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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
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 ..."
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Cited by 14 (5 self)
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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
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.
On multiple learning schemata in conflict driven solvers
"... Abstract. In this preliminary paper we describe a general approach for multiple learning in conflictdriven SAT solvers. The proposed formulation of the conflict analysis task turns out to be expressive enough to reckon with different orthogonal generalizations of the standard learning schemata, suc ..."
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Abstract. In this preliminary paper we describe a general approach for multiple learning in conflictdriven SAT solvers. The proposed formulation of the conflict analysis task turns out to be expressive enough to reckon with different orthogonal generalizations of the standard learning schemata, such as the conjunct analysis of multiple conflicts, the generation of possibly interdependent learned clauses, the imposition of global optimality criteria. We formalize the general learning problem as a search for a collection of vertex cuts in a directed acyclic graph. Optimality of the solution may be evaluated with respect to a given global objective function intended to encode search strategies and heuristics affecting the behavior of the solver. We also outline some algorithmical solutions by exploiting standard algorithms proposed to solve cut and multicut problems on DAGs. 1
Data reduction and problem kernels
"... This report documents the program and the outcomes of Dagstuhl Seminar 12241 “Data Reduction and Problem Kernels”. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as ..."
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This report documents the program and the outcomes of Dagstuhl Seminar 12241 “Data Reduction and Problem Kernels”. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available.
DIRECTED GRAPHS: FIXEDPARAMETER TRACTABILITY & BEYOND
, 2014
"... Most interesting optimization problems on graphs are NPhard, implying that (unless P = NP) there is no polynomial time algorithm that solves all the instances of an NPhard problem exactly. However, classical complexity measures the running time as a function of only the overall input size. The pa ..."
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Most interesting optimization problems on graphs are NPhard, implying that (unless P = NP) there is no polynomial time algorithm that solves all the instances of an NPhard problem exactly. However, classical complexity measures the running time as a function of only the overall input size. The paradigm of parameterized complexity was introduced by Downey and Fellows to allow for a more refined multivariate analysis of the running time. In parameterized complexity, each problem comes along with a secondary measure k which is called the parameter. The goal of parameterized complexity is to design efficient algorithms for NPhard problems when the parameter k is small, even if the input size is large. Formally, we say that a parameterized problem is fixedparameter tractable (FPT) if instances of size n and parameter k can be solved in f (k) · nO(1) time, where f is a computable function which does not depend on n. A parameterized problem belongs to the class XP if instances of size n and parameter k can be solved in f (k) ·nO(g(k)) time, where f and g are both computable functions. In this thesis we focus on the parameterized complexity of transversal and connectivity problems on directed graphs. This research direction has been hitherto relatively
Parameterized Complexity Dichotomy for Steiner Multicut
, 2014
"... We consider the Steiner Multicut problem, which asks, given an undirected graph G, a collection T = {T1,..., Tt}, Ti ⊆ V (G), of terminal sets of size at most p, and an integer k, whether there is a set S of at most k edges or nodes such that of each set Ti at least one pair of terminals is in diffe ..."
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We consider the Steiner Multicut problem, which asks, given an undirected graph G, a collection T = {T1,..., Tt}, Ti ⊆ V (G), of terminal sets of size at most p, and an integer k, whether there is a set S of at most k edges or nodes such that of each set Ti at least one pair of terminals is in different connected components of G \S. This problem generalizes several wellstudied graph cut problems, in particular the Multicut problem, which corresponds to the case p = 2. The Multicut problem was recently shown to be fixedparameter tractable for the parameter k [Marx and Razgon, Bousquet et al., STOC 2011]. The question whether this result generalizes to Steiner Multicut motivates the present work. We answer the question that motivated this work, and in fact provide a dichotomy of the parameterized complexity of Steiner Multicut on general graphs. That is, for any combination of k, t, p, and the treewidth tw(G) as constant, parameter, or unbounded, and for all versions of the problem (edge deletion and node deletion with and without deletable terminals), we prove either that the problem is fixedparameter tractable or that the problem is hard (W[1]hard or even (para)NPcomplete). Among the many results in the paper,
Directed Multicut with linearly ordered terminals
, 2014
"... Motivated by an application in network security, we investigate the following “linear ” case of Directed Multicut. Let G be a directed graph which includes some distinguished vertices t1,..., tk. What is the size of the smallest edge cut which eliminates all paths from ti to tj for all i < j? We ..."
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Motivated by an application in network security, we investigate the following “linear ” case of Directed Multicut. Let G be a directed graph which includes some distinguished vertices t1,..., tk. What is the size of the smallest edge cut which eliminates all paths from ti to tj for all i < j? We show that this problem is fixedparameter tractable when parametrized in the cutset size p via an algorithm running in O(4ppn4) time. 1 Multicut requests as partially ordered sets The problem of finding a smallest edge cut separating vertices in a graph has received much attention over the past 50 years. Directed Multicut, one of the more general forms of this problem, encompasses numerous applications in algorithmic graph theory. Name: Directed Multicut. Instance: A directed graph G and pairs of terminal vertices {(s1, t1),..., (sk, tk)} from G.