## Width parameters beyond tree-width and their applications (2007)

### Cached

### Download Links

- [comjnl.oxfordjournals.org]
- [www.labri.fr]
- DBLP

### Other Repositories/Bibliography

Venue: | Computer Journal |

Citations: | 18 - 0 self |

### BibTeX

@ARTICLE{Hliněny07widthparameters,

author = {Petr Hliněny and Sang-il Oum and Detlef Seese and Georg Gottlob},

title = {Width parameters beyond tree-width and their applications},

journal = {Computer Journal},

year = {2007},

pages = {10--1093}

}

### OpenURL

### Abstract

Besides the very successful concept of tree-width (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 tree-width and derived dynamic programming schemes—also a number of other useful parameters like branch-width, rank-width (clique-width) or hypertree-width. 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.

### Citations

10995 | Computers and Intractability: A Guide to the Theory of NP-Completeness - GAREY, JOHNSON - 1979 |

8581 | Introduction to Algorithms - Cormen, Leiserson, et al. - 2001 |

2352 | Computational Complexity
- Papadimitriou
- 1994
(Show Context)
Citation Context ...ese colors are required to match with neighboring tiles. There are many variants and applications of the tiling problem in complexity, decidability, picture recognition and physics (see, for instance =-=[22, 23]-=-). It is not difficult to generalize tilings to covering and coloring problems for arbitrary structures with local FIGURE 4. The borderline between P and NP. THE COMPUTER JOURNAL, 2007 conditions. For... |

872 | Parameterized Complexity - Downey, Fellows - 1999 |

498 |
Finite Model Theory
- Ebbinghaus, Flum
- 1999
(Show Context)
Citation Context ...on to unary or monadic relations, or equivalently to sets. This leads to the calculus which is suitable for large classes of applications and which is the basis for many generalizations—the MSO logic =-=[66, 70, 71]-=-. DEFINITION. MSO logic is an extension of the usual FO logic by variables (usually capital letters X 1,X 2,...) running over subsets of the domain, and by the relational symbol [ for membership. The ... |

458 | Optimal Implementation of Conjunctive Queries in Relational Data 8ases - Merlin - 1977 |

418 |
Combinatorial Algorithms for Integrated Circuit Layout
- Lengauer
- 1990
(Show Context)
Citation Context ...ures are defined by hierarchical expressions, for which several algorithmic problems can be solved efficiently (see [32– 40]). These results have interesting applications in the area of VLSI circuits =-=[41, 42]-=-. One of the most prominent developments, which one could at least partially subsume to branch (B), is graphs generated by graph grammars (see, e.g. [43–48]), which are also closely related to clique-... |

392 | Network-based heuristics for constraint-satisfaction problems - Dechter, Pearl - 1987 |

366 |
Graph Minors II. Algorithmic aspects of tree width
- Robertson, Seymour
- 1986
(Show Context)
Citation Context ...idth. DEFINITION. The tree-width twd(G) of a graph G is the smallest parameter k such that G is a partial k-tree. There is an equivalent definition given by Robertson and Seymour, originating in 1983 =-=[12, 13]-=- in connection with their Graph Minors Project [14] (see also Section 1.4): First, a tree-decomposition of a graph G is a pair (T, X), where T is a tree and X is a family of vertex sets Xt # V (G), ca... |

364 |
Parameterized Complexity Theory
- Flüm, Grohe
- 2006
(Show Context)
Citation Context ...nvolved) concepts. While we assume some basic knowledge concerning graphs and different parametrized complexity topics which are covered in [These special issues on Parametrized Complexity] (see also =-=[2, 3]-=-), the sections are generally self-contained. Thus, an interested reader can pick up his favorite topic at an accessible level without the need for extensive study.sPage 2 of 37 P. HLINĚNY´ ET AL. Sin... |

301 | The undecidability of the domino problem - Berger - 1966 |

292 |
Graph Classes: a Survey
- Brandstädt, Le, et al.
- 1999
(Show Context)
Citation Context ... to describe large classes of problems in a logical or algebraic calculus, and to show that all such problems can be solved more efficiently for structures with special properties (see the monographs =-=[2, 19]-=-). 1.2.1. Unification of approaches It is one of the goals of this section to find an idea for a unified structural classification approach. We propose a heuristic criterion, which can give a first hi... |

271 |
Generalized first-order spectra and polynomial-time recognizable sets
- Fagin
- 1974
(Show Context)
Citation Context ... positive answer can be verified efficiently) contains a large number of problems that are of practical interest. To identify the right language, the following result is important. THEOREM 2.1 (Fagin =-=[64]-=-, see also [52, 65]). NP # 9SO. That means each problem Q in NP can be described in the following way: G has property Q if and only if G o 9X1 ... 9Xnw, for a formula w of the usual FO predicate calcu... |

