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15
Parameterized Complexity
, 1998
"... the rapidly developing systematic connections between FPT and useful heuristic algorithms  a new and exciting bridge between the theory of computing and computing in practice. The organizers of the seminar strongly believe that knowledge of parameterized complexity techniques and results belongs ..."
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Cited by 1213 (77 self)
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the rapidly developing systematic connections between FPT and useful heuristic algorithms  a new and exciting bridge between the theory of computing and computing in practice. The organizers of the seminar strongly believe that knowledge of parameterized complexity techniques and results belongs into the toolkit of every algorithm designer. The purpose of the seminar was to bring together leading experts from all over the world, and from the diverse areas of computer science that have been attracted to this new framework. The seminar was intended as the rst larger international meeting with a specic focus on parameterized complexity, and it hopefully serves as a driving force in the development of the eld. 1 We had 49 participants from Australia, Canada, India, Israel, New Zealand, USA, and various European countries. During the workshop 25 lectures were given. Moreover, one night session was devoted to open problems and Thursday was basically used for problem discussion
On the parameterized complexity of multipleinterval graph problems
 Theor. Comput. Sci
"... Abstract. Multipleinterval graphs are a natural generalization of interval graphs where each vertex may have more than one interval associated with it. Many applications of interval graphs also generalize to multipleinterval graphs, often allowing for more robustness in the modeling of the specifi ..."
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Cited by 50 (8 self)
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Abstract. Multipleinterval graphs are a natural generalization of interval graphs where each vertex may have more than one interval associated with it. Many applications of interval graphs also generalize to multipleinterval graphs, often allowing for more robustness in the modeling of the specific application. With this motivation in mind, a recent systematic study of optimization problems in multipleinterval graphs was initiated. In this sequel, we study multipleinterval graph problems from the perspective of parameterized complexity. The problems under consideration are kIndependent Set, kDominating Set, and kClique, which are all known to be W[1]hard for general graphs, and NPcomplete for multipleinterval graphs. We prove that kClique is in FPT, while kIndependent Set and kDominating Set are both W[1]hard. We also prove that kIndependent Dominating Set, a hybrid of the two above problems, is also W[1]hard. Our hardness results hold even when each vertex is associated with at most two intervals, and all intervals have unit length. Furthermore, as an interesting byproduct of our hardness results, we develop a useful technique for showing W[1]hardness via a reduction from the kMulticolored Clique problem, a variant of kClique. We believe this technique has interest in its own right, as it should help in simplifying W[1]hardness results which are notoriously hard to construct and technically tedious.
How to compare arcannotated sequences : the alignment hierarchy
 In 13th Symposium on String Processing and Information Retrieval (SPIRE’06), volume 4209 of LNCS
, 2006
"... Abstract. We describe a new unifying framework to express comparison of arcannotated sequences, which we call alignment of arcannotated sequences. We first prove that this framework encompasses main existing models, which allows us to deduce complexity results for several cases from the literature ..."
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Cited by 19 (9 self)
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Abstract. We describe a new unifying framework to express comparison of arcannotated sequences, which we call alignment of arcannotated sequences. We first prove that this framework encompasses main existing models, which allows us to deduce complexity results for several cases from the literature. We also show that this framework gives rise to new relevant problems that have not been studied yet. We provide a thorough analysis of these novel cases by proposing two polynomial time algorithms and an NPcompleteness proof. This leads to an almost exhaustive study of alignment of arcannotated sequences.
Optimization problems in multipleinterval graphs
 In Proceedings of the 18th annual Symposium On Discrete Algorithms (SODA
, 2007
"... Multipleinterval graphs are a natural generalization of interval graphs where each vertex may have more then one interval associated with it. We initiate the study of optimization problems in multipleinterval graphs by considering three classical problems: Minimum Vertex Cover, Minimum Dominating ..."
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Cited by 18 (5 self)
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Multipleinterval graphs are a natural generalization of interval graphs where each vertex may have more then one interval associated with it. We initiate the study of optimization problems in multipleinterval graphs by considering three classical problems: Minimum Vertex Cover, Minimum Dominating Set, and Maximum Clique. We describe applications for each one of these problems, and then proceed to discuss approximation algorithms for them. Our results can be summarized as follows: Let t be the number of intervals associated with each vertex in a given multipleinterval graph. For Minimum Vertex Cover, we give a (2 − 1/t)approximation algorithm which also works when a tinterval representation of our given graph is absent. Following this, we give a t 2approximation algorithm for Minimum Dominating Set which adapts well to more general variants of the problem. We then proceed to prove that Maximum Clique is NPhard already for 3interval graphs, and provide a (t 2 −t+ 1)/2approximation algorithm for general values of t ≥ 2, using bounds proven for the socalled transversal number of tinterval families.
On the approximability of comparing genomes with duplicates
, 2009
"... A central problem in comparative genomics consists in computing a (dis)similarity measure between two genomes, e.g. in order to construct a phylogenetic tree. A large number of such measures has been proposed in the recent past: number of reversals, number of breakpoints, number of common or conser ..."
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Cited by 16 (8 self)
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A central problem in comparative genomics consists in computing a (dis)similarity measure between two genomes, e.g. in order to construct a phylogenetic tree. A large number of such measures has been proposed in the recent past: number of reversals, number of breakpoints, number of common or conserved intervals etc. In their initial definitions, all these measures suppose that genomes contain no duplicates. However, we now know that genes can be duplicated within the same genome. One possible approach to overcome this difficulty is to establish a onetoone correspondence (i.e. a matching) between genes of both genomes, where the correspondence is chosen in order to optimize the studied measure. Then, after a gene relabeling according to this matching and a deletion of the unmatched signed genes, two genomes without duplicates are obtained and the measure can be computed. In this paper, we are interested in three measures (number of breakpoints,
Improved Algorithms for Largest Cardinality 2Interval Pattern Problem
 JOURNAL OF COMBINATORIAL OPTIMIZATION
"... The 2Interval Pattern problem is to find the largest constrained pattern in a set of 2intervals. The constrained pattern is a subset of the given 2intervals such that any pair of them are Rcomparable, where model R ⊆ { <, ⊏, ()}. The problem stems from the study of general representation of ..."
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Cited by 3 (0 self)
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The 2Interval Pattern problem is to find the largest constrained pattern in a set of 2intervals. The constrained pattern is a subset of the given 2intervals such that any pair of them are Rcomparable, where model R ⊆ { <, ⊏, ()}. The problem stems from the study of general representation of RNA secondary structures. In this paper, we give three improved algorithms for different models. Firstly, an O(n log n + L) algorithm is proposed for the case R = { ()}, where L = O(dn) = O(n 2) is the total length of all 2intervals (density d is the maximum number of 2intervals over any point). This improves previous O(n 2 log n) algorithm. Secondly, we use dynamic programming techniques to obtain an O(n log n+dn) algorithm for the case R = { <, ⊏}, which improves previous O(n 2) result. Finally, we present another O(n log n + L) algorithm for the case R = { ⊏, () } with disjoint support(interval ground set), which improves previous O(n 2 √ n) upper bound.
Common Structured Patterns in Linear Graphs: Approximations and Combinatorics
"... A linear graph is a graph whose vertices are linearly ordered. This linear ordering allows pairs of disjoint edges to be either preceding (<), nesting (⊏) or crossing (≬). Given a family of linear graphs, and a nonempty subset R ⊆ {<, ⊏, ≬} of these three relations, we are interested in the M ..."
