In this paper we study the parameterized complexity of approximating the parameterized counting problems contained in the class #W [P], the parameterized analogue of #P. We prove a parameterized analogue of a famous t...
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In this paper we study the parameterized complexity of approximating the parameterized counting problems contained in the class #W [P], the parameterized analogue of #P. We prove a parameterized analogue of a famous theorem of Stockmeyer claiming that approximate counting belongs to the second level of the polynomial hierarchy.
SAT and MAX SAT are among the most prominent problems for which local search algorithms have been successfully applied. A fundamental task for such an algorithm is to increase the number of clauses satisfied by a give...
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SAT and MAX SAT are among the most prominent problems for which local search algorithms have been successfully applied. A fundamental task for such an algorithm is to increase the number of clauses satisfied by a given truth assignment by flipping the truth values of at most k variables (k-flip local search). For a total number of n variables the size of the search space is of order n(k) and grows quickly in k;hence most practical algorithms use 1-flip local search only. In this paper we investigate the worst-case complexity of k-flip local search, considering k as a parameter: is it possible to search significantly faster than the trivial n(k) bound? In addition to the unbounded case we consider instances with a bounded number of literals per clause and instances where each variable occurs in a bounded number of clauses. We also consider the related problem that asks whether we can satisfy all clauses by flipping the truth values of at most k variables. (c) 2010 Elsevier B.V. All rights reserved.
Structural Properties of Graphs and Eficient Algorithms: Problems Between Parameters Dušan Knop parameterized complexity became over last two decades one of the most impor- tant subfield of computational complexity. S...
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Structural Properties of Graphs and Eficient Algorithms: Problems Between Parameters Dušan Knop parameterized complexity became over last two decades one of the most impor- tant subfield of computational complexity. Structural graph parameters (widths) play important role both in graph theory and (parameterized) algoritmh design. By studying some concrete problems we exhibit the connection between struc- tural graph parameters and parameterized tractability. We do this by examining tractability and hardness results for the Target Set Selection, Minimum Length Bounded Cut, and other problems. In the Minimum Length Bounded Cut problem we are given a graph, source, sink, and a positive integer L and the task is to remove edges from the graph such that the distance between the source and the sink exceeds L in the resulting graph. We show that an optimal solution to the Minimum Length Bounded Cut problem can be computed in time f(k)n, where f is a computable function and k denotes the tree-depth of the input graph. On the other hand we prove that (under assumption that FPT ̸= W[1]) no such algorithm can exist if the parameter k is the tree-width of the input graph. Currently only few such problems are known. The Target Set Selection problem exibits the same phenomenon for the vertex cover number and...
We study possible winner problems related to the uncovered set and the Banks set on partial tournaments from the viewpoint of parameterized complexity. We first study a problem where given a partial tournament D and a...
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We study possible winner problems related to the uncovered set and the Banks set on partial tournaments from the viewpoint of parameterized complexity. We first study a problem where given a partial tournament D and a subset X of vertices, we are asked to add some arcs to D such that all vertices in X are included in the uncovered set. We focus on two parameterizations: parameterized by |X| and parameterized by the number of arcs to be added. In addition, we study a parameterized variant of the problem which is to determine whether all vertices of X can be included in the uncovered set by reversing at most k arcs. Finally, we study some parameterizations of a possible winner problem on partial tournaments, where we are given a partial tournament D and a distinguished vertex p, and asked whether D has a maximal transitive subtournament with p being the 0-indegree vertex. These parameterized problems are related to the Banks set. We achieve results, -hardness results as well as results along with a kernelization lower bound for the problems studied in this paper.
In the MAXIMUM COMMON INDUCED SUBGRAPH problem (henceforth MCIS), given two graphs G(1) and G(2), one looks for a graph with the maximum number of vertices being both an induced subgraph of G(1) and G(2). MCIS is amon...
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In the MAXIMUM COMMON INDUCED SUBGRAPH problem (henceforth MCIS), given two graphs G(1) and G(2), one looks for a graph with the maximum number of vertices being both an induced subgraph of G(1) and G(2). MCIS is among the most studied classical NP-hard problems. It remains NP-hard on many graph classes including forests. In this paper, we study the parameterized complexity of MCIS. As a generalization" of CLIQUE, it is W[1]-hard parameterized by the size of the solution. Being NP-hard even on forests, most structural parameterizations are intractable. One has to go as far as parameterizing by the size of the minimum vertex cover to get some tractability. Indeed, when parameterized by k := vc(G(1)) + vc(G(2)) the sum of the vertex cover number of the two input graphs, the problem was shown to be fixed-parameter tractable, with an algorithm running in time 2(o(k logk)) We complement this result by showing that, unless the ETH fails, it cannot be solved in time 2(o(k log k)) This kind of tight lower bound has been shown for a few problems and parameters but, to the best of our knowledge, not for the vertex cover number. We also show that MCIS does not have a polynomial kernel when parameterized by k, unless NP subset of coNP/poly. Finally, we study MCIS and its connected variant MCCIS on some special graph classes and with respect to other structural parameters. (C) 2017 Elsevier B.V. All rights reserved.
