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46
Parameterized Complexity: A Framework for Systematically Confronting Computational Intractability
 DIMACS Series in Discrete Mathematics and Theoretical Computer Science
, 1997
"... In this paper we give a programmatic overview of parameterized computational complexity in the broad context of the problem of coping with computational intractability. We give some examples of how fixedparameter tractability techniques can deliver practical algorithms in two different ways: (1) by ..."
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Cited by 72 (15 self)
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In this paper we give a programmatic overview of parameterized computational complexity in the broad context of the problem of coping with computational intractability. We give some examples of how fixedparameter tractability techniques can deliver practical algorithms in two different ways: (1) by providing useful exact algorithms for small parameter ranges, and (2) by providing guidance in the design of heuristic algorithms. In particular, we describe an improved FPT kernelization algorithm for Vertex Cover, a practical FPT algorithm for the Maximum Agreement Subtree (MAST) problem parameterized by the number of species to be deleted, and new general heuristics for these problems based on FPT techniques. In the course of making this overview, we also investigate some structural and hardness issues. We prove that an important naturally parameterized problem in artificial intelligence, STRIPS Planning (where the parameter is the size of the plan) is complete for W [1]. As a corollary, this implies that kStep Reachability for Petri Nets is complete for W [1]. We describe how the concept of treewidth can be applied to STRIPS Planning and other problems of logic to obtain FPT results. We describe a surprising structural result concerning the top end of the parameterized complexity hierarchy: the naturally parameterized Graph kColoring problem cannot be resolved with respect to XP either by showing membership in XP, or by showing hardness for XP without settling the P = NP question one way or the other.
Parameterized Computational Feasibility
 Feasible Mathematics II
, 1994
"... Many natural computational problems have input consisting of two or more parts. For example, the input might consist of a graph and a positive integer. For many natural problems we may view one of the inputs as a parameter and study how the complexity of the problem varies if the parameter is he ..."
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Cited by 61 (20 self)
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Many natural computational problems have input consisting of two or more parts. For example, the input might consist of a graph and a positive integer. For many natural problems we may view one of the inputs as a parameter and study how the complexity of the problem varies if the parameter is held fixed. For many applications of computational problems involving such a parameter, only a small range of parameter values is of practical significance, so that fixedparameter complexity is a natural concern. In studying the complexity of such problems, it is therefore important to have a framework in which we can make qualitative distinctions about the contribution of the parameter to the complexity of the problem. In this paper we survey one such framework for investigating parameterized computational complexity and present a number of new results for this theory.
Linear Time Computable Problems and FirstOrder Descriptions
, 1996
"... this article is a proof that each FO problem can be solved in linear time if only relational structures of bounded degree are considered. The basic idea of the proof is a localization technique based on a method that was originally developed by Hanf (Hanf 1965) to show that the elementary theories o ..."
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Cited by 31 (2 self)
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this article is a proof that each FO problem can be solved in linear time if only relational structures of bounded degree are considered. The basic idea of the proof is a localization technique based on a method that was originally developed by Hanf (Hanf 1965) to show that the elementary theories of two structures are equal under certain conditions, i.e., that two structures agree on all firstorder sentences. Fagin, Stockmeyer and Vardi (Fagin et al. 1993) developed a variant of this technique, which is applicable in descriptive complexity theory to classes of finite relational structures of uniformly bounded degree. Variants of this result can also be found in Gaifman (1982) (see also Thomas (1991)). The essential content of this result, which is also called the HanfSphere Lemma, is that two relational structures of bounded degree satisfy the same firstorder sentences of a certain quantifierrank if both contain, up to a certain number m, the same number of isomorphism types of substructures of a bounded radius r. In addition, a technique of model interpretability from Rabin (1965) (see also Arnborg et al. (1991), Seese (1992), Compton and Henson (1987) and Baudisch et al. (1982)) is adapted to descriptive complexity classes, and proved to be useful for reducing the case of an arbitrary class of relational structures to a class of structures consisting only of the domain and one binary irreflexive and symmetric relation, i.e., the class of simple graphs. It is shown that the class of simple graphs is lintimeuniversal with respect to firstorder logic, which shows that many problems on descriptive complexity classes, described in languages extending firstorder logic for arbitrary structures, can be reduced to problems on simple graphs. This paper is organized as f...
On Disjoint Cycles
 International Journal of Foundations of Computer Science
, 1990
"... It is shown, that for each constant k _ 1, the following problems can be solved in O(n) time: given a graph G, determine whether G has k vertex disjoint cycles, determine whether G has k edge disjoint cycles, determine whether G has a feedback vertex set of size _ k. Also, every class G, that is ..."
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Cited by 29 (4 self)
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It is shown, that for each constant k _ 1, the following problems can be solved in O(n) time: given a graph G, determine whether G has k vertex disjoint cycles, determine whether G has k edge disjoint cycles, determine whether G has a feedback vertex set of size _ k. Also, every class G, that is closed under minor taking, or that is closed under immersion taking, and that does not contain the graph formed by taking the disjoint union of k copies of Ks, has an O(n) membership test algorithm.
Constructive Linear Time Algorithms for Branchwidth
, 1997
"... We prove that, for any fixed k, one can construct a linear time algorithm that checks if a graph has branchwidth k and, if so, outputs a branch decomposition of minimum width. 1 Introduction This paper considers the problem of finding branch decompositions of graphs with small branchwidth. The noti ..."
