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10
Branch and Tree Decomposition Techniques for Discrete Optimization
, 2005
"... This chapter gives a general overview of two emerging techniques for discrete optimization that have footholds in mathematics, computer science, and operations research: branch decompositions and tree decompositions. Branch decompositions and tree decompositions along with their respective connectiv ..."
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Cited by 21 (3 self)
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This chapter gives a general overview of two emerging techniques for discrete optimization that have footholds in mathematics, computer science, and operations research: branch decompositions and tree decompositions. Branch decompositions and tree decompositions along with their respective connectivity invariants, branchwidth and treewidth, were first introduced to aid in proving the Graph Minors Theorem, a wellknown conjecture (Wagner’s conjecture) in graph theory. The algorithmic importance of branch decompositions and tree decompositions for solving NPhard problems modelled on graphs was first realized by computer scientists in relation to formulating graph problems in monadic second order logic. The dynamic programming techniques utilizing branch decompositions and tree decompositions, called branch decomposition and tree decomposition based algorithms, fall into a class of algorithms known as fixedparameter tractable algorithms and have been shown to be effective in a practical setting for NPhard problems such as minimum domination, the travelling salesman problem, general minor containment, and frequency assignment problems.
Exact algorithms and applications for Treelike Weighted Set Cover
 JOURNAL OF DISCRETE ALGORITHMS
, 2006
"... We introduce an NPcomplete special case of the Weighted Set Cover problem and show its fixedparameter tractability with respect to the maximum subset size, a parameter that appears to be small in relevant applications. More precisely, in this practically relevant variant we require that the given ..."
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Cited by 12 (6 self)
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We introduce an NPcomplete special case of the Weighted Set Cover problem and show its fixedparameter tractability with respect to the maximum subset size, a parameter that appears to be small in relevant applications. More precisely, in this practically relevant variant we require that the given collection C of subsets of a some base set S should be “treelike.” That is, the subsets in C can be organized in a tree T such that every subset onetoone corresponds to a tree node and, for each element s of S, the nodes corresponding to the subsets containing s induce a subtree of T. This is equivalent to the problem of finding a minimum edge cover in an edgeweighted acyclic hypergraph. Our main result is an algorithm running in O(3 k ·mn) time where k denotes the maximum subset size, n: = S, and m: = C. The algorithm also implies a fixedparameter tractability result for the NPcomplete Multicut in Trees problem, complementing previous approximation results. Our results find applications in computational biology in phylogenomics and for saving memory in tree decomposition based graph algorithms.
Confronting intractability via parameters
, 2011
"... One approach to confronting computational hardness is to try to understand the contribution of various parameters to the running time of algorithms and the complexity of computational tasks. Almost no computational tasks in real life are specified by their size alone. It is not hard to imagine that ..."
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Cited by 3 (0 self)
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One approach to confronting computational hardness is to try to understand the contribution of various parameters to the running time of algorithms and the complexity of computational tasks. Almost no computational tasks in real life are specified by their size alone. It is not hard to imagine that some parameters contribute more intractability than others and it seems reasonable to develop a theory of computational complexity which seeks to exploit this fact. Such a theory should be able to address the needs of practicioners in algorithmics. The last twenty years have seen the development of such a theory. This theory has a large number of successes in terms of a rich collection of algorithmic techniques both practical and theoretical, and a finegrained intractability theory. Whilst the theory has been widely used in a number of areas of applications including computational biology, linguistics, VLSI design, learning theory and many others, knowledge of the area is highly varied. We hope that this article will show both the basic theory and point at the wide array of techniques available. Naturally the treatment is condensed, and the reader who wants more should go to the texts of Downey and Fellows [125], Flum and Grohe [155], Niedermeier [240], and the upcoming undergraduate text Downey and Fellows [127].
Developing FixedParameter Algorithms to Solve Combinatorially Explosive Biological Problems
"... Fixedparameter algorithms can efficiently find optimal solutions to some computationally hard (NPhard) problems. We survey five main practical techniques to develop such algorithms. Each technique is circumstantiated by case studies of applications to biological problems. We also present other kno ..."
