This paper surveys parameterized complexity results for hard geometric algorithmic problems. It includes fixed-parameter tractable problems in graph drawing, geometric graphs, geometric covering and several other area...
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This paper surveys parameterized complexity results for hard geometric algorithmic problems. It includes fixed-parameter tractable problems in graph drawing, geometric graphs, geometric covering and several other areas, together with an overview of the algorithmic techniques used. Fixed-parameter intractability results are surveyed as well. Finally, we give some directions for future research.
Let be a family of graphs. In the classical -VERTEX DELETION problem, given a graph G and a positive integer k, the objective is to check whether there exists a subset S of at most k vertices such that G- S is in . In...
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Let be a family of graphs. In the classical -VERTEX DELETION problem, given a graph G and a positive integer k, the objective is to check whether there exists a subset S of at most k vertices such that G- S is in . In this paper, we introduce the conflict free version of this classical problem, namely CONFLICT FREE -VERTEX DELETION (CF--VD), and study this problem from the viewpoint of classical and parameterized complexity. In the CF--VD problem, given two graphs G and H on the same vertex set and a positive integer k, the objective is to determine whether there exists a set S. V (G), of size at most k, such that G - S is in and H[S] is edgeless. Initiating a systematic study of these problems is one of the main conceptual contribution of this work. We obtain several results on the conflict free versions of several classical problems. Our first result shows that if is characterized by a finite family of forbidden induced subgraphs then CF--VD is Fixed Parameter Tractable (FPT). Furthermore, we obtain improved algorithms for conflict free versions of several well studied problems. Next, we show that if is characterized by a "well-behaved" infinite family of forbidden induced subgraphs, then CF--VD is W[1]-hard. Motivated by this hardness result, we consider the parameterized complexity of CF--VD when H is restricted to well studied families of graphs. In particular, we show that the conflict free version of several well-known problems such as FEEDBACK VERTEX SET, ODD CYCLE TRANSVERSAL, CHORDAL VERTEX DELETION and INTERVAL VERTEX DELETION are FPT when H belongs to the families of d-degenerate graphs and nowhere dense graphs.
Recently, the shared center (SC) problem has been proposed as a mathematical model for inferring the allele-sharing status of a given set of individuals using a database of confirmed haplotypes as reference. The probl...
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Recently, the shared center (SC) problem has been proposed as a mathematical model for inferring the allele-sharing status of a given set of individuals using a database of confirmed haplotypes as reference. The problem was proved to be NP-complete and a ratio-2 polynomial-time approximation algorithm was designed for its minimization version (called the closest shared center (CSC) problem). In this paper, we consider the parameterized complexity of the SC problem. First, we show that the SC problem is W[1]-hard with parameters d and n, where d and n are the radius and the number of (diseased or normal) individuals in the input, respectively. Then, we present two asymptotically optimal parameterized algorithms for the problem and apply them to linkage analysis.
We study the well-known Vertex Cover problem parameterized above and below tight bounds. We show that two of the parameterizations (both were suggested by Mahajan et al. in J. Comput. Syst. Sci. 75(2):137-153, 2009) a...
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We study the well-known Vertex Cover problem parameterized above and below tight bounds. We show that two of the parameterizations (both were suggested by Mahajan et al. in J. Comput. Syst. Sci. 75(2):137-153, 2009) are fixed-parameter tractable and two other parameterizations are W[1]-hard (one of them is, in fact, W[2]-hard).
The paper surveys parameterized algorithms and complexities for computational tasks on biopolymer sequences, including the problems of longest common subsequence, shortest common supersequence, pairwise sequence align...
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The paper surveys parameterized algorithms and complexities for computational tasks on biopolymer sequences, including the problems of longest common subsequence, shortest common supersequence, pairwise sequence alignment, multiple sequencing alignment, structure-sequence alignment and structure-structure alignment. Algorithm techniques, built on the structural-unit level as well as on the residue level, are discussed.
In this paper, we investigate the complexity of Maximum Independent Set (MIS) in the class of H-free graphs, that is, graphs excluding a fixed graph as an induced subgraph. Given that the problem remains NP-hard for m...
