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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.
On the tractability of maximal strip recovery
 Erratum in Journal of Computational Biology, 18:129, 2011. Cited
"... Abstract. Given two genomic maps G and H represented by a sequence of n gene markers, a strip (syntenic block) is a sequence of distinct markers of length at least two which appear as subsequences in the input maps, either directly or in reversed and negated form. The problem Maximal Strip Recovery ..."
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Abstract. Given two genomic maps G and H represented by a sequence of n gene markers, a strip (syntenic block) is a sequence of distinct markers of length at least two which appear as subsequences in the input maps, either directly or in reversed and negated form. The problem Maximal Strip Recovery (MSR) is to find two subsequences G ′ and H ′ of G and H, respectively, such that the total length of disjoint strips in G ′ and H ′ is maximized (or, conversely, the number of markers hence deleted, is minimized). Previously, besides some heuristic solutions, a factor4 polynomialtime approximation is known for the MSR problem; moreover, several close variants of MSR, MSRd (with d>2 input maps), MSRDU (with marker duplications) and MSRWT (with markers weighted) are all shown to be NPcomplete. Before this work, the complexity of the original MSR problem was left open. In this paper, we solve the open problem by showing that MSR is NPcomplete, using a polynomial time reduction from OneinThree 3SAT. We also solve the MSR problem and its variants exactly with FPT algorithms, i.e., showing that MSR is fixedparameter tractable. Let k be the minimum number of markers deleted in various versions of MSR, the running time of our
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