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20
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.
On the Complexity of DNA Physical Mapping
, 1994
"... The Physical Mapping Problem is to reconstruct the relative position of fragments (clones) of DNA along the genome from information on their pairwise overlaps. We show that two simplified versions of the problem belong to the class of NPcomplete problems, which are conjectured to be computationa ..."
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Cited by 41 (7 self)
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The Physical Mapping Problem is to reconstruct the relative position of fragments (clones) of DNA along the genome from information on their pairwise overlaps. We show that two simplified versions of the problem belong to the class of NPcomplete problems, which are conjectured to be computationally intractable. In one version all clones have equal length, and in another, clone lengths may be arbitrary. The proof uses tools from graph theory and complexity.
An Alternative Method to Crossing Minimization on Hierarchical Graphs
 SIAM J. Optimization
, 1997
"... . A common method for drawing directed graphs is, as a first step, to partition the vertices into a set of k levels and then, as a second step, to permute the vertices within the levels such that the number of crossings is minimized. We suggest an alternative method for the second step, namely, remo ..."
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Cited by 28 (0 self)
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. A common method for drawing directed graphs is, as a first step, to partition the vertices into a set of k levels and then, as a second step, to permute the vertices within the levels such that the number of crossings is minimized. We suggest an alternative method for the second step, namely, removing the minimal number of edges such that the resulting graph is klevel planar. For the final diagram the removed edges are reinserted into a klevel planar drawing. Hence, instead of considering the klevel crossing minimization problem, we suggest solving the klevel planarization problem. In this paper we address the case k = 2. First, we give a motivation for our approach. Then, we address the problem of extracting a 2level planar subgraph of maximum weight in a given 2level graph. This problem is NPhard. Based on a characterization of 2level planar graphs, we give an integer linear programming formulation for the 2level planarization problem. Moreover, we define and investigate t...
On the Parameterized Complexity of Layered Graph Drawing
 PROC. 5TH ANNUAL EUROPEAN SYMP. ON ALGORITHMS (ESA '01
, 2001
"... We consider graph drawings in which vertices are assigned to layers and edges are drawn as straight linesegments between vertices on adjacent layers. We prove that graphs admitting crossingfree hlayer drawings (for fixed h) have bounded pathwidth. We then use a path decomposition as the basis for ..."
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Cited by 21 (9 self)
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We consider graph drawings in which vertices are assigned to layers and edges are drawn as straight linesegments between vertices on adjacent layers. We prove that graphs admitting crossingfree hlayer drawings (for fixed h) have bounded pathwidth. We then use a path decomposition as the basis for a lineartime algorithm to decide if a graph has a crossingfree hlayer drawing (for fixed h). This algorithm is extended to solve a large number of related problems, including allowing at most k crossings, or removing at most r edges to leave a crossingfree drawing (for fixed k or r). If the number of crossings or deleted edges is a nonfixed parameter then these problems are NPcomplete. For each setting, we can also permit downward drawings of directed graphs and drawings in which edges may span multiple layers, in which case the total span or the maximum span of edges can be minimized. In contrast to the socalled Sugiyama method for layered graph drawing, our algorithms do not assume a preassignment of the vertices to layers.
Towards Constructing Physical Maps by Optical Mapping: An Effective, Simple, Combinatorial Approach (Extended Abstract)
, 1997
"... ) S. Muthukrishnan Laxmi Parida y Abstract We initiate the complexity study of physical mapping with the emerging technology of Optical Mapping (OM) pioneered by the team lead by David Schwartz at the W. M. Keck Laboratory for Biomolecular Imaging, Dept of Chemistry, NYU. In currently popular e ..."
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Cited by 16 (6 self)
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) S. Muthukrishnan Laxmi Parida y Abstract We initiate the complexity study of physical mapping with the emerging technology of Optical Mapping (OM) pioneered by the team lead by David Schwartz at the W. M. Keck Laboratory for Biomolecular Imaging, Dept of Chemistry, NYU. In currently popular electrophoretic approaches, information about the relative ordering of the fragments comprising the DNA molecule is lost, thus leading to difficult computational problems of composing the fragments in to a physical map depicting their relative order. In contrast, the relative ordering of the pieces is readily obtained in OM. However, OM faces serious technological challenges as it has low resolution and is faultprone. We take a combinatorial approach to the problem of constructing physical maps from the erroneous data generated by OM. We identify two abstract problems in this context, namely, the Exclusive Binary FlipCut and Exclusive Weighted FlipCut problems. For both, we present polynom...
