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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.
Tractability of Parameterized Completion Problems on Chordal, Strongly Chordal and Proper Interval Graphs
, 1994
"... We study the parameterized complexity of three NPhard graph completion problems. The MINIMUM FILLIN problem is to decide if a graph can be triangulated by adding at most k edges. We develop O(c m) and O(k mn + f(k)) algorithms for this problem on a graph with n vertices and m edges. Here f(k ..."
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Cited by 57 (5 self)
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We study the parameterized complexity of three NPhard graph completion problems. The MINIMUM FILLIN problem is to decide if a graph can be triangulated by adding at most k edges. We develop O(c m) and O(k mn + f(k)) algorithms for this problem on a graph with n vertices and m edges. Here f(k) is exponential in k and the constants hidden by the bigO notation are small and do not depend on k. In particular, this implies that the problem is fixedparameter tractable (FPT). The PROPER
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 46 (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.
Pathwidth, Bandwidth and Completion Problems to Proper Interval Graphs with Small Cliques
 SIAM Journal on Computing
, 1996
"... We study two related problems motivated by molecular biology: ffl Given a graph G and a constant k, does there exist a supergraph G of G which is a unit interval graph and has clique size at most k? ffl Given a graph G and a proper kcoloring c of G, does there exist a supergraph We show th ..."
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Cited by 32 (6 self)
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We study two related problems motivated by molecular biology: ffl Given a graph G and a constant k, does there exist a supergraph G of G which is a unit interval graph and has clique size at most k? ffl Given a graph G and a proper kcoloring c of G, does there exist a supergraph We show that those problems are polynomial for fixed k. On the other hand we prove that the first problem is equivalent to deciding if the bandwidth of G is at most k \Gamma 1. Hence, it is NPhard, and W [t]hard for all t. We also show that the second problem is W [1]hard. This implies that for fixed k, both of the problems are unlikely to have an O(n ) algorithm, where ff is a constant independent of k.
Algorithms for Optical Mapping
 In Proceedings of the 2nd Annual International Conference on Computational Molecular Biology
, 1998
"... Optical mapping is a novel technique for determining the restriction sites on a DNA molecule by directly observing a number of partially digested copies of the molecule under a light microscope. The problem is complicated by uncertainty as to the orientation of the molecules and by erroneous detecti ..."
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Cited by 9 (1 self)
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Optical mapping is a novel technique for determining the restriction sites on a DNA molecule by directly observing a number of partially digested copies of the molecule under a light microscope. The problem is complicated by uncertainty as to the orientation of the molecules and by erroneous detection of cuts. In this paper we study the problem of constructing a restriction map based on optical mapping data. We give several variants of a polynomial reconstruction algorithm, as well as an algorithm that is exponential in the number of cut sites, and hence is appropriate only for small number of cut sites. We give a simple probabilistic model for data generation and for the errors and prove probabilistic upper and lower bounds on the number of molecules needed by each algorithm in order to obtain a correct map, expressed as a function of the number of cut sites and the error parameters. To the best of our knowledge, this is the first probabilistic analysis of algorithms for the problem. ...
Bounded Degree Interval Sandwich Problems
 Algorithmica
, 1999
"... The problems of Interval Sandwich (IS) and Intervalizing Colored Graphs (ICG) have received a lot of attention recently, due to their applicability to DNA physical mapping problems with ambiguous data. Most of the results obtained so far on the problems were hardness results. Here we study the probl ..."
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Cited by 7 (1 self)
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The problems of Interval Sandwich (IS) and Intervalizing Colored Graphs (ICG) have received a lot of attention recently, due to their applicability to DNA physical mapping problems with ambiguous data. Most of the results obtained so far on the problems were hardness results. Here we study the problems under assumptions of sparseness, which hold in the biological context. We prove that both problems are polynomial when either (1) the input graph degree and the solution graph clique size are bounded, or (2) the solution graph degree is bounded. In particular, this implies that ICG is polynomial on bounded degree graphs for every fixed number of colors, in contrast with the recent result of Bodlaender and de Fluiter.
Refining Restriction Enzyme Genome Maps
 Constraints, V
, 1997
"... A genome map is an ordering of a set of clones according to their believed position on a DNA string. Simple heuristics for genome map assembly based on single restriction enzyme with complete digestion data can lead to inaccuracies and ambiguities. This paper presents a method that adds additional c ..."
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Cited by 6 (1 self)
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A genome map is an ordering of a set of clones according to their believed position on a DNA string. Simple heuristics for genome map assembly based on single restriction enzyme with complete digestion data can lead to inaccuracies and ambiguities. This paper presents a method that adds additional constraint checking to the assembly process. An automaton is presented that for any genome map produces a refined genome map where both the clones and the restriction fragments in each clone are ordered satisfying natural constraints called step constraints. Any genome map that cannot be refined is highly likely to be inaccurate and can be eliminated as a possibility. 1 Introduction Deoxyribonucleic acid or DNA, the genetic material that encodes the blueprint of any living organism, is composed of a string of nucleotides that are adenine (A), thymine (T), cytosine (C), and guanine (G). Clones are copies of random substrings of a given DNA string. Clones may overlap in a clone database. Restr...
Genomic Database Applications in DISCO
"... . A genome map is an ordering of a set of clones according to their believed position on a DNA string. Simple heuristics for genome map assembly based on single restriction enzyme with complete digestion data can lead to inaccuracies and ambiguities. This paper presents a method that adds additional ..."
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. A genome map is an ordering of a set of clones according to their believed position on a DNA string. Simple heuristics for genome map assembly based on single restriction enzyme with complete digestion data can lead to inaccuracies and ambiguities. This paper presents a method that adds additional constraint checking to the assembly process. An automaton is presented that for any genome map produces a refined genome map where both the clones and the restriction fragments in each clone are ordered satisfying natural constraints called step constraints. Any genome map that cannot be refined is highly likely to be inaccurate and can be eliminated as a possibility. 1 Introduction Deoxyribonucleic acid or DNA, the genetic material that encodes the blueprint of any living organism, is composed of a string of nucleotides that are adenine (A), thymine (T), cytosine (C), and guanine (G). Clones are copies of random substrings of a given DNA string. Clones may overlap in a clone database. Res...