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87
New Upper Bounds for Maximum Satisfiability
 Journal of Algorithms
, 1999
"... The (unweighted) Maximum Satisfiability problem (MaxSat) is: given a boolean formula in conjunctive normal form, find a truth assignment that satisfies the most number of clauses. This paper describes exact algorithms that provide new upper bounds for MaxSat. We prove that MaxSat can be solved i ..."
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Cited by 38 (2 self)
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The (unweighted) Maximum Satisfiability problem (MaxSat) is: given a boolean formula in conjunctive normal form, find a truth assignment that satisfies the most number of clauses. This paper describes exact algorithms that provide new upper bounds for MaxSat. We prove that MaxSat can be solved in time O(F  1.3803 K ), where F  is the length of a formula F in conjunctive normal form and K is the number of clauses in F . We also prove the time bounds O(F 1.3995 k ), where k is the maximum number of satisfiable clauses, and O(1.1279 F  ) for the same problem. For Max2Sat this implies a bound of O(1.2722 K ). # An extended abstract of this paper was presented at the 26th International Colloquium on Automata, Languages, and Programming (ICALP'99), LNCS 1644, SpringerVerlag, pages 575584, held in Prague, Czech Republic, July 1115, 1999. + Supported by a Feodor Lynen fellowship (1998) of the Alexander von HumboldtStiftung, Bonn, and the Center for Discrete Ma...
Fixed parameter algorithms for planar dominating set and related problems
, 2000
"... We present an algorithm that constructively produces a solution to the kdominating set problem for planar graphs in time O(c √ kn), where c = 36√34. To obtain this result, we show that the treewidth of a planar graph with domination number γ(G) is O ( � γ(G)), and that such a tree decomposition ca ..."
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Cited by 35 (10 self)
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We present an algorithm that constructively produces a solution to the kdominating set problem for planar graphs in time O(c √ kn), where c = 36√34. To obtain this result, we show that the treewidth of a planar graph with domination number γ(G) is O ( � γ(G)), and that such a tree decomposition can be found in O ( � γ(G)n) time. The same technique can be used to show that the kface cover problem (find a size k set of faces that cover all vertices of a given plane graph) can be solved √ k in O(c1 n + n2) time, where c1 = 236√34 and k is the size of the face cover set. Similar results can be obtained in the planar case for some variants of kdominating set, e.g., kindependent dominating set and kweighted dominating set. Keywords. NPcomplete problems, fixed parameter tractability, planar graphs, planar dominating set, face cover, outerplanarity, treewidth.
FPT is PTime Extremal Structure I.
 In Proc. of Algorithms and Complexity in Durham (ACiD),
, 2005
"... ABSTRACT. We describe a broad program of research in parameterized complexity, and hows this plays out for the MAX LEAF SPANNING TREE problem. ..."
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Cited by 31 (2 self)
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ABSTRACT. We describe a broad program of research in parameterized complexity, and hows this plays out for the MAX LEAF SPANNING TREE problem.
Graph separators: a parameterized view
 Journal of Computer and System Sciences
, 2001
"... Graph separation is a wellknown tool to make (hard) graph problems accessible to a divide and conquer approach. We show how to use graph separator theorems in combination with (linear) problem kernels in order to develop xed parameter algorithms for many wellknown NPhard (planar) graph problems. ..."
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Cited by 29 (12 self)
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Graph separation is a wellknown tool to make (hard) graph problems accessible to a divide and conquer approach. We show how to use graph separator theorems in combination with (linear) problem kernels in order to develop xed parameter algorithms for many wellknown NPhard (planar) graph problems. We coin the key notion of glueable select&verify graph problems and derive from that a prospective way to easily check whether a planar graph problem will allow for a xed parameter algorithm of running time c p
Parameterized Complexity: The Main Ideas and Connections to Practical Computing
, 2002
"... The purposes of this paper are two: (1) to give an exposition of the main ideas of parameterized complexity, and (2) to discuss the connections of parameterized complexity to the systematic design of heuristics and approximation algorithms. ..."
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Cited by 25 (6 self)
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The purposes of this paper are two: (1) to give an exposition of the main ideas of parameterized complexity, and (2) to discuss the connections of parameterized complexity to the systematic design of heuristics and approximation algorithms.
Geometric Separation and Exact Solutions for the Parameterized Independent Set Problem on Disk Graphs
, 2002
"... We consider the parameterized problem, whether for a given set D of n disks (of bounded radius ratio) in the Euclidean plane there exists a set of k nonintersecting disks. We expose an algorithm running in time n , that isto our knowledgethe rst algorithm for this problem with running t ..."
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Cited by 24 (2 self)
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We consider the parameterized problem, whether for a given set D of n disks (of bounded radius ratio) in the Euclidean plane there exists a set of k nonintersecting disks. We expose an algorithm running in time n , that isto our knowledgethe rst algorithm for this problem with running time bounded by an exponential with a sublinear exponent. For precision disk graphs of bounded radius ratio, we show that the problem is xed parameter tractable with respect to parameter k.
Coordinatized kernels and catalytic reductions: An improved FPT algorithm for max leaf spanning tree and other problems
 In Proc. of 20th FSTTCS
, 2000
"... Abstract. We describe some new, simple and apparently general methods for designing FPT algorithms, and illustrate how these can be used to obtain a significantly improved FPT algorithm for the Maximum Leaf Spanning Tree problem. Furthermore, we sketch how the methods can be applied to a number of ..."
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Cited by 22 (1 self)
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Abstract. We describe some new, simple and apparently general methods for designing FPT algorithms, and illustrate how these can be used to obtain a significantly improved FPT algorithm for the Maximum Leaf Spanning Tree problem. Furthermore, we sketch how the methods can be applied to a number of other wellknown problems, including the parametric dual of Dominating Set (also known as Nonblocker), Matrix Domination, Edge Dominating Set, and Feedback Vertex Set for Undirected Graphs. The main payoffs of these new methods are in improved functions f(k) in the FPT running times, and in general systematic approaches that seem to apply to a wide variety of problems. 1
Solving Large FPT Problems On Coarse Grained Parallel Machines
"... Fixedparameter tractability(FPT) techniques have recently been successful in solving NPcomplete problem instances of practical importance which were too large to be solved with previous methods. In this paper we show how to enhance this approach through the addition of parallelism, thereby allowin ..."
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Cited by 22 (1 self)
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Fixedparameter tractability(FPT) techniques have recently been successful in solving NPcomplete problem instances of practical importance which were too large to be solved with previous methods. In this paper we show how to enhance this approach through the addition of parallelism, thereby allowing even larger problem instances to be solved in practice. More precisely, we demonstrate the potential of parallelism when applied to the bounded tree search phase of FPT algorithms. We apply our methodology to the kVertex Cover problem which has important applications, e.g., in multiple sequence alignments for computational biochemistry. We have implemented our parallel FPT method and application specific "plugin" code for the kVertex Cover problem using C and the MPI communication library, and tested it on a network of 10 Sun SPARC workstations. This is the first experimental examination of parallel FPT techniques. In our experiments, we obtain excellent speedup results. Not only do we achieve a speedup of p in most cases, many cases even exhibit a super linear speedup. The latter result implies that our parallel methods, when simulated on a single processor, also yield a significant improvement over existing sequential methods.
An Improved FixedParameterTractable Algorithm for Vertex Cover
, 1999
"... Given a graph G = (V; E), Vertex Cover asks for a smallest subset V 0 ` V such that for each edge (a; b) in G a 2 V 0 or b 2 V 0 . We present an improved fixedparameter tractable algorithm when the problem is parameterized by the size k of V 0 . The algorithm has a complexity of O(kn + max ..."
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Cited by 19 (5 self)
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Given a graph G = (V; E), Vertex Cover asks for a smallest subset V 0 ` V such that for each edge (a; b) in G a 2 V 0 or b 2 V 0 . We present an improved fixedparameter tractable algorithm when the problem is parameterized by the size k of V 0 . The algorithm has a complexity of O(kn + maxf(1:25542) k k 2 ; (1:2906) k 2:5kg). We improve the klam value by 16 to k = 157. 1 Introduction In 1972, Karp has shown that the following problem is NPcomplete [10]. Problem 1.1 Vertex Cover Instance: A graph G = (V; E), a positive integer k. Question: Does G have a vertex cover of size k? (I.e. does there exist a subset V 0 ` V , jV j k, such that for each (x; y) 2 E either x or y belongs to V 0 ?) Though NPcomplete, the following parameterized version was one of the first problems shown to be fixedparameter tractable [6, 9]. Problem 1.2 kVertex Cover Instance: A graph G = (V; E), a positive integer k. Parameter: k. Question: Does G have a vertex cover of size k?...
Gene Trees and Species Trees: The GeneDuplication Problem is FixedParameter Tractable
 Proceedings of the 6th International Workshop on Algorithms and Data Structures (WADS'99
, 1999
"... The GeneDuplication Problem is the problem of computing the optimal species tree for a given set of gene trees under the GeneDuplication Model (first introduced by Goodman et al.). The problem is known to be NPcomplete. We give a fixedparameter tractable algorithm solving the problem parameteriz ..."
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Cited by 18 (0 self)
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The GeneDuplication Problem is the problem of computing the optimal species tree for a given set of gene trees under the GeneDuplication Model (first introduced by Goodman et al.). The problem is known to be NPcomplete. We give a fixedparameter tractable algorithm solving the problem parameterized by the number of gene duplications necessary to rectify the gene trees with the species tree. 1 Introduction When trying to resolve the tree of life one usually wants to compute the phylogenetic relationships between the organisms based on the data provided by the DNA or protein sequences of families of homologous genes. A species tree or evolutionary tree for a given set of taxa is a complete rooted binary tree built over the set of taxa representing the phylogenetic relationships between the taxa. A gene tree is a complete rooted binary tree formed over a family of homologous genes for the set of taxa. Gene trees for different gene families do not necessarily agree [GCMRM79, F88, BE90, ...