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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 1218 (75 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
Four Strikes against Physical Mapping of DNA
 JOURNAL OF COMPUTATIONAL BIOLOGY
, 1993
"... Physical Mapping is a central problem in molecular biology ... and the human genome project. The problem is to reconstruct the relative position of fragments of DNA along the genome from information on their pairwise overlaps. We show that four simplified models of the problem lead to NPcomplete ..."
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Cited by 67 (8 self)
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Physical Mapping is a central problem in molecular biology ... and the human genome project. The problem is to reconstruct the relative position of fragments of DNA along the genome from information on their pairwise overlaps. We show that four simplified models of the problem lead to NPcomplete decision problems: Colored unit interval graph completion, the maximum interval (or unit interval) subgraph, the pathwidth of a bipartite graph, and the kconsecutive ones problem for k >= 2. These models have been chosen to reflect various features typical in biological data, including false negative and positive errors, small width of the map and chimericism.
Scheduling Split Intervals
, 2002
"... We consider the problem of scheduling jobs that are given as groups of nonintersecting segments on the real line. Each job Jj is associated with an interval, Ij, which consists of up to t segments, for some t _) 1, a of their segments intersect. Such jobs show up in a I.I Problem Statement and Mo ..."
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Cited by 58 (5 self)
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We consider the problem of scheduling jobs that are given as groups of nonintersecting segments on the real line. Each job Jj is associated with an interval, Ij, which consists of up to t segments, for some t _) 1, a of their segments intersect. Such jobs show up in a I.I Problem Statement and Motivation. We wide range of applications, including the transmission consider the problem of scheduling jobs that are given of continuousmedia data, allocation of linear resources as groups of nonintersecting segments on the real line. (e.g. bandwidth in linear processor arrays), and in Each job Jj is associated with a tinterval, Ij, which
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 52 (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.
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.
Multitrack Interval Graphs
, 1995
"... . A dtrack interval is a union of d intervals, one each from d parallel lines. The intersection graphs of dtrack intervals are the unions of d interval graphs. The multitrack interval number or simply track number of a graph G is the minimum number of interval graphs whose union is G. The trac ..."
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Cited by 11 (2 self)
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. A dtrack interval is a union of d intervals, one each from d parallel lines. The intersection graphs of dtrack intervals are the unions of d interval graphs. The multitrack interval number or simply track number of a graph G is the minimum number of interval graphs whose union is G. The track number for Km;n is determined by proving that the arboricity of Km;n equals its "caterpillar arboricity". Recognition of graphs with track number 2 is shown to be NPcomplete. 1. Introduction Combinatorial properties of interval systems were investigated as early as the 1930's by Tibor Gallai. His two beautiful unpublished remarks are now phrased as saying that interval graphs and their complements are perfect graphs. In 1968, Gallai suggested considering more general set systems consisting of unions of d intervals, one each from d parallel lines. Such set systems have been called separated dintervals, since the d parallel lines can be viewed as d disjoint host intervals on a single li...
Dotted Interval Graphs and High Throughput Genotyping
"... We introduce a generalization of interval graphs, which we call dotted interval graphs (DIG). A dotted interval graph is an intersection graph of arithmetic progressions (=dotted intervals). Coloring of dotted intervals graphs naturally arises in the context of high throughput genotyping. We study t ..."
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Cited by 6 (1 self)
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We introduce a generalization of interval graphs, which we call dotted interval graphs (DIG). A dotted interval graph is an intersection graph of arithmetic progressions (=dotted intervals). Coloring of dotted intervals graphs naturally arises in the context of high throughput genotyping. We study the properties of dotted interval graphs, with a focus on coloring. We show that any graph is a DIG but that DIGd graphs, i.e. DIGs in which the arithmetic progressions have a jump of at most d, form a strict hierarchy. We show that coloring DIGd graphs is NPcomplete even for d = 2. For any fixed d, we provide a 7 8d approximation for the coloring of DIGd graphs.
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
Strip Graphs: Recognition and Scheduling ⋆
"... Abstract. We consider the class of strip graphs, a generalization of interval graphs. Intervals are assigned to rows such that two vertices have an edge between them if either their intervals intersect or they belong to the same row. We show that recognition of the class of strip graphs is NPcomple ..."
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Abstract. We consider the class of strip graphs, a generalization of interval graphs. Intervals are assigned to rows such that two vertices have an edge between them if either their intervals intersect or they belong to the same row. We show that recognition of the class of strip graphs is NPcomplete even if all intervals are of length 2. Strip graphs are important to the study of job selection, where we need an equivalence relation to connect multiple intervals that belong to the same job. The problem we consider is Job Interval Selection (JISP) on m machines. In the singlemachine case, this is equivalent to Maximum Independent Set on strip graphs. For m machines, the problem is to choose a maximum number of intervals, one from each job, such that the resulting choices form an mcolorable interval graph. We show the singlemachine case to be fixedparameter tractable in terms of the maximum number of overlapping rows. We also use a concatenation operation on strip graphs to reduce the mmachine case to the 1machine case. This shows that mmachine JISP is fixedparameter tractable in the total number of jobs.