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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
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.
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.
Mapping ancestral genomes with massive gene loss: A matrix sandwich problem
 BIOINFORMATICS VOL. 27 ISMB 2011, PAGES I257–I265
, 2011
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Variants of the ConsecutiveOnes Property Motivated by the Reconstruction of Ancestral Species
, 2012
"... The polynomialtime decidable ConsecutiveOnes Property (C1P) of binary matrices, formally introduced in 1965 by Fulkerson and Gross [52], has since found applications in many areas. In this thesis, we propose and study several variants of this property that are motivated by the reconstruction of an ..."
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The polynomialtime decidable ConsecutiveOnes Property (C1P) of binary matrices, formally introduced in 1965 by Fulkerson and Gross [52], has since found applications in many areas. In this thesis, we propose and study several variants of this property that are motivated by the reconstruction of ancestral species. We first propose the Gapped C1P, or the (k,δ)ConsecutiveOnes Property ((k,δ)C1P): a binary matrix M has the (k,δ)C1P for integers k and δ if the columns of M can be permuted such that each row contains at most k blocks of1’s and no two neighboring blocks of 1’s are separated by a gap of more than δ 0’s. The C1P is equivalent to the (1,0)C1P. We show that for every bounded and unbounded k ≥ 2,δ ≥ 1,(k,δ)̸=(2,1), deciding the(k,δ)C1P is NPcomplete [55]. We also provide an algorithm for a relevant case of the (2,1)C1P. We then study the(k,δ)C1P with a bound d on the maximum number of1’s in any row (the maximum degree) of M. We show that the(d,k,δ)ConsecutiveOnes Property ((d,k,δ)C1P) is polynomialtime decidable when all three parameters are fixed constants. Since fixing d also fixes k(k≤d), the only case left to consider
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, 2013
"... 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 ..."
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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:
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