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Cited by 2 (1 self)
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A linear graph is a graph whose vertices are linearly ordered. This linear ordering allows pairs of disjoint edges to be either preceding (<), nesting (⊏) or crossing (≬). Given a family of linear graphs, and a nonempty subset R ⊆ {<, ⊏, ≬} of these three relations, we are interested in the Maximum Common Structured Pattern (MCSP) problem: Find a maximum size edgedisjoint graph, with edgepairs all comparable by one of the relations in R, that occurs as a subgraph in each of the linear graphs of the family. In this paper, we generalize the framework of Davydov and Batzoglou by considering patterns comparable by all possible subsets R ⊆ {<, ⊏, ≬}. This is motivated by the fact that many biological applications require considering crossing structures, and by the fact that different combinations of the relations above give rise to different generalizations of natural combinatorial problems. Our results can be summarized as follows: We give tight hardness results for the MCSP problem for {<, ≬}structured patterns and {⊏, ≬}structured patterns. Furthermore, we prove that the problem is approximable within ratios: (i) 2H (k) for {<, ≬}structured patterns, (ii) k 1/2 for {⊏, ≬}structured patterns, and (iii) O ( √ k lg k) for {<, ⊏, ≬}structured patterns, where k is the size of the optimal solution and H (k) = �k i=1 1/i is the kth harmonic number. Along the way, we provide combinatorial results concerning the different types of structured patterns that are of independent interest in their own right.
kGap interval graphs
 IN: PROC. OF THE 10TH LATIN AMERICAN THEORETICAL INFORMATICS SYMPOSIUM (LATIN) (2012). AVAILABLE AT: ARXIV:1112.3244
, 2012
"... We initiate the study of a new parameterization of graph problems. In a multiple interval representation of a graph, each vertex is associated to at least one interval of the real line, with an edge between two vertices if and only if an interval associated to one vertex has a nonempty intersectio ..."
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We initiate the study of a new parameterization of graph problems. In a multiple interval representation of a graph, each vertex is associated to at least one interval of the real line, with an edge between two vertices if and only if an interval associated to one vertex has a nonempty intersection with an interval associated to the other vertex. A graph on n vertices is a kgap interval graph if it has a multiple interval representation with at most n + k intervals in total. In order to scale up the nice algorithmic properties of interval graphs (where k = 0), we parameterize graph problems by k, and find FPT algorithms for several
Delta: a Toolset for the Structural Analysis of Biological Sequences on a 3D Triangular Lattice
"... The lattice approach to biological structural analysis was made popular by the HP model for protein folding, but had not been used previously for RNA secondary structure prediction. We introduce the Delta toolset for the structural analysis of biological sequences on a 3D triangular lattice. The Del ..."
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The lattice approach to biological structural analysis was made popular by the HP model for protein folding, but had not been used previously for RNA secondary structure prediction. We introduce the Delta toolset for the structural analysis of biological sequences on a 3D triangular lattice. The Delta toolset includes a proofofconcept RNA folding program that is both fast and accurate in predicting the secondary structures with pseudoknots of short RNA sequences.
Finding Common Structured Patterns in Linear Graphs
, 2009
"... A linear graph is a graph whose vertices are linearly ordered. This linear ordering allows pairs of disjoint edges to be either preceding (<), nesting (⊏) or crossing (≬). Given a family of linear graphs, and a nonempty subset R ⊆ {<, ⊏, ≬}, we are interested in the Maximum Common Structured ..."
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A linear graph is a graph whose vertices are linearly ordered. This linear ordering allows pairs of disjoint edges to be either preceding (<), nesting (⊏) or crossing (≬). Given a family of linear graphs, and a nonempty subset R ⊆ {<, ⊏, ≬}, we are interested in the Maximum Common Structured Pattern (MCSP) problem: find a maximum size edgedisjoint graph, with edgepairs all comparable by one of the relations in R, that occurs as a subgraph in each of the linear graphs of the family. The MCSP problem generalizes many structurecomparison and structureprediction problems that arise in computational molecular biology. We give tight hardness results for the MCSP problem for {<, ≬}structured patterns and {⊏, ≬}structured patterns. Furthermore, we prove that the problem is approximable within ratios: (i) 2H (k) for {<, ≬}structured patterns, (ii) k1/2 for {⊏, ≬}structured patterns, and (iii) O ( √ k log k) for {<, ⊏, ≬}structured patterns, where k is the size of the optimal solution and H (k) = ∑k i=1