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On the Parameterized Complexity of the fixed Alphabet Shortest Common Supersequence and Longest Common Subsequence Problems
, 2003
"... INTRODUCTION The Shortest Common Supersequence (SCS) and the Longest Common Subsequence (LCS) are classical problems in computer science. Shortest Common Supersequence (SCS) integer . is a supersequence Longest Common Subsequence (LCS) integer . is a subsequence The LCS and (not so m ..."
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INTRODUCTION The Shortest Common Supersequence (SCS) and the Longest Common Subsequence (LCS) are classical problems in computer science. Shortest Common Supersequence (SCS) integer . is a supersequence Longest Common Subsequence (LCS) integer . is a subsequence The LCS and (not so much) the SCS problems have been extensively studied over the last 30 years (see [7] and references). They are both known to be NPcomplete [8, 9]. In particular the case where the number of sequences is 2 has been studied in detail (see [7] and references). A string a is a supersequence of a string b if we can delete some characters in a such that the remaining string is equal to b, e.g. \1234" is a supersequence of \13". A string a is a subsequence of a string b if b is a supersequence of a, e.g. \13" is a subsequence of \1234". 1.1. Sequence Comparison in Bioinformatics With the recent availability of large amounts of molecular sequence data, the LCS and related problems received
Parametric Complexity of Sequence Assembly: Theory and Applications to Next Generation Sequencing
"... In recent years a flurry of new DNA sequencing technologies have altered the landscape of genomics, providing a vast amount of sequence information at a fraction of the costs that were previously feasible. The task of assembling these sequences into a genome has, however, still remained an algorithm ..."
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In recent years a flurry of new DNA sequencing technologies have altered the landscape of genomics, providing a vast amount of sequence information at a fraction of the costs that were previously feasible. The task of assembling these sequences into a genome has, however, still remained an algorithmic challenge that is in practice answered by heuristic solutions. In order to design better assembly algorithms and exploit the characteristics of sequence data from new technologies we need an improved understanding of the parametric complexity of the assembly problem. In this work, we provide a first theoretical study in this direction, exploring the connections between repeat complexity, read lengths, overlap lengths and coverage in determining the “hard ” instances of the assembly problem. Our work suggests at least two ways in which existing assemblers can be extended in a rigorous fashion, in addition to delineating directions for future theoretical investigations. 1
Fixedparameter complexity of minimum profile problems
 In Proceedings IWPEC 2006
, 2006
"... The profile of a graph is an integervalued parameter defined via vertex orderings; it is known that the profile of a graph equals the smallest number of edges of an interval supergraph. Since computing the profile of a graph is an NPhard problem, we consider parameterized versions of the problem. ..."
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Cited by 14 (8 self)
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The profile of a graph is an integervalued parameter defined via vertex orderings; it is known that the profile of a graph equals the smallest number of edges of an interval supergraph. Since computing the profile of a graph is an NPhard problem, we consider parameterized versions of the problem. Namely, we study the problem of deciding whether the profile of a connected graph of order n is at most n − 1 + k, considering k as the parameter; this is a parameterization above guaranteed value, since n − 1 is a tight lower bound for the profile. We present two fixedparameter algorithms for this problem. The first algorithm is based on a forbidden subgraph characterization of interval graphs. The second algorithm is based on two simple kernelization rules which allow us to produce a kernel with linear number of vertices and edges. For showing the correctness of the second algorithm we need to establish structural properties of graphs with small profile which are of independent interest. 1
On parameterized intractability: hardness and completeness
, 2007
"... We study the theory and techniques developed in the research of parameterized intractability, emphasizing on parameterized hardness and completeness that imply (stronger) computational lower bounds for natural computational problems. Moreover, the fundamentals of the structural properties in paramet ..."
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Cited by 5 (1 self)
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We study the theory and techniques developed in the research of parameterized intractability, emphasizing on parameterized hardness and completeness that imply (stronger) computational lower bounds for natural computational problems. Moreover, the fundamentals of the structural properties in parameterized complexity theory, relationships to classical complexity theory and more recent developments in the area are also introduced.
Breakpoint Medians and Breakpoint Phylogenies: A FixedParameter Approach
, 2002
"... With breakpoint distance, the genome rearrangement field delivered one of the currently most popular measures in phylogenetic studies for related species. Here, BREAK POINT MEDIAN, which is NPcomplete already for three given species (whose genomes are represented as signed orderings), is the core ..."
