all the O(n k) possibilities. Therefore, the chordal deletion problem parameterized by the number k of vertices to be deleted is fixedparameter tractable. This answers an open question of Cai [2]. 1
Parameterized Complexity of Finding Subgraphs with Hereditary Properties
, 2002
"... We consider the parameterized complexity of the following problem under the flamework introduced by Downey and Fellows[4]: Given a graph G, an integer parmneter : and a nontrivial hereditary property H, are there vertices of G that induce a subgraph with property H? This problem has been proved ..."
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Cited by 15 (4 self)
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We consider the parameterized complexity of the following problem under the flamework introduced by Downey and Fellows[4]: Given a graph G, an integer parmneter : and a nontrivial hereditary property H, are there vertices of G that induce a subgraph with property H? This problem has been proved NPhard by Lewis and Yanna kakis[9]. e show that if H includes all independent sets but not all cliques or vice versa, then the problem is hard for the parameterized class kV[1] and is fixed parameter tractable otherwise. In the ibrmer case, if the tbrbidden set of the property is finite, we show, in fact, that the probleln is W[1]complete (see [] for definitions). Our prooil, both of the tractability as well as the hardness ones, involve clever use of Ramsey nmnbers.
Algorithm Engineering for Optimal Graph Bipartization
, 2009
"... We examine exact algorithms for the NPhard Graph Bipartization problem. The task is, given a graph, to find a minimum set of vertices to delete to make it bipartite. Based on the “iterative compression ” method introduced by Reed, Smith, and Vetta in 2004, we present new algorithms and experimental ..."
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Cited by 14 (3 self)
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We examine exact algorithms for the NPhard Graph Bipartization problem. The task is, given a graph, to find a minimum set of vertices to delete to make it bipartite. Based on the “iterative compression ” method introduced by Reed, Smith, and Vetta in 2004, we present new algorithms and experimental results. The worstcase time complexity is improved. Based on new structural insights, we give a simplified correctness proof. This also allows us to establish a heuristic improvement that in particular speeds up the search on dense graphs. Our best algorithm can solve all instances from a testbed from computational biology within minutes, whereas established methods are only able to solve about half of the instances within reasonable time.
Approximation Algorithms for Submodular Set Cover with Applications
 IEICE Trans. Inf. Syst
, 2000
"... Introduction We start with the set cover( SC ) problem. Given a finite set M and a family N of subsets of M , a subfamily S of N is called a set cover if every element of M appears in some subset in S; in other words, the union of all subsets in S coincides with M . Each set in N is associated w ..."
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Cited by 13 (0 self)
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Introduction We start with the set cover( SC ) problem. Given a finite set M and a family N of subsets of M , a subfamily S of N is called a set cover if every element of M appears in some subset in S; in other words, the union of all subsets in S coincides with M . Each set in N is associated with a (U"C2T0"'# e) cost, and the cost of a family is the sum of costs of subsets in it. The set cover problem then asks to find a minimum cost set cover. As a special case when all the costs associated with sets are identical, it is called the unit cost set cover, and it is one of the basic NPcomplete optimization problems presented by Karp [17]. he problem is also equivalent to the hitting set problem and the dominating set problem on gra
Satisfiability Allows No Nontrivial Sparsification Unless The PolynomialTime Hierarchy Collapses
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY, REPORT NO. 38 (2010)
, 2010
"... Consider the following twoplayer communication process to decide a language L: The first player holds the entire input x but is polynomially bounded; the second player is computationally unbounded but does not know any part of x; their goal is to cooperatively decide whether x belongs to L at small ..."
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Cited by 13 (0 self)
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Consider the following twoplayer communication process to decide a language L: The first player holds the entire input x but is polynomially bounded; the second player is computationally unbounded but does not know any part of x; their goal is to cooperatively decide whether x belongs to L at small cost, where the cost measure is the number of bits of communication from the first player to the second player. For any integer d ≥ 3 and positive real ǫ we show that if satisfiability for nvariable dCNF formulas has a protocol of cost O(n d−ǫ) then coNP is in NP/poly, which implies that the polynomialtime hierarchy collapses to its third level. The result even holds when the first player is conondeterministic, and is tight as there exists a trivial protocol for ǫ = 0. Under the hypothesis that coNP is not in NP/poly, our result implies tight lower bounds for parameters of interest in several areas, namely sparsification, kernelization in parameterized complexity, lossy compression, and probabilistically checkable proofs. By reduction, similar results hold for other NPcomplete problems. For the vertex cover problem on nvertex duniform hypergraphs, the above statement holds for any integer d ≥ 2. The case d = 2 implies that no NPhard vertex deletion problem based on a graph property that is inherited by subgraphs can have kernels consisting of O(k 2−ǫ) edges unless coNP is in NP/poly, where k denotes the size of the deletion set. Kernels consisting of O(k 2) edges are known for several problems in the class, including vertex cover, feedback vertex set, and boundeddegree deletion.