267 | The classical decision problem - Borger, Gradel, et al. - 1997 |

252 |
Graph minors XIII. The disjoint paths problem
- Robertson, Seymour
- 1995
(Show Context)
Citation Context ...g whether G contains a minor isomorphic to H with both G, H on the input is NP-hard since one can test, say, the existence of a Hamiltonian cycle in G in this way.) THEOREM 1.6 (Robertson and Seymour =-=[61]-=-). For each fixed H, there is an algorithm testing in time O(n 3 ) whether an n-vertex graph G has a minor isomorphic to H. Hence, from Theorem 1.5, every minor-closed property of graphs can be decide... |

251 | The clustering for constraint networks - Dechter, Pearl - 1999 |

232 | Easy problems for tree-decomposable graphs
- Arnborg, Lagergren, et al.
- 1991
(Show Context)
Citation Context ...7, 80]. Compton and Henson [81] used it to deduce lower bounds for the complexity of theories. It has later been adapted to linear and polynomial time computable decision and optimization problems in =-=[55, 78, 79]-=-. (See also [58, 75, 82], where a variant of it is called a transduction). A key result for interpretability is as follows. WIDTH PARAMETERS AND THEIR APPLICATIONS Page 9 of 37 THE COMPUTER JOURNAL, 2... |

230 | Languages that capture complexity classes
- Immerman
- 1987
(Show Context)
Citation Context ... ..., Xn, so that these relations have a certain property w over the structure G, which can be expressed in FO predicate logic FO. This is one of the key results of descriptive complexity theory (see =-=[65, 68]-=-), which tries to capture complexity classes, like P or NP, through the expressive power of special logical languages. We call a decision problem Q L-definable over the class K of structures if there ... |

192 | A linear-time algorithm for finding tree-decompositions of small treewidth - Bodlaender - 1996 |

174 | Complexity and Approximation - Ausiello, Crescenzi, et al. - 1999 |

165 | A sufficient condition for backtrack-bounded search - Freuder - 1985 |

163 | Graph minors X: Obstructions to tree-decomposition
- Robertson, Seymour
- 1991
(Show Context)
Citation Context ...with large grids’ in a more precise way as ‘there is a fixed integer k such that the input graph does not have a k-grid Qk (Fig. 5) as a minor’ leads to the class of graphs of bounded tree-width (see =-=[29]-=- and Theorem 1.4). This theorem, and also its newer generalizations to matroids [30] (Section 3.4) or to vertex-minors [31] (Section 4.2), provide the structural basis to explain why the exclusion of ... |

157 | A linear recognition algorithm for cographs - Corneil, Perl, et al. - 1985 |

152 | Linear time algorithms for NP-hard problems restricted to partial k-trees - Arnborg, Proskurowski - 1989 |

145 | F.: A comparison of structural CSP decomposition methods - Gottlob, Leone, et al. - 2000 |

142 | The expression of graph properties and graph transformations in monadic second-order logic, in
- Courcelle
- 1997
(Show Context)
Citation Context ...on [81] used it to deduce lower bounds for the complexity of theories. It has later been adapted to linear and polynomial time computable decision and optimization problems in [55, 78, 79]. (See also =-=[58, 75, 82]-=-, where a variant of it is called a transduction). A key result for interpretability is as follows. WIDTH PARAMETERS AND THEIR APPLICATIONS Page 9 of 37 THE COMPUTER JOURNAL, 2007 THEOREM 2.3 ([55, 78... |

138 | Graph theory, volume 173 of Graduate Texts in Mathematics - Diestel - 2005 |

128 | Hypertree decompositions and tractable queries - Gottlob, Leone, et al. |

121 |
Graph minors. I. Excluding a forest
- Robertson, Seymour
- 1983
(Show Context)
Citation Context ...idth. DEFINITION. The tree-width twd(G) of a graph G is the smallest parameter k such that G is a partial k-tree. There is an equivalent definition given by Robertson and Seymour, originating in 1983 =-=[12, 13]-=- in connection with their Graph Minors Project [14] (see also Section 1.4): First, a tree-decomposition of a graph G is a pair (T, X), where T is a tree and X is a family of vertex sets Xt # V (G), ca... |

120 | Complement reducible graphs - Corneil, Lerchs, et al. - 1981 |

117 |
Graph Theory, Graduate Texts
- Diestel
- 1996
(Show Context)
Citation Context ...is connected by a path. A connected graph is a tree if it contains no cycles. For missing basic notation and terminology on graphs and algorithms, we refer the reader to classical text books, such as =-=[4, 5]-=- for instance. 1.1. Tree-shaped structures: motivation Many algorithmic problems of practical or theoretical interest are NP-hard and to date efficient solutions in polynomial time have not been found... |

112 | Linear time solvable optimization problems on graphs of bounded clique-width
- Courcelle, Makowsky, et al.
(Show Context)
Citation Context ...an also be allowed in such problems, since it is expressible in terms of maximization and negation which both are allowed in evaluation terms. The LinEMSOL optimization problems over K, introduced in =-=[59]-=-, essentially follow the definition of linear EMSO optimization problems from [55, 78], but restrict the constants to integers and do not allow arbitrary evaluation relations. A problem to compute the... |

