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32
PC trees and circularones arrangements
 Theoretical Computer Science
"... A 01 matrix has the consecutiveones property if its columns can be ordered so that the ones in every row are consecutive. It has the circularones property if its columns can be ordered so that, in every row, either the ones or the zeros are consecutive. PQ trees are used for representing all cons ..."
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Cited by 37 (4 self)
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A 01 matrix has the consecutiveones property if its columns can be ordered so that the ones in every row are consecutive. It has the circularones property if its columns can be ordered so that, in every row, either the ones or the zeros are consecutive. PQ trees are used for representing all consecutiveones orderings of the columns of a matrix that has the consecutiveones property. We give an analogous structure, called a PC tree, for representing all circularones orderings of the columns of a matrix that has the circularones property. No such representation has been given previously. In contrast to PQ trees, PC trees are unrooted. We obtain a much simpler algorithm for computing PQ trees that those that were previously available, by adding a zero column, x, to a matrix, computing the PC tree, and then picking the PC tree up by x to root it. 1
Interval bigraphs and circular arc graphs
 J. Graph Theory
"... Abstract We prove that the complements of interval bigraphs are precisely those circular arc graphs of clique covering number two which admit a representation without two arcs covering the whole circle. We give another characterization of interval bigraphs, in terms of a vertex ordering, that we hop ..."
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Cited by 24 (5 self)
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Abstract We prove that the complements of interval bigraphs are precisely those circular arc graphs of clique covering number two which admit a representation without two arcs covering the whole circle. We give another characterization of interval bigraphs, in terms of a vertex ordering, that we hope may prove helpful in finding a more efficient recognition algorithm than presently known. We use these results to show equality, amongst bipartite graphs, of several classes of structured graphs (proper interval bigraphs, complements of proper circular arc graphs, asteroidaltriplefree graphs, permutation graphs, and cocomparability graphs). Our results verify a conjecture of Lundgren and disprove a conjecture of M&quot;uller. 1 Background A graph H is an interval graph if it is the intersection graph of a family of intervals Iv, v 2 V (H). (Two vertices v; v 0 are adjacent in H if and only if Iv and Iv0 intersect.) If the
Fragments of Order
, 2003
"... Highdimensional collections of 01 data occur in many applications. The attributes in such data sets are typically considered to be unordered. However, in many cases there is a natural total or partial order # underlying the variables of the data set. Examples of variables for which such orders exi ..."
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Cited by 13 (2 self)
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Highdimensional collections of 01 data occur in many applications. The attributes in such data sets are typically considered to be unordered. However, in many cases there is a natural total or partial order # underlying the variables of the data set. Examples of variables for which such orders exist include terms in documents, courses in enrollment data, and paleontological sites in fossil data collections. The observations in such applications are flat, unordered sets; however, the data sets respect the underlying ordering of the variables. By this we mean that if A # B # C are three variables respecting the underlying ordering #, and both of variables A and C appear in an observation, then, up to noise levels, variable B also appears in this observation. Similarly, if A1 # A2 # # A l1 # A l is a longer sequence of variables, we do not expect to see many observations for which there are indices i < j < k such that A i and Ak occur in the observation but A j does not.
Approximation and FixedParameter Algorithms for Consecutive Ones Submatrix Problems
 JOURNAL OF COMPUTER AND SYSTEM SCIENCES
"... We develop an algorithmically useful refinement of a forbidden submatrix characterization of 0/1matrices fulfilling the Consecutive Ones Property (C1P). This characterization finds applications in new polynomialtime approximation algorithms and fixedparameter tractability results for the NPhard ..."
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Cited by 13 (0 self)
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We develop an algorithmically useful refinement of a forbidden submatrix characterization of 0/1matrices fulfilling the Consecutive Ones Property (C1P). This characterization finds applications in new polynomialtime approximation algorithms and fixedparameter tractability results for the NPhard problem to delete a minimum number of rows or columns from a 0/1matrix such that the remaining submatrix has the C1P.
A test for the consecutive ones property on noisy data – application to physical mapping and sequence assembly
 J. Comput. Biol
, 2003
"... 2 A (0,1)matrix satisfies the consecutive ones property (COP) for the rows if there exists a column permutation such that the ones in each row of the resulting matrix are consecutive. The consecutive ones test is useful for DNA sequence assembly, for example, in the STS content mapping of YAC libra ..."