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In this paper, we investigate the complexity of Maximum Independent Set (MIS) in the class of H-free graphs, that is, graphs excluding a fixed graph as an induced subgraph. Given that the problem remains NP-hard for most graphs H, we study its fixed-parameter tractability and make progress towards a dichotomy between FPT and W[1]-hard cases. We first show that MIS remains W[1]-hard in graphs forbidding simultaneously K-1,K-4, any finite set of cycles of length at least 4, and any finite set of trees with at least two branching vertices. In particular, this answers an open question of Dabrowski et al. concerning C-4-free graphs. Then we extend the polynomial algorithm of Alekseev when H is a disjoint union of edges to an FPT algorithm when H is a disjoint union of cliques. We also provide a framework for solving several other cases, which is a generalization of the concept of iterative expansion accompanied by the extraction of a particular structure using Ramsey's theorem. Iterative expansion is a maximization version of the so-called iterative compression. We believe that our framework can be of independent interest for solving other similar graph problems. Finally, we present positive and negative results on the existence of polynomial (Turing) kernels for several graphs H.
We study the Minimum Length-Bounded Cut problem where the task is to find a set of edges of a graph such that after removal of this set, the shortest path between two prescribed vertices is at least L + 1 long. We sho...
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We study the Minimum Length-Bounded Cut problem where the task is to find a set of edges of a graph such that after removal of this set, the shortest path between two prescribed vertices is at least L + 1 long. We show the problem can be computed in FPT time with respect to L and the tree-width of the input graph G as parameters and with linear dependence of vertical bar V(G)vertical bar (i.e., in time f (L, tw(G))vertical bar V(G)vertical bar for a computable function f). We derive an FPT algorithm for a more general multi-commodity length-bounded cut problem when additionally parameterized by the number of terminals. For the former problem we show a W[1]-hardness result when the parameterization is done by the path-width only (instead of the tree-width) and that this problem does not admit polynomial kernel when parameterized by path-width and L. We also derive an FPT algorithm for the Minimum LENGTH-BOUNDED Cut problem when parameterized by the tree-depth, thus showing an interesting behavior for this problem and parameters tree-depth and path-width.
After the number of vertices, Vertex Cover Number is the largest of the classical graph parameters and has more and more frequently been used as a separate parameter in parameterized problems, including problems that ...
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After the number of vertices, Vertex Cover Number is the largest of the classical graph parameters and has more and more frequently been used as a separate parameter in parameterized problems, including problems that are not directly related to the Vertex Cover Number. Here we consider the TREEWIDTH and PATHWIDTH problems parameterized by k, the size of a minimum vertex cover of the input graph. We show that the PATHWIDTH and TREEWIDTH can be computed in O*(3(k)) time. This complements recent polynomial kernel results for TREEWIDTH and PATHWIDTH parameterized by the Vertex Cover Number. (C) 2014 Elsevier B.V. All rights reserved.
In this paper we study the notion of parameterized exponential time complexity. We show that a parameterized problem can be solved in parameterized 2(o(f(k)))p(n) time if and only if it is solvable in time O(2(delta f...
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In this paper we study the notion of parameterized exponential time complexity. We show that a parameterized problem can be solved in parameterized 2(o(f(k)))p(n) time if and only if it is solvable in time O(2(delta f(k))q(n)) for any constant delta>0, where p and q are polynomials. We then illustrate how this equivalence can be used to show that special instances of parameterized NP-hard problems are as difficult as the general instances. For example, we show that the PLANAR DOMINATING SET problem on degree-3 graphs can be solved in 2(0(root k))p(n) parameterized time if and only if the general PLANAR DOMINATING SET problem can. Apart from their complexity theoretic implications, our results have some interesting algorithmic implications as well. (C) 2009 Elsevier B.V. All rights reserved.
We study (vertex-disjoint) packings of paths of length two (i.e., of P-2' s) in graphs under a parameterized perspective. Starting from a maximal P-2-packing P of size j we use extremal combinatorial arguments for...
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We study (vertex-disjoint) packings of paths of length two (i.e., of P-2' s) in graphs under a parameterized perspective. Starting from a maximal P-2-packing P of size j we use extremal combinatorial arguments for determining how many vertices of P appear in some P-2-packing of size (j + 1) (if such a packing exists). We prove that one can 'reuse' 2.5j vertices. We also show that this bound is asymptotically sharp. Based on a WIN-WIN approach, we build an algorithm which decides, given a graph, if a P-2-packing of size at least k exists in time O*(2.448(3k)).
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