## An Efficient Graph Algorithm for Dominance Constraints (2003)

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Venue: | JOURNAL OF ALGORITHMS |

Citations: | 37 - 15 self |

### BibTeX

@ARTICLE{Althaus03anefficient,

author = {Ernst Althaus and Denys Duchier and Alexander Koller and Kurt Mehlhorn and Joachim Niehren and Sven Thiel},

title = {An Efficient Graph Algorithm for Dominance Constraints},

journal = {JOURNAL OF ALGORITHMS},

year = {2003},

volume = {48},

pages = {194--219}

}

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### Abstract

Dominance constraints are logical descriptions of trees that are widely used in computational linguistics. Their general satisfiability problem is known to be NP-complete. Here we identify normal dominance constraints and present an efficient graph algorithm for testing their satisfiablity in deterministic polynomial time. Previously, no polynomial time algorithm was known.

### Citations

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655 | LEDA: A Platform for Combinatorial and Geometric Computing
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(Show Context)
Citation Context ... satisfiability test to enumerate solutions. We apply a choice rule exhaustively while checking for satisfiability after each step. Both procedures have been implemented in C++ using the LEDA library =-=[17]-=- and applied to scope ambiguities in natural language semantics in the CLLS framework [11,18,19]. To complement our results, we finally investigate a close variant of the configuration problem of domi... |

267 | Minimal Recursion Semantics: An introduction - Copestake, Flickinger, et al. - 1999 |

179 |
Dealing with ambiguity by underspecification. Construction, representation, and deduction
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- 1993
(Show Context)
Citation Context .... Section 11 concludes and discusses further work. 2 Motivation As one application of dominance constraints in computational linguistics, we will give a brief introduction to scope underspecification =-=[18,22,23,20]-=-. This application is concerned with coping with ambiguous sentences such as the following: (1) Every linguist speaks two languages. Sentence (1) is ambiguous because it has two different possible mea... |

166 |
Generalized finite automata theory with an application to a decision problem of second-order logic
- Thatcher, Wright
- 1968
(Show Context)
Citation Context ...earch Centre (SFB) 378 of the Deutsche Forschungsgemeinschaft. Preprint submitted to Elsevier Science 29 April 2003sDominance based tree descriptions were first used in automata theory in the sixties =-=[3]-=-, rediscovered in computational linguistics in the early eighties [4], and investigated from a logical point of view in the early nineties [5]. Since then, they have found numerous applications in com... |

141 | Predicate logic unplugged
- Bos
- 1995
(Show Context)
Citation Context ...g of 3-partition are in solved form, but they may well be unconfigurable. The relevance of this result is again in its relation to computational linguistics. There are alternative approaches to scope =-=[Bos96]-=- which require that the holes and roots of the trees must be paired uniquely: The roots must be “plugged” into the holes, and every hole must be plugged. This corresponds to making the holes open leav... |

112 | Monotonic Semantic Interpretation - Alshawi, Crouch - 1992 |

107 |
Dtheory: Talking about talking about trees
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(Show Context)
Citation Context ...nt submitted to Elsevier Science 29 April 2003sDominance based tree descriptions were first used in automata theory in the sixties [3], rediscovered in computational linguistics in the early eighties =-=[4]-=-, and investigated from a logical point of view in the early nineties [5]. Since then, they have found numerous applications in computational linguistics: they have been used for grammar formalisms [6... |

103 | D-tree grammars - Rambow, Vijay-Shanker, et al. - 1995 |

98 | Records for logic programming
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(Show Context)
Citation Context ... results on satisfiability. The variant of dominance constraints we employ describes constructor trees – ground terms over a signature of function symbols – rather than feature trees as considered in =-=[25,5,26,27]-=-. 3.1 Trees and Constructor Trees We assume a finite or infinite signature Σ with function symbols f, g, . . ., each of which is equipped with an arity ar(f) ≥ 0. Constants are function symbols of ari... |