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Cited by 11 (2 self)
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2 A (0,1)matrix satisfies the consecutive ones property (COP) for the rows if there exists a column permutation such that the ones in each row of the resulting matrix are consecutive. The consecutive ones test is useful for DNA sequence assembly, for example, in the STS content mapping of YAC library, and in the Bactig assembly based on STS as well as EST markers. The linear time algorithm by Booth and Lueker (1976) for this problem has a serious drawback: the data must be errorfree. However, laboratory work is never flawless. We devised a new iterative clustering algorithm for this problem, which has the following advantages: 1. If the original matrix satisfies the COP, then the algorithm will produce a column ordering realizing it without any fillin. 2. Under moderate assumptions, the algorithm can accommodate the following four types of errors: FNs, FPs and NPs and CCs. Note that in some cases (low quality EST marker identification), NPs occur because of repeat sequences. 3. In case some local data is too noisy, our algorithm could likely discover that and suggest additional lab work that could reduce the degree of ambiguity in that part. 4. A unique feature of our algorithm is that, rather than forcing all probes to be included and ordered in the final arrangement, our algorithm would delete some probes. Thus, it could produce more than one contig. The gaps are created mostly by noisy columns. In summary, we have modified previous rigid algorithms for testing consecutive ones property into one that can accommodate clustering techniques, and produces satisfactory approximate probe orderings for most data. 3 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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Cited by 11 (1 self)
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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.
Minimal Conflicting Sets for the Consecutive Ones Property in ancestral genome reconstruction
, 2009
"... 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 compara ..."
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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.
A Synthesis on Partition Refinement: a useful Routine for Strings, Graphs, Boolean Matrices and Automata
 In Proc. Fifteenth STACS
, 1998
"... Partition refinement techniques are used in many algorithms. This tool allows efficient computation of equivalence relations and is somehow dual to unionfind algorithms. The goal of this paper is to propose a single routine to quickly implement all these already known algorithms and to solve a ..."
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Cited by 9 (7 self)
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Partition refinement techniques are used in many algorithms. This tool allows efficient computation of equivalence relations and is somehow dual to unionfind algorithms. The goal of this paper is to propose a single routine to quickly implement all these already known algorithms and to solve a large class of potentially new problems. Our framework yields to a unique scheme for correctness proofs and complexity analysis. Various examples are presented to show the different ways of using this routine. 1 Introduction A partition of a finite set E is a collection of disjoint subsets of E called classes whose union is E. Refining a partition consists in splitting its classes into smaller classes. Partition refinement techniques have been studied in four main papers [7, 15, 13, 6]. Hopcroft [7] may be the very first designer of such a technique. He used it in order to minimize the number of states of a deterministic finite automaton. Spinrad [15] investigated the graph partitioning ...
Approximability and parameterized complexity of consecutive ones submatrix problems
 IN PROC. 4TH TAMC, VOLUME 4484 OF LNCS
, 2007
"... We develop a refinement of a forbidden submatrix characterization of 0/1matrices fulfilling the Consecutive Ones Property (C1P). This novel characterization finds applications in new polynomialtime approximation algorithms and fixedparameter tractability results for the problem to find a maximum ..."
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Cited by 8 (4 self)
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We develop a refinement of a forbidden submatrix characterization of 0/1matrices fulfilling the Consecutive Ones Property (C1P). This novel characterization finds applications in new polynomialtime approximation algorithms and fixedparameter tractability results for the problem to find a maximumsize submatrix of a 0/1matrix such that the submatrix has the C1P. Moreover, we achieve a problem kernelization based on simple data reduction rules and provide several search tree algorithms. Finally, we derive inapproximability results.
On Planar Supports for Hypergraphs
"... A graph G is a support for a hypergraph H = (V, S) if the vertices of G correspond to the vertices of H such that for each hyperedge Si ∈ S the subgraph of G induced by Si is connected. G is a planar support if it is a support and planar. Johnson and Pollak [9] proved that it is NPcomplete to decide ..."
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Cited by 6 (0 self)
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A graph G is a support for a hypergraph H = (V, S) if the vertices of G correspond to the vertices of H such that for each hyperedge Si ∈ S the subgraph of G induced by Si is connected. G is a planar support if it is a support and planar. Johnson and Pollak [9] proved that it is NPcomplete to decide if a given hypergraph has a planar support. In contrast, there are polynomial time algorithms to test whether a given hypergraph has a planar support that is a path, cycle, or tree. In this paper we present an algorithm which tests in polynomial time if a given hypergraph has a planar support that is a tree where the maximal degree of each vertex is bounded. Our algorithm is constructive and computes a support if it exists. Furthermore, we prove that it is already NPhard to decide if a hypergraph has a 2outerplanar support.