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Kernels in planar digraphs
 In Optimization Online. Mathematical Programming Society
, 2001
"... A set S of vertices in a digraph D = (V, A) is a kernel if S is independent and every vertex in V − S has an outneighbor in S. We show that there exist O(n2 19.1 √ k + n 4)time and O(2 19.1 √ k k 9 + n 2)time algorithms for checking whether a planar digraph D of order n has a kernel with at most k ..."
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Cited by 16 (1 self)
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A set S of vertices in a digraph D = (V, A) is a kernel if S is independent and every vertex in V − S has an outneighbor in S. We show that there exist O(n2 19.1 √ k + n 4)time and O(2 19.1 √ k k 9 + n 2)time algorithms for checking whether a planar digraph D of order n has a kernel with at most k vertices. Moreover, if D has a kernel of size at most k, the algorithms find such a kernel of minimal size. 1
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 Experiments: The New (and Old) Methodology
 J. Univ. Comput. Sci
, 2001
"... The last twenty years have seen enormous progress in the design of algorithms, but little of it has been put into practice. Because many recently developed algorithms are hard to characterize theoretically and have large runningtime coefficients, the gap between theory and practice has widened over ..."
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Cited by 9 (4 self)
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The last twenty years have seen enormous progress in the design of algorithms, but little of it has been put into practice. Because many recently developed algorithms are hard to characterize theoretically and have large runningtime coefficients, the gap between theory and practice has widened over these years. Experimentation is indispensable in the assessment of heuristics for hard problems, in the characterization of asymptotic behavior of complex algorithms, and in the comparison of competing designs for tractable problems. Implementation, although perhaps not rigorous experimentation, was characteristic of early work in algorithms and data structures. Donald Knuth has throughout insisted on testing every algorithm and conducting analyses that can predict behavior on actual data; more recently, Jon Bentley has vividly illustrated the difficulty of implementation and the value of testing. Numerical analysts have long understood the need for standardized test suites to ensure robustness, precision and efficiency of numerical libraries. It is only recently, however, that the algorithms community has shown signs of returning to implementation and testing as an integral part of algorithm development. The emerging disciplines of experimental algorithmics and algorithm engineering have revived and are extending many of the approaches used by computing pioneers such as Floyd and Knuth and are placing on a formal basis many of Bentley's observations. We reflect on these issues, looking back at the last thirty years of algorithm development and forward to new challenges: designing cacheaware algorithms, algorithms for mixed models of computation, algorithms for external memory, and algorithms for scientific research.
Constructing optimal trees from quartets
 Journal of Algorithms
, 2001
"... We present fast new algorithms for constructing phylogenetic trees from quartets Ž resolved trees on four leaves.. The problem is central to divideandconquer approaches to phylogenetic analysis and has been receiving considerable attention from the computational biology community. Most formulation ..."
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Cited by 7 (1 self)
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We present fast new algorithms for constructing phylogenetic trees from quartets Ž resolved trees on four leaves.. The problem is central to divideandconquer approaches to phylogenetic analysis and has been receiving considerable attention from the computational biology community. Most formulations of the problem are NPhard. Here we consider a number of constrained versions that have polynomial time solutions. The main result is an algorithm for determining bounded degree trees with optimal quartet weight, subject to the constraint that the splits in the tree come from a given collection, for example, the splits in the aligned sequence data. The algorithm can search an exponentially large number of phylogenetic trees in polynomial time. We present applications of this algorithm to a number of problems in phylogenetics, including sequence analysis, construction of trees from phylogenetic networks, and consensus methods. � 2001 Academic Press Key Words: quartets; phylogenetic trees; algorithms; consensus; networks. 1.
Spanning directed trees with many leaves
 SIAM J. Discrete Math
"... Abstract. The Directed Maximum Leaf OutBranching problem is to find an outbranching (i.e. a rooted oriented spanning tree) in a given digraph with the maximum number of leaves. In this paper, we obtain two combinatorial results on the number of leaves in outbranchings. We show that – every strong ..."
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Cited by 6 (4 self)
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Abstract. The Directed Maximum Leaf OutBranching problem is to find an outbranching (i.e. a rooted oriented spanning tree) in a given digraph with the maximum number of leaves. In this paper, we obtain two combinatorial results on the number of leaves in outbranchings. We show that – every strongly connected nvertex digraph D with minimum indegree at least 3 has an outbranching with at least (n/4) 1/3 − 1 leaves; – if a strongly connected digraph D does not contain an outbranching with k leaves, then the pathwidth of its underlying graph UG(D) is O(k log k). Moreover, if the digraph is acyclic, the pathwidth is at most 4k. The last result implies that it can be decided in time 2 O(k log2 k) · n O(1) whether a strongly connected digraph on n vertices has an outbranching with at least k leaves. On acyclic digraphs the running time of our algorithm is 2 O(k log k) · n O(1). 1
Using Nondeterminism to Design Efficient Deterministic Algorithms
 Algorithmica
, 2001
"... In this paper, we illustrate how nondeterminism can be used conveniently and e#ectively in designing e#cient deterministic algorithms. ..."
