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11
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 55 (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
"... We consider the problem of scheduling jobs that are given as groups of nonintersecting segments on the real line. Each job J j is associated with an interval, I j , which consists of up to t segments, for some t 1, a positive weight, w j , and two jobs are in conflict if any of their segments inte ..."
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Cited by 51 (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 J j is associated with an interval, I j , which consists of up to t segments, for some t 1, a positive weight, w j , and two jobs are in conflict if any of their segments intersect. Such jobs show up in a wide range of applications, including the transmission of continuousmedia data, allocation of linear resources (e.g. bandwidth in linear processor arrays), and in computational biology/geometry. The objective is to schedule a subset of nonconflicting jobs of maximum total weight.
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 21 (2 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 13 (4 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...
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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Cited by 1 (0 self)
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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.
THE TOTAL INTERVALNUMBER OF A GRAPH, II: TREES AND COMPLEXITY
"... A multipleinterval representation of a simple graph G assigns each vertex aunion of disjoint real intervals, such that vertices are adjacent if and only if their assigned sets intersect. The total interval number I(G) isthe minimum of the total number of intervals used in any such representation of ..."
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A multipleinterval representation of a simple graph G assigns each vertex aunion of disjoint real intervals, such that vertices are adjacent if and only if their assigned sets intersect. The total interval number I(G) isthe minimum of the total number of intervals used in any such representation of G. For trianglefree graphs, I(G) = m + t(G), where m is the number of edges in G and t(G) isthe minimum number of pairwise edgedisjoint trails such that every edge of G has an endpoint in at least one of the trails. This yields the NPcompleteness of testing I(G) = m + 1, even for trianglefree 3regular planar graphs, and an alternative proof that HAMILTONIAN CYCLE is NPcomplete for line graphs. It also yields a lineartime algorithm to compute I(G) for trees and a characterization of the trees requiring m + t intervals, for fixed t. Further corollaries include the Aigner/Andreae bound of I(G) ≤⎣(5n − 3)/4 ⎦ for nvertex trees (achieved bysubdividing every edge of a star), a characterization of the extremal trees, and a shorter proof of the extremal bound ⎣(5m + 2)/4 ⎦ for connected graphs. Ke ywords: intersection graphs, intervals, trees, extremal problem