Multiniche crowding in genetic algorithms and its application to the assembly of DNA restrictionfragments
 Evolutionary Computation
, 1995
"... The determination of the sequence of all nucleotide basepairs in a DNA molecule, from restrictionfragment data, is a complex task and can be posed as the problem of finding the optima of a multimodal function. A genetic algorithm that uses multiniche crowding permits us to do this. Performance o ..."
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Cited by 14 (3 self)
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The determination of the sequence of all nucleotide basepairs in a DNA molecule, from restrictionfragment data, is a complex task and can be posed as the problem of finding the optima of a multimodal function. A genetic algorithm that uses multiniche crowding permits us to do this. Performance of this algorithm is first tested using a standard suite of test functions. The algorithm is next tested using two data sets obtained from the Human Genome Project at the Lawrence Livermore National Laboratory. The new method holds promise in automating the sequencing computations.
A FixedParameter Approach to TwoLayer Planarization
, 2002
"... A bipartite graph is biplanar if the vertices can be placed on two parallel lines (layers) in the plane such that there are no edge crossings when edges are drawn straight. The 2Layer Planarization problem asks if k edges can be deleted from a given graph G so that the remaining graph is biplanar ..."
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Cited by 12 (4 self)
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A bipartite graph is biplanar if the vertices can be placed on two parallel lines (layers) in the plane such that there are no edge crossings when edges are drawn straight. The 2Layer Planarization problem asks if k edges can be deleted from a given graph G so that the remaining graph is biplanar. This problem is NPcomplete, as is the 1Layer Planarization problem in which the permutation of the vertices in one layer is fixed. We give the following fixed parameter tractability results: an O(k ·6 k +G) algorithm for 2Layer Planarization and an O(3 k ·G) algorithm for 1Layer Planarization, thus achieving linear time for fixed k.
Representing Graphs by Disks and Balls (a survey of recognitioncomplexity results)
"... . Practical applications, like radio frequency assignments, led to the denition of disk intersection graphs in the plane, called shortly disk graphs. If the disks in the representation are not allowed to overlap, we speak about disk contact graphs (coin graphs). In this paper we survey recogniti ..."
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Cited by 12 (1 self)
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. Practical applications, like radio frequency assignments, led to the denition of disk intersection graphs in the plane, called shortly disk graphs. If the disks in the representation are not allowed to overlap, we speak about disk contact graphs (coin graphs). In this paper we survey recognitioncomplexity results for disk intersection and contact graphs in the plane. In particular, we refer a classical result by Koebe about disk contact representations, and works of Breu and Kirkpatrick about boundedratio disk representations. We prove that the recognition of diskintersection graphs (in the unbounded ratio case) is NPhard. This result is proved in a more general setting of noncrossing arcconnected sets. We also show some partial results concerning recognition of ball intersection and contact graphs in higher dimensions. In particular, we prove that the recognition of unitball contact graphs is NPhard in dimensions 3; 4, and 8 (24). 1 Introduction 1.1 Intersect...
An Efficient Fixed Parameter Tractable Algorithm for 1Sided Crossing Minimization
 ALGORITHMICA
, 2004
"... We give an O(ϕ k · n 2) fixed parameter tractable algorithm for the 1SIDED CROSSING MINIMIZATION problem. The constant ϕ in the running time is the golden ratio ϕ = (1 + √ 5)/2 ≈ 1.618. The constant k is the parameter of the problem: the number of allowed edge crossings. ..."
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Cited by 11 (4 self)
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We give an O(ϕ k · n 2) fixed parameter tractable algorithm for the 1SIDED CROSSING MINIMIZATION problem. The constant ϕ in the running time is the golden ratio ϕ = (1 + √ 5)/2 ≈ 1.618. The constant k is the parameter of the problem: the number of allowed edge crossings.
Interval graph limits
 In preparation
"... Abstract. We work out the graph limit theory for dense interval graphs. The theory developed departs from the usual description of a graph limit as a symmetric function W (x, y) on the unit square, with x and y uniform on the interval (0, 1). Instead, we fix a W and change the underlying distributio ..."
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Cited by 10 (4 self)
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Abstract. We work out the graph limit theory for dense interval graphs. The theory developed departs from the usual description of a graph limit as a symmetric function W (x, y) on the unit square, with x and y uniform on the interval (0, 1). Instead, we fix a W and change the underlying distribution of the coordinates x and y. We find choices such that our limits are continuous. Connections to random interval graphs are given, including some examples. We also show a continuity result for the chromatic number and clique number of interval graphs. Some results on uniqueness of the limit description are given for general graph limits. 1.