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Cited by 2 (0 self)
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Fixedparameter algorithms can efficiently find optimal solutions to some computationally hard (NPhard) problems. We survey five main practical techniques to develop such algorithms. Each technique is circumstantiated by case studies of applications to biological problems. We also present other known bioinformaticsrelated applications and give pointers to experimental results. Key Words: Computationally hard problems; combinatorial explosions; discrete problems; fixedparameter tractability; optimal solutions. 1
Technical Communications of ICLP
, 2015
"... Abstract Dynamic programming (DP) on tree decompositions is a well studied approach for solving hard problems efficiently. Stateoftheart implementations usually rely on tables for storing information, and algorithms specify how the tuples are manipulated during traversal of the decomposition. Ho ..."
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Abstract Dynamic programming (DP) on tree decompositions is a well studied approach for solving hard problems efficiently. Stateoftheart implementations usually rely on tables for storing information, and algorithms specify how the tuples are manipulated during traversal of the decomposition. However, a major bottleneck of such tablebased algorithms is relatively high memory consumption. The goal of the doctoral thesis herein discussed is to mitigate performance and memory shortcomings of such algorithms. The idea is to replace tables with an efficient data structure that no longer requires to enumerate intermediate results explicitly during the computation. To this end, Binary Decision Diagrams (BDDs) and related concepts are studied with respect to their applicability in this setting. Besides native support for efficient storage, from a conceptual point of view BDDs give rise to an alternative approach of how DP algorithms are specified. Instead of tuplebased manipulation operations, the algorithms are specified on a logical level, where sets of models can be conjointly updated. The goal of the thesis is to provide a general toolset for problems that can be solved efficiently via DP on tree decompositions.
Roman Domination on Graphs of Bounded Treewidth
 THE 24TH WORKSHOP ON COMBINATORIAL MATHEMATICS AND COMPUTATION THEORY
"... A roman domination function on a graph G = (V,E) is a function f: V →{0, 1,2} satisfying that every vertex u with f(u) = 0 has a neighbor v such that f(v) =2. The weight of a roman domination function is the value � v∈V f(v).The roman domination number of a graph G is the minimum weight of all poss ..."
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A roman domination function on a graph G = (V,E) is a function f: V →{0, 1,2} satisfying that every vertex u with f(u) = 0 has a neighbor v such that f(v) =2. The weight of a roman domination function is the value � v∈V f(v).The roman domination number of a graph G is the minimum weight of all possible roman domination functions on G. In this paper, we show that the roman domination number for a graph with bounded treewidth can be computed in linear time.
Global Defensive Alliances in Star Graphs
 THE 24TH WORKSHOP ON COMBINATORIAL MATHEMATICS AND COMPUTATION THEORY
"... A defensive alliance in a graph G =(V,E) is a set of vertices S ⊆ V where for each v ∈ S, at least half of the vertices in the closed neighborhood of v are in S. A defensive alliance S is called global if every vertex in V (G) \ S is adjacent to at least one member of the defensive alliance S. In t ..."
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A defensive alliance in a graph G =(V,E) is a set of vertices S ⊆ V where for each v ∈ S, at least half of the vertices in the closed neighborhood of v are in S. A defensive alliance S is called global if every vertex in V (G) \ S is adjacent to at least one member of the defensive alliance S. In this paper, we derive an upper bound to the size of the minimum global defensive alliances in star graphs.
WEIGHTED DOMINATION NUMBER OF CACTUS GRAPHS
, 2016
"... Abstract: In this paper we propose a linear algorithm for calculating the weighted domination number of a vertexweighted cactus. The algorithm is based on the well known depth first search (DFS) structure. Our algorithm needs less than 12n + 5b additions and 9n + 2b minoperations where n is the n ..."
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Abstract: In this paper we propose a linear algorithm for calculating the weighted domination number of a vertexweighted cactus. The algorithm is based on the well known depth first search (DFS) structure. Our algorithm needs less than 12n + 5b additions and 9n + 2b minoperations where n is the number of vertices and b is the number of blocks in the cactus.
Graph Minors and Parameterized Algorithm Design
"... Abstract. The Graph Minors Theory, developed by Robertson and Seymour, has been one of the most influential mathematical theories in parameterized algorithm design. We present some of the basic algorithmic techniques and methods that emerged from this theory. We discuss its direct metaalgorithmic ..."
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Abstract. The Graph Minors Theory, developed by Robertson and Seymour, has been one of the most influential mathematical theories in parameterized algorithm design. We present some of the basic algorithmic techniques and methods that emerged from this theory. We discuss its direct metaalgorithmic consequences, we present the algorithmic applications of core theorems such as the gridexclusion theorem, and we give a brief description of the irrelevant vertex technique.