Results 1  10
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30
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 55 (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.
Approximating Treewidth, Pathwidth, Frontsize, and Shortest Elimination Tree
, 1995
"... Various parameters of graphs connected to sparse matrix factorization and other applications can be approximated using an algorithm of Leighton et al. that finds vertex separators of graphs. The approximate values of the parameters, which include minimum front size, treewidth, pathwidth, and minimum ..."
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Cited by 52 (4 self)
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Various parameters of graphs connected to sparse matrix factorization and other applications can be approximated using an algorithm of Leighton et al. that finds vertex separators of graphs. The approximate values of the parameters, which include minimum front size, treewidth, pathwidth, and minimum elimination tree height, are no more than O(logn) (minimum front size and treewidth) and O(log^2 n) (pathwidth and minimum elimination tree height) times the optimal values. In addition, we show that unless P = NP there are no absolute approximation algorithms for any of the parameters.
Efficient and Constructive Algorithms for the Pathwidth and Treewidth of Graphs
, 1993
"... In this paper we give, for all constants k, l, explicit algorithms, that given a graph G = (V; E) with a treedecomposition of G with treewidth at most l, decide whether the treewidth (or pathwidth) of G is at most k, and if so, find a treedecomposition or (pathdecomposition) of G of width at most ..."
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Cited by 49 (11 self)
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In this paper we give, for all constants k, l, explicit algorithms, that given a graph G = (V; E) with a treedecomposition of G with treewidth at most l, decide whether the treewidth (or pathwidth) of G is at most k, and if so, find a treedecomposition or (pathdecomposition) of G of width at most k, and that use O(V) time. In contrast with previous solutions, our algorithms do not rely on nonconstructive reasoning, and are single exponential in k and l. This result can be combined with a result of Reed [37], yielding explicit O(n log n) algorithms for the problem, given a graph G, to determine whether the treewidth (or pathwidth) of G is at most k, and if so, to find a tree (or path)decomposition of width at most k (k constant). Also, Bodlaender [13] has used the result of this paper to obtain linear time algorithms for these problems. We also show that for all constants k, there exists a polynomial time algorithm, that, when given a graph G = (V; E) with treewidth k, computes the pathwidth of G and a minimum path decomposition of G.
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 40 (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
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 29 (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.
Computing Optimal Linear Layouts of Trees in Linear Time
 Proc. ESA 2000, number 1879
, 1999
"... We present a linear time algorithm which, given a tree, computes a linear layout optimal with respect to vertex separation. As a consequence optimal edge search strategies, optimal node search strategies, and optimal interval augmentations can be computed also in O(n) for trees. This improves the ru ..."
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Cited by 14 (0 self)
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We present a linear time algorithm which, given a tree, computes a linear layout optimal with respect to vertex separation. As a consequence optimal edge search strategies, optimal node search strategies, and optimal interval augmentations can be computed also in O(n) for trees. This improves the running time of former algorithms from O(n log n) to O(n) and answers two related open questions raised in [7] and [15].
Memory Requirements for Table Computations in Partial kTree Algorithms
"... This paper addresses memory requirement issues arising in implementations of algorithms on graphs of bounded treewidth. Such dynamic programming algorithms require a large data table for each vertex of a treedecomposition T of the input graph. We give a lineartime algorithm that finds the trav ..."
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Cited by 13 (2 self)
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This paper addresses memory requirement issues arising in implementations of algorithms on graphs of bounded treewidth. Such dynamic programming algorithms require a large data table for each vertex of a treedecomposition T of the input graph. We give a lineartime algorithm that finds the traversal order of T minimizing the number of tables stored simultaneously. We show that this minimum value is lowerbounded by the pathwidth of T plus one, and upper bounded by twice the pathwidth of T plus one. We also give a lineartime algorithm finding the depthfirst traversal order minimizing the sum of the sizes of tables stored simultaneously.
Parameterized Algorithms for Directed Maximum Leaf Problems
 Proc. ICALP 2007, LNCS 4596
, 2007
"... Abstract. We prove that finding a rooted subtree with at least k leaves in a digraph is a fixed parameter tractable problem. A similar result holds for finding rooted spanning trees with many leaves in digraphs from a wide family L that includes all strong and acyclic digraphs. This settles complete ..."
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Cited by 12 (7 self)
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Abstract. We prove that finding a rooted subtree with at least k leaves in a digraph is a fixed parameter tractable problem. A similar result holds for finding rooted spanning trees with many leaves in digraphs from a wide family L that includes all strong and acyclic digraphs. This settles completely an open question of Fellows and solves another one for digraphs in L. Our algorithms are based on the following combinatorial result which can be viewed as a generalization of many results for a ‘spanning tree with many leaves ’ in the undirected case, and which is interesting on its own: If a digraph D ∈ L of order n with minimum indegree at least 3 contains a rooted spanning tree, then D contains one with at least (n/2) 1/5 − 1 leaves. 1
Algorithms and Obstructions for LinearWidth and Related Search Parameters
 Discrete Applied Mathematics
, 1997
"... The linearwidth of a graph G is defined to be the smallest integer k such that the edges of G can be arranged in a linear ordering (e 1 ; : : : ; e r ) in such a way that for every i = 1; : : : ; r \Gamma 1, there are at most k vertices incident to edges that belong both to fe 1 ; : : : ; e i g an ..."
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Cited by 11 (5 self)
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The linearwidth of a graph G is defined to be the smallest integer k such that the edges of G can be arranged in a linear ordering (e 1 ; : : : ; e r ) in such a way that for every i = 1; : : : ; r \Gamma 1, there are at most k vertices incident to edges that belong both to fe 1 ; : : : ; e i g and to fe i+1 ; : : : ; e r g. In this paper, we give a set of 57 graphs and prove that it is the set of the minimal forbidden minors for the class of graphs with linearwidth at most two. Our proof also gives a linear time algorithm that either reports that a given graph has linearwidth more than two or outputs an edge ordering of minimum linearwidth. We further prove a structural connection between linearwidth and the mixed search number which enables us to determine, for any k 1, the set acyclic forbidden minors for the class of graphs with linearwidth k. Moreover, due to this connection, our algorithm can be transfered to two linear time algorithms that check whether a graph has mixe...