103 | Graph minors XX: Wagner’s conjecture
- Robertson, Seymour
(Show Context)
Citation Context ... that a graph G is planar if and only if G has no minor isomorphic to K5 or K3,3, or a graph G is series–parallel if and only if K4 is not a minor of G. The celebrated result of Robertson and Seymour =-=[62]-=- can be stated as follows. THEOREM 1.5 (Robertson and Seymour [62]). Every minorclosed class of finite graphs can be characterized by a finite list of forbidden minors. 1.4.2. Minors and efficient alg... |

98 | Conjunctive Query Containment Revisited - Chekuri, Rajaraman - 1997 |

90 | Automatic structures - Blumensath, Grädel - 2000 |

86 |
C.: Introduction to Algorithms, 2nd edn
- Cormen, Leiserson, et al.
- 2001
(Show Context)
Citation Context ...is connected by a path. A connected graph is a tree if it contains no cycles. For missing basic notation and terminology on graphs and algorithms, we refer the reader to classical text books, such as =-=[4, 5]-=- for instance. 1.1. Tree-shaped structures: motivation Many algorithmic problems of practical or theoretical interest are NP-hard and to date efficient solutions in polynomial time have not been found... |

83 | Decomposing constraint satisfaction problems using database techniques - Gyssens, Jeavons, et al. - 1994 |

80 | Monadic second-order evaluations on tree-decomposable graphs
- Courcelle, Mosbah
- 1993
(Show Context)
Citation Context ...on [81] used it to deduce lower bounds for the complexity of theories. It has later been adapted to linear and polynomial time computable decision and optimization problems in [55, 78, 79]. (See also =-=[58, 75, 82]-=-, where a variant of it is called a transduction). A key result for interpretability is as follows. WIDTH PARAMETERS AND THEIR APPLICATIONS Page 9 of 37 THE COMPUTER JOURNAL, 2007 THEOREM 2.3 ([55, 78... |

80 | Distance-hereditary graphs - Bandelt, Mulder - 1986 |

77 | Degrees of acyclicity for hypergraphs and relational database schemes - Fagin - 1983 |

74 | Deciding first-order properties of locally tree-decomposable structures
- Frick, Grohe
(Show Context)
Citation Context ...e a tiling inside a problem P (using the grid-structure) on the right. THE COMPUTER JOURNAL, 2007 linear time for structures of finite degree [51, 52] and for structures of locally bounded tree-width =-=[53, 54]-=-. A further strong general result, in this context, can be obtained in combination with (A). If we restrict the input to graphs of bounded tree-width, and only consider problems definable in simple ex... |

74 | The Complexity of Acyclic Conjunctive Queries - Gottlob, Leone, et al. - 2001 |

69 | Handle-rewriting hypergraph grammars - Courcelle, Engelfriet, et al. - 1993 |

68 | An algebraic theory of graph reductions - Arnborg, Courcelle, et al. - 1993 |

67 | Dominating sets in planar graphs: branch-width and exponential speed-up - Fomin, Thilikos - 2003 |

67 | The first order properties of products of algebraic systems - Feferman, Vaught - 1959 |

66 | The complexity of first-order and monadic second-order logic revisited
- Frick, Grohe
- 2002
(Show Context)
Citation Context ...e a tiling inside a problem P (using the grid-structure) on the right. THE COMPUTER JOURNAL, 2007 linear time for structures of finite degree [51, 52] and for structures of locally bounded tree-width =-=[53, 54]-=-. A further strong general result, in this context, can be obtained in combination with (A). If we restrict the input to graphs of bounded tree-width, and only consider problems definable in simple ex... |

65 | Automatic Generation of Linear-Time Algorithms from Predicate Calculus - Borie, Parker, et al. |

60 | On the fixed parameter complexity of graph enumeration problems definable in monadic second order logic
- Courcelle, Makowsky, et al.
- 2001
(Show Context)
Citation Context ...) is solvable in linear time for graphs having clique-width at most k for fixed k if the k-expression defining the graph is given with the input. We will describe how to find a k-expression later. In =-=[76]-=-, Courcelle et al. showed that MS1 counting problems, enumerating the truth assignments of a fixed MS 1 logic formula, are solvable in linear time as well under the same condition. Not only MS1 proble... |

57 |
Recurring dominoes;; making the highly undecidable highly understandable
- Harel
- 1983
(Show Context)
Citation Context ...s in a way that it can be verified whether two neighboring tiles match together (horizontally and vertically). A simple illustration of this idea is shown in Fig. 5. Detailed examples can be found in =-=[26, 27]-=-. 1.2.3. Reducing problem complexity An analysis of the above idea leads us to three canonical possibilities to reduce the complexity of a problem P on a class K of structures; either by (i) avoiding ... |