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59
Additive approximation for edgedeletion problems
 Proc. of FOCS 2005
, 2005
"... A graph property is monotone if it is closed under removal of vertices and edges. In this paper we consider the following algorithmic problem, called the edgedeletion problem; given a monotone property P and a graph G, compute the smallest number of edge deletions that are needed in order to turn G ..."
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A graph property is monotone if it is closed under removal of vertices and edges. In this paper we consider the following algorithmic problem, called the edgedeletion problem; given a monotone property P and a graph G, compute the smallest number of edge deletions that are needed in order to turn G into a graph satisfying P. We denote this quantity by E ′ P (G). The first result of this paper states that the edgedeletion problem can be efficiently approximated for any monotone property. • For any fixed ɛ> 0 and any monotone property P, there is a deterministic algorithm, which given a graph G = (V, E) of size n, approximates E ′ P (G) in linear time O(V  + E) to within an additive error of ɛn2. Given the above, a natural question is for which monotone properties one can obtain better additive approximations of E ′ P. Our second main result essentially resolves this problem by giving a precise characterization of the monotone graph properties for which such approximations exist. 1. If there is a bipartite graph that does not satisfy P, then there is a δ> 0 for which it is
Interval completion with few edges
 In STOC’07—Proceedings of the 39th Annual ACM Symposium on Theory of Computing
, 2007
"... We present an algorithm with runtime O(k 2k n 3 m) for the following NPcomplete problem [8, problem GT35]: Given an arbitrary graph G on n vertices and m edges, can we obtain an interval graph by adding at most k new edges to G? This resolves the longstanding open question [17, 6, 24, 13], first p ..."
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We present an algorithm with runtime O(k 2k n 3 m) for the following NPcomplete problem [8, problem GT35]: Given an arbitrary graph G on n vertices and m edges, can we obtain an interval graph by adding at most k new edges to G? This resolves the longstanding open question [17, 6, 24, 13], first posed by Kaplan, Shamir and Tarjan, of whether this problem could be solved in time f(k) · n O(1). The problem has applications in Physical Mapping of DNA [11] and in Profile Minimization for Sparse Matrix Computations [9, 25]. For the first application, our results show tractability for the case of a small number k of false negative errors, and for the second, a small number k of zero elements in the envelope. Our algorithm performs bounded search among possible ways of adding edges to a graph to obtain an interval graph, and combines this with a greedy algorithm when graphs of a certain structure are reached by the search. The presented result is surprising, as it was not believed that a bounded search tree algorithm would suffice to answer the open question affirmatively.
Homomorphisms in graph property testing  A survey
 Electronic Colloquium on Computational Complexity (ECCC), Report
, 2005
"... on the occasion of his 60 th birthday ..."
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Parameterized Complexity Analysis in Computational Biology
 Comput. Appl. Biosci
, 1995
"... Many computational problems in biology involve parameters for which a small range of values cover important applications. We argue that for many problems in this setting, parameterized computational complexity rather than NPcompleteness is the appropriate tool for studying apparent intractability. ..."
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Cited by 9 (4 self)
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Many computational problems in biology involve parameters for which a small range of values cover important applications. We argue that for many problems in this setting, parameterized computational complexity rather than NPcompleteness is the appropriate tool for studying apparent intractability. At issue in the theory of parameterized complexity is whether a problem can be solved in time O(n ff ) for each fixed parameter value, where ff is a constant independent of the parameter. In addition to surveying this complexity framework, we describe a new result for the Longest common subsequence problem. In particular, we show that the problem is hard for W [t] for all t when parameterized by the number of strings and the size of the alphabet. Lower bounds on the complexity of this basic combinatorial problem imply lower bounds on more general sequence alignment and consensus discovery problems. We also describe a number of open problems pertaining to the parameterized complexity of pro...
Minimizing Phylogenetic Number to find Good Evolutionary Trees
, 1996
"... Inferring phylogenetic trees is a fundamental problem in computational biology. We present a new objective criterion, the phylogenetic number, for evaluating evolutionary trees for species defined by biomolecular sequences or other qualitative characters. The phylogenetic number of a tree T is the m ..."
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Cited by 9 (2 self)
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Inferring phylogenetic trees is a fundamental problem in computational biology. We present a new objective criterion, the phylogenetic number, for evaluating evolutionary trees for species defined by biomolecular sequences or other qualitative characters. The phylogenetic number of a tree T is the maximum number of times that any given character state arises in T . By contrast, the classical parsimony criterion measures the total number of times that different character states arise in T . We consider the following related problems: finding the tree with minimum phylogenetic number, and computing the phylogenetic number of a given topology in which only the leaves are labeled by species. When the number of states is bounded (as is the case for biomolecular sequence characters), we can solve the second problem in polynomial time. Given the topology for an evolutionary tree, we can also compute a phylogeny with phylogenetic number 2 (when one exists) for an arbitrary number of states. Th...
