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21
Deciding FirstOrder Properties of Locally TreeDecomposable Graphs
 In Proc. 26th ICALP
, 1999
"... . We introduce the concept of a class of graphs being locally treedecomposable. There are numerous examples of locally treedecomposable classes, among them the class of planar graphs and all classes of bounded valence or of bounded treewidth. We show that for each locally treedecomposable cl ..."
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Cited by 75 (13 self)
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. We introduce the concept of a class of graphs being locally treedecomposable. There are numerous examples of locally treedecomposable classes, among them the class of planar graphs and all classes of bounded valence or of bounded treewidth. We show that for each locally treedecomposable class C of graphs and for each property ' of graphs that is denable in rstorder logic, there is a linear time algorithm deciding whether a given graph G 2 C has property '. 1 Introduction It is an important task in the theory of algorithms to nd feasible instances of otherwise intractable algorithmic problems. A notion that has turned out to be extremely useful in this context is that of treewidth of a graph. 3Colorability, Hamiltonicity, and many other NPcomplete properties of graphs can be decided in linear time when restricted to graphs whose treewidth is bounded by a xed constant (see [Bod97] for a survey). Courcelle [Cou90] proved a metatheorem, which easily implies numer...
The Complexity of Firstorder and Monadic Secondorder Logic Revisited
 Annals of Pure and Applied Logic
, 2002
"... The modelchecking problem for a logic L on a class C of structures asks whether a given Lsentence holds in a given structure in C. In this paper, we give superexponential lower bounds for fixedparameter tractable modelchecking problems for firstorder and monadic secondorder logic. We show tha ..."
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Cited by 63 (6 self)
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The modelchecking problem for a logic L on a class C of structures asks whether a given Lsentence holds in a given structure in C. In this paper, we give superexponential lower bounds for fixedparameter tractable modelchecking problems for firstorder and monadic secondorder logic. We show that unless PTIME = NP, the modelchecking problem for monadic secondorder logic on finite words is not solvable in time f(k) · p(n), for any elementary function f and any polynomial p. Here k denotes the size of the input sentence and n the size of the input word. We prove the same result for firstorder logic under a stronger complexity theoretic assumption from parameterized complexity theory. Furthermore, we prove that the modelchecking problems for firstorder logic on structures of degree 2 and of bounded degree d ≥ 3 are not solvable in time 2 2o(k) · p(n) (for degree 2) and 2 22o(k) · p(n) (for degree d), for any polynomial p, again under an assumption from parameterized complexity theory. We match these lower bounds by corresponding upper bounds. 1.
Locally excluding a minor
, 2007
"... We introduce the concept of locally excluded minors. Graph classes locally excluding a minor generalise the concept of excluded minor classes but also of graph classes with bounded local treewidth and graph classes with bounded expansion. We show that firstorder modelchecking is fixedparameter t ..."
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Cited by 34 (13 self)
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We introduce the concept of locally excluded minors. Graph classes locally excluding a minor generalise the concept of excluded minor classes but also of graph classes with bounded local treewidth and graph classes with bounded expansion. We show that firstorder modelchecking is fixedparameter tractable on any class of graphs locally excluding a minor. This strictly generalises analogous results by Flum and Grohe on excluded minor classes and Frick and Grohe on classes with bounded local treewidth. As an important consequence of the proof we obtain fixedparameter algorithms for problems such as dominating or independent set on graph classes excluding a minor, where now the parameter is the size of the dominating set and the excluded minor. We also study graph classes with excluded minors, where the minor may grow slowly with the size of the graphs and show that again, firstorder modelchecking is fixedparameter tractable on any such class of graphs.
Fixedparameter tractability, definability, and model checking
 SIAM Journal on Computing
, 2001
"... In this article, we study parameterized complexity theory from the perspective of logic, or more specifically, descriptive complexity theory. We propose to consider parameterized modelchecking problems for various fragments of firstorder logic as generic parameterized problems and show how this ap ..."
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Cited by 30 (12 self)
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In this article, we study parameterized complexity theory from the perspective of logic, or more specifically, descriptive complexity theory. We propose to consider parameterized modelchecking problems for various fragments of firstorder logic as generic parameterized problems and show how this approach can be useful in studying both fixedparameter tractability and intractability. For example, we establish the equivalence between the modelchecking for existential firstorder logic, the homomorphism problem for relational structures, and the substructure isomorphism problem. Our main tractability result shows that modelchecking for firstorder formulas is fixedparameter tractable when restricted to a class of input structures with an excluded minor. On the intractability side, for everyØ�we prove an equivalence between modelchecking for firstorder formulas withØquantifier alternations and the parameterized halting problem for alternating Turing machines withØalternations. We discuss the close connection between this alternation hierarchy and Downey and Fellows ’ Whierarchy. On a more abstract level, we consider two forms of definability, called Fagin definability and slicewise definability, that are appropriate for describing parameterized problems. We give a characterization of the class FPT of all fixedparameter tractable problems in terms of slicewise definability in finite variable least fixedpoint logic, which is reminiscent of the ImmermanVardi Theorem characterizing the class PTIME in terms of definability in least fixedpoint logic. 1
Width parameters beyond treewidth and their applications
 Computer Journal
, 2007
"... Besides the very successful concept of treewidth (see [Bodlaender, H. and Koster, A. (2007) Combinatorial optimisation on graphs of bounded treewidth. These are special issues on Parameterized Complexity]), many concepts and parameters measuring the similarity or dissimilarity of structures compare ..."
