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Measuring Indifference: Unit Interval Vertex Deletion
 In Proc. WG 2010, LNCS 6410:232–243
"... Abstract. Making a graph unit interval by a minimum number of vertex deletions is NPhard. The problem is motivated by applications in seriation and measuring indifference between data items. We present a fixedparameter algorithm based on the iterative compression technique that finds in O((14k + 14 ..."
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Abstract. Making a graph unit interval by a minimum number of vertex deletions is NPhard. The problem is motivated by applications in seriation and measuring indifference between data items. We present a fixedparameter algorithm based on the iterative compression technique that finds in O((14k + 14) k+1 kn 6) time a set of k vertices whose deletion from an nvertex graph makes it unit interval. Additionally, we show that making a graph chordal by at most k vertex deletions is NPcomplete even on {claw,net,tent}free graphs. 1
TwoLayer Planarization Parameterized by Feedback Edge Set
"... Given an undirected graph G and an integer k ≥ 0, the NPhard 2Layer Planarization problem asks whether G can be transformed into a forest of caterpillar trees by removing at most k edges. Since transforming G into a forest of caterpillar trees requires breaking every cycle, the size f of a minimum ..."
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Given an undirected graph G and an integer k ≥ 0, the NPhard 2Layer Planarization problem asks whether G can be transformed into a forest of caterpillar trees by removing at most k edges. Since transforming G into a forest of caterpillar trees requires breaking every cycle, the size f of a minimum feedback edge set is a natural parameter with f ≤ k. We improve on previous fixedparameter tractability results with respect to k by presenting a problem kernel with O(f) vertices and edges and a new searchtree based algorithm, both with about the same worstcase bounds for f as the previous results for k, although we expect f to be smaller than k for a wide range of input instances.
A faster algorithm for finding minimum Tucker submatrices
"... Abstract. A binary matrix has the Consecutive Ones Property (C1P) if its columns can be ordered in such a way that all 1s on each row are consecutive. Algorithmic issues of the C1P are central in computational molecular biology, in particular for physical mapping and ancestral genome reconstruction. ..."
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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 1s on each row are consecutive. Algorithmic issues of the C1P are central in computational molecular biology, in particular for physical mapping and ancestral genome reconstruction. In 1972, Tucker gave a characterization of matrices that have the C1P by a set of forbidden submatrices, and a substantial amount of research has been devoted to the problem of efficiently finding such a minimum size forbidden submatrix. This paper presents a new O( ∆ 3 m 2 (m ∆ + n 3)) time algorithm for this particular task for a m×n binary matrix with at most ∆ 1entries per row, thereby improving the O( ∆ 3 m 2 (mn + n 3)) time algorithm of Dom et al. [17]. 1
Mapping ancestral genomes with massive gene loss: A matrix sandwich problem
 BIOINFORMATICS VOL. 27 ISMB 2011, PAGES I257–I265
, 2011
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ON CERTIFICATES THAT A MATRIX DOES NOT HAVE THE CONSECUTIVE ONES PROPERTY
, 2011
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A TIGHT BOUND ON THE LENGTH OF ODD CYCLES IN THE INCOMPATIBILITY GRAPH OF A NONC1P MATRIX
, 1109
"... Abstract. A binary matrix has the consecutive ones property (C1P) if it is possible to order the columns so that all 1s are consecutive in every row. In [McConnell, SODA 2004 768777] the notion of incompatibility graph of a binary matrix was introduced and it was shown that odd cycles of this graph ..."
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Abstract. A binary matrix has the consecutive ones property (C1P) if it is possible to order the columns so that all 1s are consecutive in every row. In [McConnell, SODA 2004 768777] the notion of incompatibility graph of a binary matrix was introduced and it was shown that odd cycles of this graph provide a certificate that a matrix does not have the consecutive ones property. A bound of k + 2 was claimed for the smallest odd cycle of a nonC1P matrix with k columns. In this note we show that this result can be obtained simply and directly via Tucker patterns, and that the correct bound is k + 2 when k is even, but k + 3 when k is odd. 1.
A PolynomialTime Algorithm for Finding a Minimal Conflicting Set Containing a Given Row
, 2011
"... Abstract. A binary matrix has the Consecutive Ones Property (C1P) if there exists a permutation of its columns (i.e. a sequence of column swappings) such that in the resulting matrix the 1s are consecutive in every row. A Minimal Conflicting Set (MCS) of rows is a set of rows R that does not have th ..."
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Abstract. A binary matrix has the Consecutive Ones Property (C1P) if there exists a permutation of its columns (i.e. a sequence of column swappings) such that in the resulting matrix the 1s are consecutive in every row. A Minimal Conflicting Set (MCS) of rows is a set of rows R that does not have the C1P, but such that any proper subset of R has the C1P. In [5], Chauve et al. gave a O( ∆ 2 m max(4,∆+1) (n + m + e)) time algorithm to decide if a row of a m × n binary matrix with at most ∆ 1s per row belongs to at least one MCS of rows. Answering a question raised in [2], [5] and [25], we present the first polynomialtime algorithm to decide if a row of a m × n binary matrix belongs to at least one MCS of rows. 1
Variants of the Consecutive Ones property: Algorithms, Complexity and Applications in Genomics
"... The Consecutive Ones property (C1P) is a well studied structural property concerning binary matrices, with important applications in genome assembly. While the original decision problem of detecting C1P matrices is solvable in polynomial time, decision and optimization problems on generalizations of ..."
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The Consecutive Ones property (C1P) is a well studied structural property concerning binary matrices, with important applications in genome assembly. While the original decision problem of detecting C1P matrices is solvable in polynomial time, decision and optimization problems on generalizations of the property are usually intractable. These generalizations are extremely important concepts used as models in genomics, and the limits of tractability of both decision and optimization problems concerning them is a pivotal question. The following document examines generalizations of the C1P and tractability results for the same. These include fixedparameter tractability, optimization problems on these variants, and some proposed new approaches to the original C1P problem, which we hope may lead to interesting results. It also discusses the application of the algorithms developed to genomic data sets. Chapter 1
Title of thesis:
, 2013
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this work may be reproduced, without authorization, under the conditions for "Fair Dealing". Therefore, limited reproduction of this work for the purposes of private study, research, criticism, review, and news reporting is likely to be in accordance with the law, particularly if cited appropriately. APPROVAL Name: Degree: