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29
Two Strikes Against Perfect Phylogeny
 PROC. OF 19TH INTERNATIONAL COLLOQUIUM ON AUTOMATA LANGUAGES AND PROGRAMMING
, 1992
"... One of the major efforts in molecular biology is the computation of phylo genies for species sets. A longstanding open problem in this area is called the Perfect Phylogeny problem. For almost two decades the complexity of this problem remained open, with progress limited to polynomial time algor ..."
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Cited by 113 (27 self)
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One of the major efforts in molecular biology is the computation of phylo genies for species sets. A longstanding open problem in this area is called the Perfect Phylogeny problem. For almost two decades the complexity of this problem remained open, with progress limited to polynomial time algorithms for a few special cases, and many relaxations of the problem shown to be NPComplete. From an applications point of view, the problem is of interest both in its general forin, where the number of characters may vary, and in its fixedparameter form. The Perfect Phylogeny problem has been shown to be equivalent to the problem of triangulating colored graphs[30]. It has also been shown recently that for a given fixed number of characters the yesinstances have bounded treewidth[45], opening the possibility of applying methodologies for bounded treewidth to the fixedparameter form of the problem. We show that the Perfect Phylogeny problem is difficult in two different ways. We show
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.
Approximation Algorithms for Classes of Graphs Excluding SingleCrossing Graphs as Minors
"... Many problems that are intractable for general graphs allow polynomialtime solutions for structured classes of graphs, such as planar graphs and graphs of bounded treewidth. ..."
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Cited by 25 (16 self)
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Many problems that are intractable for general graphs allow polynomialtime solutions for structured classes of graphs, such as planar graphs and graphs of bounded treewidth.
The Hardness of Perfect Phylogeny, Feasible Register Assignment and Other Problems on Thin Colored Graphs
"... In this paper, we consider the complexity of a number of combinatorial problems; namely, Intervalizing Colored Graphs (DNA physical mapping), Triangulating Colored Graphs (perfect phylogeny), (Directed) (Modified) Colored Cutwidth, Feasible Register Assignment and Module Allocation for graphs of bou ..."
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Cited by 17 (4 self)
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In this paper, we consider the complexity of a number of combinatorial problems; namely, Intervalizing Colored Graphs (DNA physical mapping), Triangulating Colored Graphs (perfect phylogeny), (Directed) (Modified) Colored Cutwidth, Feasible Register Assignment and Module Allocation for graphs of bounded pathwidth. Each of these problems has as a characteristic a uniform upper bound on the tree or path width of the graphs in "yes"instances. For all of these problems with the exceptions of Feasible Register Assignment and Module Allocation, a vertex or edge coloring is given as part of the input. Our main results are that the parameterized variant of each of the considered problems is hard for the complexity classes W [t] for all t 2 N. We also show that Intervalizing Colored Graphs, Triangulating Colored Graphs, and Colored Cutwidth are NPComplete. 1 Introduction This paper focuses on a number of graph decision problems which share the characteristic that all have a uniform upper bo...
Exponential speedup of fixed parameter algorithms on K_3,3minorfree or K_5minorfree graphs
 in The 13th Anual International Symposium on Algorithms and Computation—ISAAC 2002
, 2002
"... We present a fixed parameter algorithm that constructively solves the kdominating set problem on graphs excluding one of the K5 or K3,3 as a minor in time O(3 6 √ 34k n O(1)). In fact, we present our algorithm for any Hminorfree graph where H is a singlecrossing graph (can be drawn on the plane ..."
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Cited by 14 (6 self)
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We present a fixed parameter algorithm that constructively solves the kdominating set problem on graphs excluding one of the K5 or K3,3 as a minor in time O(3 6 √ 34k n O(1)). In fact, we present our algorithm for any Hminorfree graph where H is a singlecrossing graph (can be drawn on the plane with at most one crossing) and obtain the algorithm for K3,3(K5)minorfree graphs as a special case. As a consequence, we extend our results to several other problems such as vertex cover, edge dominating set, independent set, cliquetransversal set, kernels in digraphs, feedback vertex set and a series of vertex removal problems. Our work generalizes and extends the recent result of exponential speedup in designing fixedparameter algorithms on planar graphs due to Alber et al. to other (nonplanar) classes of graphs.
Treewidth and Minimum Fillin on DTrapezoid Graphs
, 1998
"... We show that the minimum llin and the minimum interval graph completion of a dtrapezoid graph can be computed in time O(n d). We also show that the treewidth and the pathwidth of a dtrapezoid graph can be computed in time O(n tw(G)^{d1}). In both cases, d is supposed to be a fixed positive integ ..."
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Cited by 9 (3 self)
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We show that the minimum llin and the minimum interval graph completion of a dtrapezoid graph can be computed in time O(n d). We also show that the treewidth and the pathwidth of a dtrapezoid graph can be computed in time O(n tw(G)^{d1}). In both cases, d is supposed to be a fixed positive integer and it is required that a suitable intersection model of the given dtrapezoid graph is part of the input. As a consequence, each of the four graph parameters can be computed in time O(n^2) for trapezoid graphs and thus for permutation graphs even if no intersection model is part of the input.
Fast approximation schemes for K_{3,3}minorfree or K_5minorfree graphs
"... As the class of graphs of bounded treewidth is of limited size, we need to solve NPhard problems for wider classes of graphs than this class. Eppstein introduced a new concept which can be considered as a generalization of bounded treewidth. A graph G has locally bounded treewidth if for each verte ..."
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Cited by 5 (5 self)
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As the class of graphs of bounded treewidth is of limited size, we need to solve NPhard problems for wider classes of graphs than this class. Eppstein introduced a new concept which can be considered as a generalization of bounded treewidth. A graph G has locally bounded treewidth if for each vertex v of G, the treewidth of the subgraph of G induced on all vertices of distance at most r from v is only a function of r, called local treewidth. So far the only graphs determined to have small local treewidth are planar graphs. In this paper, we prove that any graph excluding one of K5 or K3;3 as a minor has local treewidth bounded by 3k + 4. As a result, we can design practical polynomialtime approximation schemes for both minimization and maximization problems on these classes of nonplanar graphs.
Exponential Speedup of FixedParameter Algorithms for Classes of Graphs Excluding SingleCrossing Graphs as Minors
, 2002
"... We present a fixedparameter algorithm that constructively solves the kdominating set problem on any class of graphs excluding a singlecrossing graph (a graph that can be drawn in the plane with at most one crossing) as a minor in O(4 9.55 √ k n O(1) ) time. Examples of such graph classes are the ..."
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Cited by 5 (4 self)
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We present a fixedparameter algorithm that constructively solves the kdominating set problem on any class of graphs excluding a singlecrossing graph (a graph that can be drawn in the plane with at most one crossing) as a minor in O(4 9.55 √ k n O(1) ) time. Examples of such graph classes are the K3,3minorfree graphs and the K5minorfree graphs. As a consequence, we extend our results to several other problems such as vertex cover, edge dominating set, independent set, cliquetransversal set, kernels in digraphs, feedback vertex set, and a collection of vertexremoval problems. Our work generalizes and extends the recent results of exponential speedup in designing fixedparameter algorithms on planar graphs due to Alber et al. to other (nonplanar) classes of graphs.