84 | Feature Constraint Logic for Unification Grammars
- Smolka
- 1989
(Show Context)
Citation Context ... results on satisfiability. The variant of dominance constraints we employ describes constructor trees – ground terms over a signature of function symbols – rather than feature trees as considered in =-=[25,5,26,27]-=-. 3.1 Trees and Constructor Trees We assume a finite or infinite signature Σ with function symbols f, g, . . ., each of which is equipped with an arity ar(f) ≥ 0. Constants are function symbols of ari... |

81 | The Constraint Language for Lambda Structures - Egg, Koller, et al. - 2001 |

79 | Using descriptions of trees in a tree adjoining grammar - Vijay-Shanker - 1993 |

47 | A first-order axiomatization of the theory of finite trees
- Backofen, Rogers, et al.
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(Show Context)
Citation Context ... were first used in automata theory in the sixties [TW67], rediscovered in computational linguistics in the early eighties [MHF83], and investigated from a logical point of view in the early nineties =-=[BRVS95]-=-. Since then, they have found numerous applications in computational linguistics: they have been used for grammar formalisms [VS92, RVSW95, DT99, Per00], in natural language semantics [Mus95, ENRX98],... |

41 |
An improper treatment of quantification in ordinary English
- Hobbs
- 1983
(Show Context)
Citation Context ...the the language of dominance constraints and recall known results on satisfiability. The variant of dominance constraints we employ describes constructor trees – ground 5 The following sentence from =-=[Hob83]-=-, which is interesting both in form and in content, has around 200 readings: “Many people feel that most sentences exhibit too few quantifier scope ambiguities for much effort to be devoted to this pr... |

40 | Constraints over lambdastructures in semantic underspeci - Egg, Niehren, et al. - 1998 |

40 |
An improved algorithm for transitive closure on acyclic digraphs
- Simon
- 1988
(Show Context)
Citation Context ...chain of length k. 8.1 Incremental Redundancy Elimination The enumeration algorithm in Section 5 has to maintain the transitive reduction of the dominance graph G. This can be done in time O(nm) (see =-=[29,30]-=-). But for all recursive calls of the algorithm the reduction can be computed much faster, only the top-level needs to do the full-fledged reduction. This is because the instances on which recursive c... |

37 | Describing discourse semantics
- Gardent, Webber
- 1998
(Show Context)
Citation Context ...d numerous applications in computational linguistics: they have been used for grammar formalisms [VS92, RVSW95, DT99, Per00], in natural language semantics [Mus95, ENRX98], and for discourse analysis =-=[GW98]-=-. The two most important computational tasks for dominance constraints are satisfiability testing – does the constraint describe a tree? – and enumerating solutions, i.e. the described trees. But as s... |

37 | Dominance constraints: Algorithms and complexity
- Koller, Niehren, et al.
- 1998
(Show Context)
Citation Context ... important computational tasks for dominance constraints are satisfiability testing – does the constraint describe a tree? – and enumerating solutions, i.e. the described trees. But as shown recently =-=[KNT01]-=-, testing satisfiability is an NP-complete problem. Earlier attempts at processing dominance constraints [Cor94, VSWR95, DN00] all suffer from this fact. This has shed doubt on their practical usefuln... |

36 | Parsing with tree descriptions: a constraint-based approach - Duchier, Thater - 1999 |

27 | On determining the consistency of partial descriptions of trees - Cornell - 1994 |

22 | Generalized automata theory with an application to a decision-problem of second-order logic - Thatcher, Wright - 1968 |

21 | Dealing with Ambiguities by Underspeci Construction, Representation and Deduction - Reyle - 1993 |

20 | Dominance constraints with set operators - Duchier, Niehren - 2000 |

19 | A polynomial-time fragment of dominance constraints
- Koller, Mehlhorn, et al.
- 2000
(Show Context)
Citation Context ...tion problem (investigated in [ADK + 01]). Second, we show that the configurability of dominance graphs is linear time equivalent to the satisfiability of normal dominance constraints (first shown in =-=[KMN00]-=-). Third, we characterize the configurability of dominance graphs as the absence of certain cycles, which we finally test for by reduction to a matching problem. We also discuss how to use the efficie... |

