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14
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 ..."
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Cited by 32 (6 self)
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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.
Multicut is FPT
 In STOC
, 2011
"... Let G = (V,E) be a graph on n vertices and R be a set of pairs of vertices in V called requests. A multicut is a subset F of E such that every request xy of R is separator by F, i.e.every xypath of G intersects F. We show that there exists an O(f(k)nc) algorithm which decides if there exists a mult ..."
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Let G = (V,E) be a graph on n vertices and R be a set of pairs of vertices in V called requests. A multicut is a subset F of E such that every request xy of R is separator by F, i.e.every xypath of G intersects F. We show that there exists an O(f(k)nc) algorithm which decides if there exists a multicut of size at most k. In other words, the MULTICUT problem parameterized by the solution size k is FixedParameter Tractable. 1
Subset feedback vertex set is fixedparameter tractable
, 2011
"... The classical FEEDBACK VERTEX SET problem asks, for a given undirected graph G and an integer k, to find a set of at most k vertices that hits all the cycles in the graph G. FEEDBACK VERTEX SET has attracted a large amount of research in the parameterized setting, and subsequent kernelization and fi ..."
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Cited by 9 (2 self)
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The classical FEEDBACK VERTEX SET problem asks, for a given undirected graph G and an integer k, to find a set of at most k vertices that hits all the cycles in the graph G. FEEDBACK VERTEX SET has attracted a large amount of research in the parameterized setting, and subsequent kernelization and fixedparameter algorithms have been a rich source of ideas in the field. In this paper we consider a more general and difficult version of the problem, named SUBSET FEEDBACK VERTEX SET (SUBSETFVS in short) where an instance comes additionally with a set S ⊆ V of vertices, and we ask for a set of at most k vertices that hits all simple cycles passing through S. Because of its applications in circuit testing and genetic linkage analysis SUBSETFVS was studied from the approximation algorithms perspective by Even et al. [SICOMP’00, SIDMA’00]. The question whether the SUBSETFVS problem is fixedparameter tractable was posed independently by Kawarabayashi and Saurabh in 2009. We answer this question affirmatively. We begin by showing that this problem is fixedparameter tractable when parametrized by S. Next we present an algorithm which reduces the given instance to 2 k n O(1) instances with the size of S bounded by O(k 3), using kernelization techniques such as the 2Expansion Lemma, Menger’s theorem and Gallai’s theorem. These two facts allow us to give a 2 O(k log k) n O(1) time algorithm solving the SUBSET FEEDBACK VERTEX SET problem, proving that it is indeed fixedparameter tractable.
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 ..."
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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.
Parameterized Tractability of Multiway Cut with Parity Constraints
"... Abstract. In this paper, we study a parity based generalization of the classical MULTIWAY CUT problem. Formally, we study the PARITY MULTIWAY CUT problem, where the input is a graph G, vertex subsets Te and To (T = Te ∪ To) called terminals, a positive integer k and the objective is to test whether ..."
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Abstract. In this paper, we study a parity based generalization of the classical MULTIWAY CUT problem. Formally, we study the PARITY MULTIWAY CUT problem, where the input is a graph G, vertex subsets Te and To (T = Te ∪ To) called terminals, a positive integer k and the objective is to test whether there exists a ksized vertex subset S such that S intersects all odd paths from v ∈ To to T \ {v} and all even paths from v ∈ Te to T \ {v}. When Te = To, this is precisely the classical MULTIWAY CUT problem. If To = ∅ then this is the EVEN MULTIWAY CUT problem and if Te = ∅ then this is the ODD MULTIWAY CUT problem. We remark that even the problem of deciding whether there is a set of at most k vertices that intersects all odd paths between a pair of vertices s and t is NPcomplete. Our primary motivation for studying this problem is the recently initiated parameterized study of parity versions of graphs minors (Kawarabayashi, Reed and Wollan, FOCS 2011) and separation problems similar to MULTIWAY CUT. The area of design of parameterized algorithms for graph separation problems has seen a lot of recent activity, which includes algorithms
Parameterized Complexity of the Anchored kCore Problem for Directed Graphs
 In FSTTCS
, 2013
"... Motivated by the study of unraveling processes in social networks, Bhawalkar, Kleinberg, Lewi, Roughgarden, and Sharma [ICALP 2012] introduced the Anchored kCore problem, where the task is for a given graph G and integers b, k, and p to find an induced subgraph H with at least p vertices (the core) ..."
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Motivated by the study of unraveling processes in social networks, Bhawalkar, Kleinberg, Lewi, Roughgarden, and Sharma [ICALP 2012] introduced the Anchored kCore problem, where the task is for a given graph G and integers b, k, and p to find an induced subgraph H with at least p vertices (the core) such that all but at most b vertices (called anchors) of H are of degree at least k. In this paper, we extend the notion of kcore to directed graphs and provide a number of new algorithmic and complexity results for the directed version of the problem. We show that • The decision version of the problem is NPcomplete for every k ≥ 1 even if the input graph is restricted to be a planar directed acyclic graph of maximum degree at most k + 2. • The problem is fixed parameter tractable (FPT) parameterized by the size of the core p for k = 1, and W[1]hard for k ≥ 2. • When the maximum degree of the graph is at most ∆, the problem is FPT parameterized by p+ ∆ if k ≥ ∆2. 1
Important separators and parameterized algorithms
, 2011
"... The notion of “important separators” and bounding the number of such separators turned out to be a very useful technique in the design of fixedparameter tractable algorithms for multi(way) cut problems. For example, the recent breakthrough result of Chen et al. [3] on the Directed Feedback Vertex ..."
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The notion of “important separators” and bounding the number of such separators turned out to be a very useful technique in the design of fixedparameter tractable algorithms for multi(way) cut problems. For example, the recent breakthrough result of Chen et al. [3] on the Directed Feedback Vertex Set problem can be also explained using this notion. In my talk, I will overview combinatorial and algorithmic results that can be obtained by studying such separators.
Parameterized Enumeration of Neighbour Strings and Kemeny Aggregations
, 2013
"... I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii In this thesis, we consider approaches to enumeration ..."
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I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii In this thesis, we consider approaches to enumeration problems in the parameterized complexity setting. We obtain competitive parameterized algorithms to enumerate all, as well as several of, the solutions for two related problems Neighbour String and Kemeny Rank Aggregation. In both problems, the goal is to find a solution that is as close as possible to a set of inputs (strings and total orders, respectively) according to some distance measure. We also introduce a notion of enumerative kernels for which there is a bijection between solutions to the original instance and solutions to the kernel, and provide such a kernel for Kemeny Rank Aggregation, improving a previous kernel for the problem. We demonstrate how several of the algorithms and notions discussed in this thesis are extensible to a group of parameterized problems, improving published results for some other problems. iii Acknowledgements I would like to thank my supervisor, Professor Naomi Nishimura, for her generous support and invaluable advice on my research. I would also like to thank Professor Jonathan Buss and Professor Timothy Chan for their helpful comments on the direction of my research. I wish to thank Professor Bin Ma for fruitful discussions on some of the results in this work. I am also grateful to my thesis committee for spending their valuable time reading this thesis, and for their suggestions which improved the content and presentation of my work. Special thanks to my family for their encouragement and love, and to the many friends I met in Waterloo, for making my PhD experience really enjoyable. iv
Algorithms for cut problems on trees
 Jinhui Xu, Boting Yang, Fenghui Zhang, Peng Zhang, and Binhai
, 2013
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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 ..."
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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.