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18
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.
Sequential Elimination Graphs
"... A graph is chordal if it does not contain any induced cycle of size greater than three. An alternative characterization of chordal graphs is via a perfect elimination ordering, which is an ordering of the vertices such that, for each vertex v, the neighbors of v that occur later than v in the order ..."
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Cited by 13 (2 self)
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A graph is chordal if it does not contain any induced cycle of size greater than three. An alternative characterization of chordal graphs is via a perfect elimination ordering, which is an ordering of the vertices such that, for each vertex v, the neighbors of v that occur later than v in the ordering form a clique. Akcoglu et al [2] define an extension of chordal graphs whereby the neighbors of v that occur later than v in the elimination order have at most k independent vertices. We refer to such graphs as sequentially kindependent graphs. Clearly this extension of chordal graphs also extends the class of (k+1)clawfree graphs. We study properties of such families of graphs, and we show that several natural classes of graphs are sequentially kindependent for small k. In particular, any intersection graph of translates of a convex object in a two dimensional plane is a sequentially 3independent graph; furthermore, any planar graph is a sequentially 3independent graph. For any fixed constant k, we develop simple, polynomial time approximation algorithms for sequentially kindependent graphs with respect to several wellstudied NPcomplete problems based on this ksequentially independent ordering. Our generalized formulation unifies and extends several previously known results. We also consider other classes of sequential elimination graphs.
Approximation Algorithms for Intersection Graphs
, 2009
"... We introduce three new complexity parameters that in some sense measure how chordallike a graph is. The similarity to chordal graphs is used to construct simple polynomialtime approximation algorithms with constant approximation ratio for many NPhard problems, when restricted to graphs for which ..."
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Cited by 8 (0 self)
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We introduce three new complexity parameters that in some sense measure how chordallike a graph is. The similarity to chordal graphs is used to construct simple polynomialtime approximation algorithms with constant approximation ratio for many NPhard problems, when restricted to graphs for which at least one of our new complexity parameters is bounded by a constant. As applications we present approximation algorithms with constant approximation ratio for maximum weighted independent set, minimum (independent) dominating set, minimum vertex coloring, maximum weighted clique, and minimum clique partition for large classes of intersection graphs.
Extracting Constrained 2Interval Subsets in 2Interval Sets
, 2007
"... 2interval sets were used in [28,29] for establishing a general representation for macroscopic describers of RNA secondary structures.In this context, we have a 2interval for each legal local fold in a given RNA sequence, and a constrained pattern made of disjoint 2intervals represents a putative ..."
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Cited by 3 (1 self)
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2interval sets were used in [28,29] for establishing a general representation for macroscopic describers of RNA secondary structures.In this context, we have a 2interval for each legal local fold in a given RNA sequence, and a constrained pattern made of disjoint 2intervals represents a putative RNA secondary structure. We focus here on the problem of extracting a constrained pattern in a set of 2intervals.More precisely, given a set of 2intervals D and a model R describing if two disjoint 2intervals in a solution can be in precedence order (<), be allowed to nest (⊏) and/or be allowed to cross (≬), we consider the problem of finding a maximum cardinality subset D ′ ⊆Dof disjoint 2intervals such that any two 2intervals in D ′ agree with R.The different combinations of restrictions on model R alter the computational complexity of the problem, and need to be examined separately. In this paper, we improve the time complexity of [29] for model R = {⊏} by giving an optimal O(n log n) time algorithm, where n is the cardinality of the 2interval set D.We also give a graphlike relaxation for model R = {⊏, ≬} that is solvable in O(n 2 √ n) time.Finally, we prove that the considered problem is NPcomplete for model R = {<, ≬} even for samelength intervals, and give a fixedparameter tractability result based on the crossing structure of D.
Parameterized complexity in multipleinterval graphs: domination
 In Proceedings of the 6th International Symposium on Parameterized and Exact Computation
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Optimization Problems in Multiple Subtree Graphs
"... We consider various optimization problems in tsubtree graphs, the intersection graphs of tsubtrees, where a tsubtree is the union of t disjoint subtrees of some tree. This graph class generalizes both the class of chordal graphs and the class of tinterval graphs, a generalization of interval gra ..."
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Cited by 2 (1 self)
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We consider various optimization problems in tsubtree graphs, the intersection graphs of tsubtrees, where a tsubtree is the union of t disjoint subtrees of some tree. This graph class generalizes both the class of chordal graphs and the class of tinterval graphs, a generalization of interval graphs that has recently been studied from a combinatorial optimization point of view. We present approximation
Maximal falsifiability: Definitions, algorithms, and applications
 In: LPAR
, 2013
"... Abstract. Similarly to Maximum Satisfiability (MaxSAT), Minimum Satisfiability (MinSAT) is an optimization extension of the Boolean Satisfiability (SAT) decision problem. In recent years, both problems have been studied in terms of exact and approximation algorithms. In addition, the MaxSAT problem ..."
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Abstract. Similarly to Maximum Satisfiability (MaxSAT), Minimum Satisfiability (MinSAT) is an optimization extension of the Boolean Satisfiability (SAT) decision problem. In recent years, both problems have been studied in terms of exact and approximation algorithms. In addition, the MaxSAT problem has been characterized in terms of Maximal Satisfiable Subsets (MSSes) and Minimal Correction Subsets (MCSes), as well as Minimal Unsatisfiable Subsets (MUSes) and minimal hitting set dualization. However, and in contrast with MaxSAT, no such characterizations exist for MinSAT. This paper addresses this issue by casting the MinSAT problem in a more general framework. The paper studies Maximal Falsifiability, the problem of computing a subsetmaximal set of clauses that can be simultaneously falsified, and shows that MinSAT corresponds to the complement of a largest subsetmaximal set of simultaneously falsifiable clauses, i.e. the solution of the Maximum Falsifiability (MaxFalse) problem. Additional contributions of the paper include novel algorithms for Maximum and Maximal Falsifiability, as well as minimal hitting set dualization results for the MaxFalse problem. Moreover, the proposed algorithms are validated on practical instances. 1
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