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CompressionBased FixedParameter Algorithms for Feedback Vertex Set and Edge Bipartization
, 2006
"... We show that the NPcomplete Feedback Vertex Set problem, which asks for the smallest set of vertices to remove from a graph to destroy all cycles, is deterministically solvable in O(c k ·m) time. Here, m denotes the number of graph edges, k denotes the size of the feedback vertex set searched for, ..."
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Cited by 47 (4 self)
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We show that the NPcomplete Feedback Vertex Set problem, which asks for the smallest set of vertices to remove from a graph to destroy all cycles, is deterministically solvable in O(c k ·m) time. Here, m denotes the number of graph edges, k denotes the size of the feedback vertex set searched for, and c is a constant. We extend this to an algorithm enumerating all solutions in O(d k ·m) time for a (larger) constant d. As a further result, we present a fixedparameter algorithm with runtime O(2 k · m 2) for the NPcomplete Edge Bipartization problem, which asks for at most k edges to remove from a graph to make it bipartite.
Almost 2SAT is fixedparameter tractable
 Journal of Computer and System Sciences
"... Abstract. We consider the following problem. Given a 2CNF formula, is it possible to remove at most k clauses so that the resulting 2CNF formula is satisfiable? This problem is known to different research communities in Theoretical Computer Science under the names ’Almost 2SAT’, ’Allbutk 2SAT’ ..."
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Cited by 43 (6 self)
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Abstract. We consider the following problem. Given a 2CNF formula, is it possible to remove at most k clauses so that the resulting 2CNF formula is satisfiable? This problem is known to different research communities in Theoretical Computer Science under the names ’Almost 2SAT’, ’Allbutk 2SAT’, ’2CNF deletion’, ’2SAT deletion’. The status of fixedparameter tractability of this problem is a longstanding open question in the area of Parameterized Complexity. We resolve this open question by proposing an algorithm which solves this problem in O(15 k ∗ k ∗ m 3) and thus we show that this problem is fixedparameter tractable. 1
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 ..."
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Cited by 19 (4 self)
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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
Improved FixedParameter Algorithms for Two Feedback Set Problems
 In Proc. 9th WADS, volume 3608 of LNCS
, 2005
"... Abstract. Settling a ten years open question, we show that the NPcomplete Feedback Vertex Set problem is deterministically solvable in O(c k ·m) time, where m denotes the number of graph edges, k denotes the size of the feedback vertex set searched for, and c is a constant. As a second result, we pr ..."
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Cited by 17 (3 self)
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Abstract. Settling a ten years open question, we show that the NPcomplete Feedback Vertex Set problem is deterministically solvable in O(c k ·m) time, where m denotes the number of graph edges, k denotes the size of the feedback vertex set searched for, and c is a constant. As a second result, we present a fixedparameter algorithm for the NPcomplete Edge Bipartization problem with runtime O(2 k · m 2). 1
Simpler Parameterized Algorithm for OCT
"... We give a simple and intuitive fixed parameter tractable algorithm for the Odd Cycle Transversal problem, running in time O(3 k · k · E  · V ). Our algorithm is best viewed as a reinterpretation of the classical Iterative Compression algorithm for Odd Cycle Transversal by Reed, Smith and Vetta ..."
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Cited by 15 (3 self)
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We give a simple and intuitive fixed parameter tractable algorithm for the Odd Cycle Transversal problem, running in time O(3 k · k · E  · V ). Our algorithm is best viewed as a reinterpretation of the classical Iterative Compression algorithm for Odd Cycle Transversal by Reed, Smith and Vetta [8]. 1
Models and Algorithms for Haplotyping Problem
 Current Bioinformatics
, 2006
"... Abstract: One of the main topics in genomics is to determine the relevance of DNA variations with some genetic disease. Single nucleotide polymorphism (SNP) is the most frequent and important form of genetic variation which involves a single DNA base. The values of a set of SNPs on a particular chro ..."
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Cited by 12 (1 self)
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Abstract: One of the main topics in genomics is to determine the relevance of DNA variations with some genetic disease. Single nucleotide polymorphism (SNP) is the most frequent and important form of genetic variation which involves a single DNA base. The values of a set of SNPs on a particular chromosome copy define a haplotype. Because of its importance in the studies of complex disease association, haplotyping is one of the central problems in bioinformatics. There are two classes of in silico haplotyping problems, i.e., single individual haplotyping and population haplotyping. In this review paper, we give an overview on the existing models and algorithms on this topic, report the recent progresses from the computational viewpoint and further discuss the future research trends. 1.