We study the Hospitals/Residents with Couples problem, a variant of the classical Stable Marriage problem. This is the extension of the Hospitals/Residents problem where residents are allowed to form pairs and submit ...
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We study the Hospitals/Residents with Couples problem, a variant of the classical Stable Marriage problem. This is the extension of the Hospitals/Residents problem where residents are allowed to form pairs and submit joint rankings over hospitals. We use the framework of parameterized complexity, considering the number of couples as a parameter. We also apply a local search approach, and examine the possibilities for giving FPT algorithms applicable in this context. Furthermore, we also investigate the matching problem containing couples that is the simplified version of the Hospitals/Residents with Couples problem modeling the case when no preferences are given. (C) 2010 Elsevier B.V. All rights reserved.
The comparison of tree structured data is widespread since trees can be used to represent wide varieties of data, such as XML data, evolutionary histories, or carbohydrate structures. Two graph-theoretical problems us...
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The comparison of tree structured data is widespread since trees can be used to represent wide varieties of data, such as XML data, evolutionary histories, or carbohydrate structures. Two graph-theoretical problems used in the comparison of such data are the problems of finding the maximum common subtree (MCT) and the minimum common supertree (MCST) of two trees. These problems generalize to the problem of finding the MCT and MCST of multiple trees (Multi-MCT and Multi-MCST, respectively). In this paper, we prove parameterized complexity hardness results for the different parameterized versions of the Multi-MCT and Multi-MCST problem under isomorphic embeddings.
For an even integer t >= 2, the Matching Connectivity matrix H-t is a matrix that has rows and columns both labeled by all perfect matchings of the complete graph on t vertices;an entry H-t[M-1, M-2] is 1 if M-1 an...
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For an even integer t >= 2, the Matching Connectivity matrix H-t is a matrix that has rows and columns both labeled by all perfect matchings of the complete graph on t vertices;an entry H-t[M-1, M-2] is 1 if M-1 and M-2 form a Hamiltonian cycle and 0 otherwise. Motivated by applications for the Hamiltonicity problem, we show that H-t has rank exactly 2(t/2-1) over GF(2). The upper bound is established by an explicit factorization of H-t as the product of two submatrices;the matchings labeling columns and rows, respectively, of the submatrices therefore form a basis X-t of H-t. The lower bound follows because the 2(t/2-1) x 2(t/2-1) submatrix with rows and columns labeled by X-t can be seen to have full rank. We obtain several algorithmic results based on the rank of H-t and the particular structure of the matchings in X-t. First, we present a 1.888(n)n(O(1)) time Monte Carlo algorithm that solves the Hamiltonicity problem in directed bipartite graphs. Second, we give a Monte Carlo algorithm that solves the problem in (2 + root 2)(pw)n(O()(1)) time when provided with a path decomposition of width pw for the input graph. Moreover, we show that this algorithm is best possible under the Strong Exponential Time Hypothesis, in the sense that an algorithm with running time (2 + root 2 - epsilon)(pw)n(O(1)), for any epsilon > 0, would imply the breakthrough result of a (2 - epsilon')(n)-time algorithm for CNF-Sat for some epsilon' > 0.
We consider the EDGE EDITING TO A CONNECTED GRAPH OF GIVEN DEGREES problem that asks, given a graph G, non-negative integers d,k and a function delta: V(G) ->{1,...,d}, whether it is possible to obtain a connected ...
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We consider the EDGE EDITING TO A CONNECTED GRAPH OF GIVEN DEGREES problem that asks, given a graph G, non-negative integers d,k and a function delta: V(G) ->{1,...,d}, whether it is possible to obtain a connected graph G' from G such that the degree of v is delta(v) for every vertex v by at most kedge editing operations. As the problem is NP-complete even if delta(v) = 2, we are interested in the parameterized complexity and show that Edge Editing to a Connected Graph of Given Degrees admits a polynomial kernel when parameterized by d + k. For the special case delta(v) = d, i.e., when the aim is to obtain a connected d-regular graph, the problem is shown to be fixed parameter tractable when parameterized by k only. (C) 2017 Elsevier Inc. All rights reserved.
We investigate the parameterized complexity of the graph editing problem called EDITING TO A GRAPH WITH A GIVEN DEGREE SEQUENCE Where the aim iS to obtain a graph With a given degree sequence sigma by at most k vertex...
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We investigate the parameterized complexity of the graph editing problem called EDITING TO A GRAPH WITH A GIVEN DEGREE SEQUENCE Where the aim iS to obtain a graph With a given degree sequence sigma by at most k vertex deletions, edge deletions and edge additions. We show that the problem is W[1]-hard when parameterized by k for any combination of the allowed editing operations. From the positive side, we show that the problem can be solved in time 2 04 *+42)n2logn for n-vertex graphs, where Delta* = max sigma, i.e., the problem is FAT when parameterized by k + A*. We also show that EDITING TO A GRAPH. WITH A GIVEN DEGREE SEQUENCE has a polynomial kernel when parameterized by k + Delta* if only edge additions are allowed, and there is no polynomial kernel unless NP subset of co-NP/poly for all other combinations of the allowed editing operations. (C) 2016 Published by Elsevier B.V.
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