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Cited by 19 (0 self)
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Besides the very successful concept of treewidth (see [Bodlaender, H. and Koster, A. (2007) Combinatorial optimisation on graphs of bounded treewidth. These are special issues on Parameterized Complexity]), many concepts and parameters measuring the similarity or dissimilarity of structures compared to trees have been born and studied over the past years. These concepts and parameters have proved to be useful tools in many applications, especially in the design of efficient algorithms. Our presented novel look at the contemporary developments of these ‘width ’ parameters in combinatorial structures delivers—besides traditional treewidth and derived dynamic programming schemes—also a number of other useful parameters like branchwidth, rankwidth (cliquewidth) or hypertreewidth. In this contribution, we demonstrate how ‘width ’ parameters of graphs and generalized structures (such as matroids or hypergraphs), can be used to improve the design of parameterized algorithms and the structural analysis in other applications on an abstract level.
Random separation: a new method for solving fixedcardinality optimization problems
 Proceedings 2nd International Workshop on Parameterized and Exact Computation, IWPEC 2006
, 2006
"... Abstract. We develop a new randomized method, random separation, for solving fixedcardinality optimization problems on graphs, i.e., problems concerning solutions with exactly a fixed number k of elements (e.g., k vertices V ′ ) that optimize solution values (e.g., the number of edges covered by V ..."
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Cited by 14 (1 self)
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Abstract. We develop a new randomized method, random separation, for solving fixedcardinality optimization problems on graphs, i.e., problems concerning solutions with exactly a fixed number k of elements (e.g., k vertices V ′ ) that optimize solution values (e.g., the number of edges covered by V ′). The key idea of the method is to partition the vertex set of a graph randomly into two disjoint sets to separate a solution from the rest of the graph into connected components, and then select appropriate components to form a solution. We can use universal sets to derandomize algorithms obtained from this method. This new method is versatile and powerful as it can be used to solve a wide range of fixedcardinality optimization problems for degreebounded graphs, graphs of bounded degeneracy (a large family of graphs that contains degreebounded graphs, planar graphs, graphs of bounded treewidth, and nontrivial minorclosed families of graphs), and even general graphs.
Algorithmic MetaTheorems
 In M. Grohe and R. Neidermeier eds, International Workshop on Parameterized and Exact Computation (IWPEC), volume 5018 of LNCS
, 2008
"... Algorithmic metatheorems are algorithmic results that apply to a whole range of problems, instead of addressing just one specific problem. This kind of theorems are often stated relative to a certain class of graphs, so the general form of a meta theorem reads “every problem in a certain class C of ..."
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Cited by 14 (2 self)
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Algorithmic metatheorems are algorithmic results that apply to a whole range of problems, instead of addressing just one specific problem. This kind of theorems are often stated relative to a certain class of graphs, so the general form of a meta theorem reads “every problem in a certain class C of problems can be solved efficiently on every graph satisfying a certain property P”. A particularly well known example of a metatheorem is Courcelle’s theorem that every decision problem definable in monadic secondorder logic (MSO) can be decided in linear time on any class of graphs of bounded treewidth [1]. The class C of problems can be defined in a number of different ways. One option is to state combinatorial or algorithmic criteria of problems in C. For instance, Demaine, Hajiaghayi and Kawarabayashi [5] showed that every minimisation problem that can be solved efficiently on graph classes of bounded treewidth and for which approximate solutions can be computed efficiently from solutions of certain subinstances, have a PTAS on any class of graphs excluding a fixed minor. While this gives a strong unifying explanation for PTAS of many
On the expression of graph properties in some fragments of monadic secondorder logic
 In Descriptive Complexity and Finite Models: Proceedings of a DIAMCS Workshop
, 1996
"... ABSTRACT: We review the expressibility of some basic graph properties in certain fragments of Monadic SecondOrder logic, like the set of MonadicNP formulas. We focus on cases where a property and its negation are both expressible in the same (or in closely related) fragments. We examine cases wher ..."
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Cited by 13 (1 self)
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ABSTRACT: We review the expressibility of some basic graph properties in certain fragments of Monadic SecondOrder logic, like the set of MonadicNP formulas. We focus on cases where a property and its negation are both expressible in the same (or in closely related) fragments. We examine cases where edge quantifications can be eliminated and cases where they cannot. We compare two logical expressions of planarity: one of them is constructive in the sense that it defines a planar embedding of the considered graph if it is planar and 3connected, and the other, logically simpler, uses the forbidden Kuratowski subgraphs.
Efficient FirstOrder ModelChecking Using Short Labels
"... We prove that there exists an O(log(n))labeling scheme for every firstorder formula with free set variables in every class of graphs that is nicely locally cwddecomposable, which contains in particular, the nicely locally treedecomposable classes. For every class of bounded expansion we prove th ..."
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Cited by 4 (1 self)
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We prove that there exists an O(log(n))labeling scheme for every firstorder formula with free set variables in every class of graphs that is nicely locally cwddecomposable, which contains in particular, the nicely locally treedecomposable classes. For every class of bounded expansion we prove that every bounded formula has an O(log(n))labeling scheme. We also prove that, for fixed k, every quantifierfree formula has an O(log(n))labeling scheme in graphs of arboricity at most k. Some of these results are extended to counting queries.