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 ..."
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Cited by 12 (6 self)
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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
Optimal Edge Deletions for Signed Graph Balancing
, 2007
"... The Balanced Subgraph problem (edge deletion variant) asks for a 2coloring of a graph that minimizes the inconsistencies with given edge labels. It has applications in social networks, systems biology, and integrated circuit design. We present an exact algorithm for Balanced Subgraph based on a com ..."
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Cited by 11 (2 self)
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The Balanced Subgraph problem (edge deletion variant) asks for a 2coloring of a graph that minimizes the inconsistencies with given edge labels. It has applications in social networks, systems biology, and integrated circuit design. We present an exact algorithm for Balanced Subgraph based on a combination of data reduction rules and a fixedparameter algorithm. The data reduction is based on finding small separators and a novel gadget construction scheme. The fixedparameter algorithm is based on iterative compression with a very effective heuristic speedup. Our implementation can solve biological realworld instances exactly for which previously only approximations [DasGupta et al., WEA 2006] were known.
Obtaining a bipartite graph by contracting few edges
, 2011
"... We initiate the study of the BIPARTITE CONTRACTION problem from the perspective of parameterized complexity. In this problem we are given a graph G on n vertices and an integer k, and the task is to determine whether we can obtain a bipartite graph from G by a sequence of at most k edge contractions ..."
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Cited by 9 (4 self)
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We initiate the study of the BIPARTITE CONTRACTION problem from the perspective of parameterized complexity. In this problem we are given a graph G on n vertices and an integer k, and the task is to determine whether we can obtain a bipartite graph from G by a sequence of at most k edge contractions. Our main result is an f (k) n O(1) time algorithm for BIPARTITE CONTRACTION. Despite a strong resemblance between BIPARTITE CONTRACTION and the classical ODD CYCLE TRANSVERSAL (OCT) problem, the methods developed to tackle OCT do not seem to be directly applicable to BIPARTITE CONTRACTION. To obtain our result, we combine several techniques and concepts that are central in parameterized complexity: iterative compression, irrelevant vertex, and important separators. To the best of our knowledge, this is the first time the irrelevant vertex technique and the concept of important separators are applied in unison. Furthermore, our algorithm may serve as a comprehensible example of the usage of the irrelevant vertex technique.
Faster Parameterized Algorithms using Linear Programming
 CoRR
"... We investigate the parameterized complexity ofVertex Cover parameterized by the di↵erence between the size of the optimal solution and the value of the linear programming (LP) relaxation of the problem. By carefully analyzing the change in the LP value in the branching steps, we argue that combining ..."
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Cited by 9 (3 self)
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We investigate the parameterized complexity ofVertex Cover parameterized by the di↵erence between the size of the optimal solution and the value of the linear programming (LP) relaxation of the problem. By carefully analyzing the change in the LP value in the branching steps, we argue that combining previously known preprocessing rules with the most straightforward branching algorithm yields an O⇤(2.618k) algorithm for the problem. Here k is the excess of the vertex cover size over the LP optimum, and we write O⇤(f(k)) for a time complexity of the form O(f(k)nO(1)). We proceed to show that a more sophisticated branching algorithm achieves a running time of O⇤(2.3146k). Following this, using previously known as well as new reductions, we give O⇤(2.3146k) algorithms for the parameterized versions of Above Guarantee Vertex Cover, Odd Cycle Transversal, Split Vertex Deletion andAlmost 2SAT, andO⇤(1.5214k) algorithms forKönig Vertex Deletion andVertex Cover parameterized by the size of the smallest odd cycle transversal and König vertex deletion set. These algorithms significantly improve the best known bounds for these problems. The most notable improvement among these is the new bound for Odd Cycle Transversal this is the first algorithm which improves upon the dependence on k of the seminal O⇤(3k) algorithm of Reed, Smith and Vetta. Finally, using our algorithm, we obtain a kernel for the standard parameterization of Vertex Cover with at most 2k c log k vertices. Our kernel is simpler than previously known kernels achieving the same size bound.