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Cited by 4 (4 self)
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In this paper, we illustrate how nondeterminism can be used conveniently and e#ectively in designing e#cient deterministic algorithms.
Parametrizing Above Guaranteed Values: MaxSat and MaxCut
, 1997
"... In this paper we investigate the parametrized complexity of the problems MaxSat and MaxCut using the framework developed by Downey and Fellows[7]. Let G be an arbitrary graph having n vertices and m edges, and let f be an arbitrary CNF formula with m clauses on n variables. We improve Cai and Chen's ..."
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Cited by 4 (0 self)
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In this paper we investigate the parametrized complexity of the problems MaxSat and MaxCut using the framework developed by Downey and Fellows[7]. Let G be an arbitrary graph having n vertices and m edges, and let f be an arbitrary CNF formula with m clauses on n variables. We improve Cai and Chen's O(2 2k m) time algorithm for determining if at least k clauses of of a cCNF formula f can be satisfied[4]; our algorithm runs in O(jf j +k 2 OE k ) time for arbitrary formulae and in O(m + kOE k ) time for cCNF formulae. We also give an algorithm for finding a cut of size at least k; our algorithm runs in O(m + n + k4 k ) time. Since it is known that G has a cut of size at least d m 2 e and that there exists an assignment to the variables of f that satisfies at least d m 2 e clauses of f , we argue that the standard parametrization of these problems is unsuitable. Nontrivial situations arise only for large parameter values, in which range the fixedparameter tractable a...
Parameterized Complexity Results for Exact Bayesian Network Structure Learning
"... Bayesian network structure learning is the notoriously difficult problem of discovering a Bayesian network that optimally represents a given set of training data. In this paper we study the computational worstcase complexity of exact Bayesian network structure learning under graph theoretic restric ..."
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Bayesian network structure learning is the notoriously difficult problem of discovering a Bayesian network that optimally represents a given set of training data. In this paper we study the computational worstcase complexity of exact Bayesian network structure learning under graph theoretic restrictions on the (directed) superstructure. The superstructure is an undirected graph that contains as subgraphs the skeletons of solution networks. We introduce the directed superstructure as a natural generalization of its undirected counterpart. Our results apply to several variants of scorebased Bayesian network structure learning where the score of a network decomposes into local scores of its nodes. Results: We show that exact Bayesian network structure learning can be carried out in nonuniform polynomial time if the superstructure has bounded treewidth, and in linear time if in addition the superstructure has bounded maximum degree. Furthermore, we show that if the directed superstructure is acyclic, then exact Bayesian network structure learning can be carried out in quadratic time. We complement these positive results with a number of hardness results. We show that both restrictions (treewidth and degree) are essential and cannot be dropped without loosing uniform polynomial time tractability (subject to a complexitytheoretic assumption). Similarly, exact Bayesian network structure learning remains NPhard for “almost acyclic ” directed superstructures. Furthermore, we show that the restrictions remain essential if we do not search for a globally optimal network but aim to improve a given network by means of at most k arc additions, arc deletions, or arc reversals (kneighborhood local search). 1.
On the MAX kVERTEX COVER problem Federico Della Croce, Vangelis Th. PaschosOn the max kvertex cover problem ∗
, 2013
"... Given a graph G(V,E) of order n and a constant k � n, the max kvertex cover problem consists of determining k vertices that cover the maximum number of edges in G. In its (standard) parameterized version, max kvertex cover can be stated as follows: “given G, k and parameter ℓ, does G contain k ver ..."
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Given a graph G(V,E) of order n and a constant k � n, the max kvertex cover problem consists of determining k vertices that cover the maximum number of edges in G. In its (standard) parameterized version, max kvertex cover can be stated as follows: “given G, k and parameter ℓ, does G contain k vertices that cover at least ℓ edges?”. We first devise moderately exponential exact algorithms for max kvertex cover, with complexity exponential to n (note that the known results concerned time bounds of the form n O(k) ) by developing a branch and reduce method based upon the measureandconquer technique. We then prove that, interestingly enough, although max kvertex cover is non fixed parameter tractable with respect to ℓ, it is fixed parameter tractable with respect to the size τ of a minimum vertex cover of G. We also point out that the same happens for a lot of wellknown problems quite different from max kvertex cover. We finally study approximation of max kvertex cover by moderately exponential algorithms. The general goal of the issue of moderately exponential approximation is to catchup on polynomial inapproximability, by providing algorithms achieving, with worstcase running times importantly smaller than those needed for exact computation, approximation ratios unachievable in polynomial time. 1