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Cited by 27 (6 self)
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We prove that, for any fixed k, one can construct a linear time algorithm that checks if a graph has branchwidth k and, if so, outputs a branch decomposition of minimum width. 1 Introduction This paper considers the problem of finding branch decompositions of graphs with small branchwidth. The notion of branchwidth has a close relationship to the more wellknown notion of treewidth, a notion that has come to play a large role in many recent investigations in algorithmic graph theory. (See Section 2 for definitions of treewidth and branchwidth.) One reason for the interest in this notion is that many graph problems can be solved by linear time algorithms, when the inputs are restricted to graphs with some uniform upper bound on their treewidth. Most of these algorithms first try to find a tree decomposition of small width, and then utilize the advantages of the tree structure of the decomposition (see [1], [4]). The branchwidth of a graph differs from its treewidth by at most a multipl...
Graphs with Branchwidth at most Three
 J. Algorithms
, 1997
"... In this paper we investigate both the structure of graphs with branchwidth at most three, as well as algorithms to recognise such graphs. We show that a graph has branchwidth at most three, if and only if it has treewidth at most three and does not contain the threedimensional binary cube graph as ..."
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Cited by 25 (1 self)
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In this paper we investigate both the structure of graphs with branchwidth at most three, as well as algorithms to recognise such graphs. We show that a graph has branchwidth at most three, if and only if it has treewidth at most three and does not contain the threedimensional binary cube graph as a minor. A set of four graphs is shown to be the obstruction set of graphs with branchwidth at most three. We give a safe and complete set of reduction rules for the graphs with branchwidth at most three. Using this set, a linear time algorithm is given that checks if a given graph has branchwidth at most three, and, if so, outputs a minimum width branch decomposition. Keywords: graph algorithms, branchwidth, obstruction set, graph minor, reduction rule. 1 Introduction This paper considers the study of the graphs with branchwidth at most three. The notion of branchwidth has a close relationship to the more wellknown notion of treewidth, a notion that has come to play a large role in many ...
On Winning Strategies With Unary Quantifiers
 J. Logic and Computation
, 1996
"... A combinatorial argument for two finite structures to agree on all sentences with bounded quantifier rank in firstorder logic with any set of unary generalized quantifiers, is given. It is known that connectivity of finite structures is neither in monadic \Sigma 1 1 nor in L !! (Q u ), where Q ..."
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Cited by 25 (6 self)
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A combinatorial argument for two finite structures to agree on all sentences with bounded quantifier rank in firstorder logic with any set of unary generalized quantifiers, is given. It is known that connectivity of finite structures is neither in monadic \Sigma 1 1 nor in L !! (Q u ), where Q u is the set of all unary generalized quantifiers. Using this combinatorial argument and a combination of secondorder EhrenfeuchtFra iss'e games developed by Ajtai and Fagin, we prove that connectivity of finite structures is not in monadic \Sigma 1 1 with any set of unary quantifiers, even if sentences are allowed to contain builtin relations of moderate degree. The combinatorial argument is also used to show that no class (if it is not in some sense trivial) of finite graphs defined by forbidden minors, is in L !! (Q u ). Especially, the class of planar graphs is not in L !! (Q u ). 1. Introduction The expressive power of firstorder logic L !! is rather limited. This is beca...
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.
A Characterization of Graphs with Vertex Cover up to Five
, 1994
"... . For the family of graphs with fixedsize vertex cover k, we present all of the forbidden minors (obstructions), for k up to five. We derive some results, including a practical finitestate recognition algorithm, needed to compute these obstructions. 1 Introduction The proof of Wagner's conjec ..."
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Cited by 23 (11 self)
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. For the family of graphs with fixedsize vertex cover k, we present all of the forbidden minors (obstructions), for k up to five. We derive some results, including a practical finitestate recognition algorithm, needed to compute these obstructions. 1 Introduction The proof of Wagner's conjecture by Robertson and Seymour (see [RS85, RS]), now known as the Graph Minor Theorem (GMT), has led to an explosion of interest in obstruction sets. Though the GMT is primarily of theoretical interest, our research group has been exploring applications of the theory. We have developed a system called VACS to help determine obstruction sets for certain graph families. One of these families, vertex cover, is the subject of this paper. We make two main contributions in this paper. First, we present a lineartime algorithm that determines the vertex cover for the class of graphs with boundedpathwidth (partial tpaths). There are related results as discussed below, but the algorithm we present ...
Unifying clustertree decompositions for reasoning in graphical models
 Artificial Intelligence
, 2005
"... The paper provides a unifying perspective of treedecomposition algorithms appearing in various automated reasoning areas such as jointree clustering for constraintsatisfaction and the cliquetree algorithm for probabilistic reasoning. Within this framework, we introduce a new algorithm, called bu ..."
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Cited by 17 (9 self)
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The paper provides a unifying perspective of treedecomposition algorithms appearing in various automated reasoning areas such as jointree clustering for constraintsatisfaction and the cliquetree algorithm for probabilistic reasoning. Within this framework, we introduce a new algorithm, called buckettree elimination (BT E), that extends Bucket Elimination (BE) to trees, and show that it can provide a speedup of n over BE for various reasoning tasks. Timespace tradeoffs of treedecomposition processing are analyzed. 1