On Physical Mapping and the Consecutive Ones Property for Sparse Matrices
 Discrete Appl. Math
, 1996
"... this paper we give a simplified model for Physical Mapping with probes that tend to occur very rarely along the DNA and show that the problem is NPcomplete even for sparse matrices. Moreover, we show that Physical Mapping with chimeric clones (a clone is chimeric if it stems from a concatenatio ..."
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this paper we give a simplified model for Physical Mapping with probes that tend to occur very rarely along the DNA and show that the problem is NPcomplete even for sparse matrices. Moreover, we show that Physical Mapping with chimeric clones (a clone is chimeric if it stems from a concatenation of several fragments of the DNA) is NPcomplete even for sparse matrices. Both problems are modeled as variants of the Consecutive Ones Problem which makes our results interesting for other application areas. 1 Supported by the Applied Mathematical Sciences program, U.S. Dept. of Energy, Office of Energy Research, and the work was performed at Sandia National Labs, operated for the U.S. DOE under contract No. DEAC0476DP00789. Preprint submitted to Elsevier Preprint 19 January 1996 1 Introduction In order to study a long DNA molecule it is necessary to break several copies of the molecule into smaller fragments. For further investigation copies
Minimal Conflicting Sets for the Consecutive Ones Property in ancestral genome reconstruction
, 912
"... Abstract. A binary matrix has the Consecutive Ones Property (C1P) if its columns can be ordered in such a way that all 1’s on each row are consecutive. A Minimal Conflicting Set is a set of rows that does not have the C1P, but every proper subset has the C1P. Such submatrices have been considered in ..."
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Cited by 8 (4 self)
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Abstract. A binary matrix has the Consecutive Ones Property (C1P) if its columns can be ordered in such a way that all 1’s on each row are consecutive. A Minimal Conflicting Set is a set of rows that does not have the C1P, but every proper subset has the C1P. Such submatrices have been considered in comparative genomics applications, but very little is known about their combinatorial structure and efficient algorithms to compute them. We first describe an algorithm that detects rows that belong to Minimal Conflicting Sets. This algorithm has a polynomial time complexity when the number of 1s in each row of the considered matrix is bounded by a constant. Next, we show that the problem of computing all Minimal Conflicting Sets can be reduced to the joint generation of all minimal true clauses and maximal false clauses for some monotone boolean function. We use these methods on simulated data related to ancestral genome reconstruction to show that computing Minimal Conflicting Set is useful in discriminating between true positive and false positive ancestral syntenies. We also study a dataset of yeast genomes and address the reliability of an ancestral genome proposal of the Saccahromycetaceae yeasts. Draft, do not distribute. Version of December 21, 2009. 1
Algorithmic Aspects of the ConsecutiveOnes Property
, 2009
"... We survey the consecutiveones property of binary matrices. Herein, a binary matrix has the consecutiveones property (C1P) if there is a permutation of its columns that places the 1s consecutively in every row. We provide an overview over connections to graph theory, characterizations, recognition ..."
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We survey the consecutiveones property of binary matrices. Herein, a binary matrix has the consecutiveones property (C1P) if there is a permutation of its columns that places the 1s consecutively in every row. We provide an overview over connections to graph theory, characterizations, recognition algorithms, and applications such as integer linear programming and solving Set Cover.
Rounding to an integral program
 In Proceedings of the 4th International Workshop on Efficient and Experimental Algorithms (WEA’05
, 2005
"... Abstract. We present a general framework for approximating several NPhard problems that have two underlying properties in common. First, the problems we consider can be formulated as integer covering programs, possibly with additional side constraints. Second, the number of covering options is rest ..."
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Abstract. We present a general framework for approximating several NPhard problems that have two underlying properties in common. First, the problems we consider can be formulated as integer covering programs, possibly with additional side constraints. Second, the number of covering options is restricted in some sense, although this property may be well hidden. Our method is a natural extension of the threshold rounding technique. 1
Interval Completion is Fixed Parameter Tractable
 IN PROCEEDINGS OF THE 39TH ANNUAL ACM SYMPOSIUM ON THEORY OF COMPUTING, STOC 2007
, 2006
"... We give an algorithm with runtime O(k 2k n 3 m) for the NPcomplete problem [GT35 in 6] of deciding whether a graph on n vertices and m edges can be turned into an interval graph by adding at most k edges. We thus prove that this problem is fixed parameter tractable (FPT), settling a longstanding o ..."
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Cited by 6 (2 self)
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We give an algorithm with runtime O(k 2k n 3 m) for the NPcomplete problem [GT35 in 6] of deciding whether a graph on n vertices and m edges can be turned into an interval graph by adding at most k edges. We thus prove that this problem is fixed parameter tractable (FPT), settling a longstanding open problem [13, 5, 19, 11]. The problem has applications in Physical Mapping of DNA [9] and in Profile Minimization for Sparse Matrix Computations [7, 20]. For the first application, our results show tractability for the case of a small number k of false negative errors, and for the second, a small number k of zero elements in the envelope.