Simpler Parameterized Algorithm for OCT

Lokshtanov, Daniel; Saurabh, Saket; Sikdar, Somnath

doi:10.1007/978-3-642-10217-2_37

Cited by 33 publications

(29 citation statements)

References 11 publications

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“…Naturally, one should look at problems where the algorithm or the the running time suggests that the best known algorithm is optimal. Possible candidates are the O(2 k ) time algorithm for STEINER TREE with k terminals [2], the O(2 k ) time randomized algorithm for k-PATH [29], and the O(2 k ) (resp., O(3 k )) time algorithms for EDGE BIPARTIZATION (resp., ODD CYCLE TRANSVERSAL) [16,22].…”

Section: Resultsmentioning

confidence: 99%

Known Algorithms on Graphs of Bounded Treewidth are Probably Optimal

Lokshtanov

Marx

Saurabh

2011

Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms

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Section: Resultsmentioning

confidence: 99%

Known Algorithms on Graphs of Bounded Treewidth are Probably Optimal

Lokshtanov

Marx

Saurabh

2011

Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms

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“…Here, f (k) is a computable function depending only on the parameter k, which confines the combinatorial explosion that is seemingly inevitable for an NP-hard problem. The most intensively studied graph modification problems involve vertexor edge-deletions as their base operation; fixed-parameter tractability has been established for the problems of transforming a graph into a forest [11,8,3], a bipartite graph [26,11,18,15,24], a chordal graph [20], a planar graph [22], or an interval graph [27,2]. Results have also been obtained for problems involving directed graphs [5] or group-labeled graphs [10,7].…”

Section: Introductionmentioning

confidence: 78%

“…An instance I = (G, k) of Bipartite Compression is solved the following way. First, we run the algorithm of Reed et al [26] to look for a set X ⊆ V (G) of size ≤ 2k such that G \ X is bipartite; the running time of the algorithm is O(3 2k knm) (see also [18]) 1 . If there is no such set, then we answer "no".…”

Section: Reduction To a Cut Problemmentioning

confidence: 99%

A Faster FPT Algorithm for Bipartite Contraction

Guillemot

Marx

2013

Parameterized and Exact Computation

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Abstract. The Bipartite Contraction problem is to decide, given a graph G and a parameter k, whether we can can obtain a bipartite graph from G by at most k edge contractions. The fixed-parameter tractability of the problem was shown by Heggernes et al. [13], with an algorithm whose running time has double-exponential dependence on k. We present a new randomized FPT algorithm for the problem, which is both conceptually simpler and achieves an improved 2 O(k 2 ) nm running time, i.e., avoiding the double-exponential dependence on k. The algorithm can be derandomized using standard techniques.

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“…This technique has been useful in resolving several other open problems in the area of parameterized complexity, including Directed Feedback Vertex Set, Almost 2-SAT, Multicut [3,32,26]. However, the algorithm for OCT had seen no further improvements in the last 9 years, though reinterpretations of the algorithm have been published [13,20]. Only recently, Lokshtanov et al [18] obtained an algorithm with an improved dependence on the parameter k. This algorithm is based on a branching guided by linear programming and runs in time O(2.32 k n O(1) ).…”

Section: Introductionmentioning

confidence: 99%

Linear Time Parameterized Algorithms via Skew-Symmetric Multicuts

Ramanujan¹,

Saurabh²

2013

Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms

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A skew-symmetric graph (D = (V, A), σ) is a directed graph D with an involution σ on the set of vertices and arcs. Flows on skew-symmetric 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, d-SkewSymmetric Multicut, where we are given a skewsymmetric graph D, a family of T of d-sized 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 strongly connected components of D = (V, A \ (X ∪ σ(X)). In this paper, we give an algorithm for d-Skew-Symmetric 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 d-Skew-Symmetric Multicut paves the way for the first linear time parameterized algorithms for several problems. We demonstrate its utility by obtaining the following linear time parameterized algorithms.• We show that Almost 2-SAT is a special case of 1-Skew-Symmetric Multicut, resulting in an algorithm for Almost 2-SAT which runs in time O(4 k k 4 ) where k is the size of the solution and is the length of the input formula. Then, using linear time parameter preserving reductions to Almost 2-SAT, we obtain algorithms for Odd Cycle Transversal and Edge Bipartization which run in time O(4 k k 4 (m+n)) and O(4 k k 5 (m+ n)) respectively where k is size of the solution, * Supported by Parameterized Approximation, ERC Starting Grant 306992.

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Simpler Parameterized Algorithm for OCT

Cited by 33 publications

References 11 publications

Known Algorithms on Graphs of Bounded Treewidth are Probably Optimal

Known Algorithms on Graphs of Bounded Treewidth are Probably Optimal

A Faster FPT Algorithm for Bipartite Contraction

Linear Time Parameterized Algorithms via Skew-Symmetric Multicuts

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