Results 1  10
of
45
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 ..."
Abstract

Cited by 98 (14 self)
 Add to MetaCart
(Show Context)
. 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
, 2004
"... ..."
(Show Context)
Locally excluding a minor
"... We introduce the concept of locally excluded minors. Graph classes locally excluding a minor are a common generalisation of the concept of excluded minor classes and of graph classes with bounded local treewidth. We show that firstorder modelchecking is fixedparameter tractable on any class of gr ..."
Abstract

Cited by 47 (13 self)
 Add to MetaCart
We introduce the concept of locally excluded minors. Graph classes locally excluding a minor are a common generalisation of the concept of excluded minor classes and of graph classes with bounded local treewidth. 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 ..."
Abstract

Cited by 37 (13 self)
 Add to MetaCart
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.
Firstorder queries on structures of bounded degree are computable with constant delay
 ACM TRANS. ON COMPUTATIONAL LOGIC
, 2007
"... ..."
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 ..."
Abstract

Cited by 27 (3 self)
 Add to MetaCart
(Show Context)
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 ..."
Abstract

Cited by 22 (6 self)
 Add to MetaCart
(Show Context)
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
"... 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 q ..."
Abstract

Cited by 14 (1 self)
 Add to MetaCart
(Show Context)
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.