16 |
Minimal Recursion Semantics. An Introduction. Manuscript, available at ftp://csli-ftp.stanford.edu/linguistics/sag/mrs.ps.gz
- Copestake, Flickinger, et al.
- 1997
(Show Context)
Citation Context ...se variant of the configuration problem of dominance graphs where closed leaves are permitted in addition. This variant is more general but also relevant for applications in computational linguistics =-=[20,21]-=-. We show that configurability is already NPcomplete for the more general dominance graphs. Nevertheless, the presented algorithms can still help to solve this alternative problem more efficiently. Pl... |

14 | Ordering constraints over feature trees
- Müller, Niehren, et al.
(Show Context)
Citation Context ... results on satisfiability. The variant of dominance constraints we employ describes constructor trees – ground terms over a signature of function symbols – rather than feature trees as considered in =-=[25,5,26,27]-=-. 3.1 Trees and Constructor Trees We assume a finite or infinite signature Σ with function symbols f, g, . . ., each of which is equipped with an arity ar(f) ≥ 0. Constants are function symbols of ari... |

13 | Unique maximum matching algorithms
- Gabow, Kaplan, et al.
- 1999
(Show Context)
Citation Context ..., and since C is edge-simple, we can conclude that C visits l only once and hence contains only one dominance edge incident to l. This proves that C is a hypernormal cycle in Gu. 16 Q2 lsGabow et al. =-=[GKT01]-=- gave an algorithm which can decide whether A contains an alternating cycle with respect to M in time O(m ′ ), where m ′ is the number of edges in A. Now, we want to bound the size of the auxiliary gr... |

11 | Tree descriptions, constraints and incrementality
- Duchier, Gardent
- 2001
(Show Context)
Citation Context ...al solved form of G(ϕ). 18sT 1 m B + 1 l B + 1 Fig. 10.1. The dominance graph constructed in the reduction of 3-partition. sketched above; “old” refers to the dominance constraint solver described in =-=[DG01]-=-. 10. Dominance Graphs with Closed Leaves. A slight extension of the configuration problem by closed leaves becomes NP-complete again. This is interesting in its own right because it shows where the f... |

10 | 2002. Processing underspecified semantic representations in the constraint language for lambda structures
- Erk, Koller, et al.
(Show Context)
Citation Context ...want to extend the presented graph algorithm to more expressive languages than normal dominance constraints. In particular, one might try to find a polynomial time fragment of parallelism constraints =-=[EKN02]-=- that are also useful for computational linguistics [EKN01]. On the other hand side, it might be worthwhile in general to investigate graph algorithms for other problems in the areas constraint progra... |

10 | A reduct-and-closure algorithm for graphs - Goralcikova, Koubek - 1979 |

9 | A axiomatization of the theory of trees - Backofen, Rogers, et al. - 1995 |

9 | Feature constraint logics for uni grammars - Smolka - 1992 |

8 | An efficient algorithm for the configuration problem of dominance graphs - Althaus, Duchier, et al. - 2000 |

5 | Two accounts of scope availability and semantic underspecification
- Willis, Manandhar
- 1999
(Show Context)
Citation Context ... we presented eliminates any doubts about the computational practicability of dominance constraints which were raised by the NP-completeness result for the general language [13] and expressed e.g. in =-=[33]-=-. First experiments confirm the efficiency of the new algorithm – it is superior to the NP algorithms especially on larger constraints. Two main directions are to be pursued in the future. On the one ... |

4 | From intuitionistic proof nets to interaction grammars - PERRIER |

3 | An ecient algorithm for the con problem of dominance graphs - Althaus, Duchier, et al. - 2001 |

3 | Processing underspeci semantic representations in the constraint language for lambda structures - Erk, Koller, et al. |

2 | An improper treatment of quanti in ordinary english - Hobbs - 1983 |