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Cited by 4 (1 self)
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With breakpoint distance, the genome rearrangement field delivered one of the currently most popular measures in phylogenetic studies for related species. Here, BREAK POINT MEDIAN, which is NPcomplete already for three given species (whose genomes are represented as signed orderings), is the core basic problem. For the important special case of three species, approximation (ratio 7/6) and exact heuristic algorithms were developed. Here, we provide an exact, fixedparameter algorithm with provable performance bounds. For instance, a breakpoint median for three signed orderings over n elements that causes at most d breakpoints can be computed in time O((2.15) n). We show the algorithm's practical usefulness through experimental studies. In particular, we demonstrate that a simple implementation of our algorithm combined with a new tree construction heuristic allows for a new approach to breakpoint phylogeny, yielding evolutionary trees that are competitive in comparison with known results developed in a recent series of papers that use clever algorithm engineering methods.
Linear Recognition of Almost (Unit) Interval Graphs
, 2014
"... Give a graph class G and a nonnegative integer k, we use G+kv, G+ke, and G−ke to denote the classes of graphs that can be obtained from some graph in G by adding k vertices, adding k edges, and deleting k edges, respectively. They are called almost (unit) interval graphs if G is the class of (unit) ..."
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Give a graph class G and a nonnegative integer k, we use G+kv, G+ke, and G−ke to denote the classes of graphs that can be obtained from some graph in G by adding k vertices, adding k edges, and deleting k edges, respectively. They are called almost (unit) interval graphs if G is the class of (unit) interval graphs. Almost (unit) interval graphs are well motivated from computational biology, where the data ought to be represented by a (unit) interval graph while we can only expect an almost (unit) interval graph for the best. For any fixed k, we give lineartime algorithms for recognizing all these classes, and in the case of membership, our algorithms provide also a specific (unit) interval graph as evidence. When k is part of the input, all the recognition problems are NPcomplete. Our results imply that all of them are fixedparameter tractable parameterized by k, thereby resolving the longstanding open problem on the parameterized complexity of recognizing (unit) interval+ke, first asked by Bodlaender et al. [Comput. Appl. Biosci., 11(1):49–57, 1995]. Moreover, our algorithms for recognizing (unit)interval+kv and (unit)interval−ke have singleexponential dependence on k and linear dependence on the graph size, which significantly improve all previous algorithms for recognizing the same classes. In particular, we show that: (n and m stand for the numbers of vertices and edges respectively in the input graph) • interval−ke can be recognized in time O(6k · (n +m)), improved from O(k2k · n3m) [Heggernes et al., STOC 2007]; • unitinterval−ke can be recognized in time O(4k · (n+m)), improved from O(16k · (m+n)) [Kaplan et al., FOCS 1994]; • interval+kv can be recognized in time O(8k · (n +m)), improved from O(10k · n9) [Cao and Marx,
The Hardness of Problems on Thin Colored Graphs
, 2000
"... In this paper, we consider the complexity of a number of combinatorial problems; namely, Intervalizing Colored Graphs (DNA physical mapping), Triangulating Colored Graphs (perfect phylogeny), (Directed) (Modified) Colored Cutwidth, Feasible Register Assignment and Module Allocation for graphs of bou ..."
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In this paper, we consider the complexity of a number of combinatorial problems; namely, Intervalizing Colored Graphs (DNA physical mapping), Triangulating Colored Graphs (perfect phylogeny), (Directed) (Modified) Colored Cutwidth, Feasible Register Assignment and Module Allocation for graphs of bounded treewidth. Each of these problems has as a characteristic a uniform upper bound on the tree or path width of the graphs in "yes"instances. For all of these problems with the exceptions of feasible register assignment and module allocation, a vertex or edge coloring is given as part of the input. Our main results are that the parameterized variant of each of the considered problems is hard for the complexity classes W [t] for all t 2 Z + . We also show that Intervalizing Colored Graphs, Triangulating Colored Graphs, and Colored Cutwidth are NPComplete.
IEEE/OSA/IAPR International Conference on Infonnatics, Electronics & Vision An Ant Colony Optimization approach to solve the Minimum String Cover Problem
"... AbstractIn this paper, we consider the problem of covering a set of strings S with a set of strings C. C is said to cover S if every string in S can be written as a concatenation of a set of strings which are elements of C. We discuss here three different variants of Ant Colony Optimization (ACO) a ..."
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AbstractIn this paper, we consider the problem of covering a set of strings S with a set of strings C. C is said to cover S if every string in S can be written as a concatenation of a set of strings which are elements of C. We discuss here three different variants of Ant Colony Optimization (ACO) and propose how we can solve the minimum string cover problem using these techniques. Our simulation results show that ACO based approach gives better solution than the existing approximation algorithm for this problem.