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16
Hybrid backtracking bounded by treedecomposition of constraint networks
 Artificial Intelligence
, 2003
"... We propose a framework for solving CSPs based both on backtracking techniques and on the notion of treedecomposition of the constraint networks. This mixed approach permits us to define a new framework for the enumeration, which we expect that it will benefit from the advantages of two approaches: ..."
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Cited by 51 (16 self)
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We propose a framework for solving CSPs based both on backtracking techniques and on the notion of treedecomposition of the constraint networks. This mixed approach permits us to define a new framework for the enumeration, which we expect that it will benefit from the advantages of two approaches: a practical efficiency of enumerative algorithms and a warranty of a limited time complexity by an approximation of the treewidth of the constraint networks. Finally, experimental results allow us to show the advantages of this approach. 1
Generating All The Minimal Separators Of A Graph
, 1999
"... We present an efficient algorithm which computes the set of minimal separators of a graph in O(n³) time per separator, thus gaining a factor of n² on the current besttime algorithms for this problem. Our process is based on a new structural result, derived from the work of Kloks and Kratsch on list ..."
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Cited by 35 (9 self)
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We present an efficient algorithm which computes the set of minimal separators of a graph in O(n³) time per separator, thus gaining a factor of n² on the current besttime algorithms for this problem. Our process is based on a new structural result, derived from the work of Kloks and Kratsch on listing all the minimal separators of a graph.
Recognizing Weakly Triangulated Graphs by Edge Separability
, 2000
"... . We apply Lekkerkerker and Boland's recognition algorithm for triangulated graphs to the class of weakly triangulated graphs. This yields a new characterization of weakly triangulated graphs, as well as a new O(m 2 ) recognition algorithm which, unlike the previous ones, is not based on the notio ..."
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Cited by 27 (13 self)
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. We apply Lekkerkerker and Boland's recognition algorithm for triangulated graphs to the class of weakly triangulated graphs. This yields a new characterization of weakly triangulated graphs, as well as a new O(m 2 ) recognition algorithm which, unlike the previous ones, is not based on the notion of a 2pair, but rather on the structural properties of the minimal separators of the graph. It also gives the strongest relationship to the class of triangulated graphs that has been established so far. CR Classification: G.2.2, F.2.2 Key words: Weakly triangulated graphs, graph recognition, graph characterization, minimal separators, triangulated graphs. 1. Introduction Weakly triangulated graphs were introduced by Hayward [11] as a natural extension of the perfect class of triangulated graphs. A graph is triangulated, or chordal, if it does not contain a chordless cycle on four or more vertices. A graph is weakly triangulated if neither the graph nor its complement contains a chordl...
A WideRange Algorithm for Minimal Triangulation from an Arbitrary Ordering
, 2003
"... We present a new algorithm, called LBTriang, which computes minimal triangulations. ..."
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Cited by 26 (19 self)
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We present a new algorithm, called LBTriang, which computes minimal triangulations.
Triangulated and Weakly Triangulated Graphs: Simpliciality in Vertices and Edges
, 2001
"... We extend Dirac's characterization by the minimal separators of a triangulated graph to a new characterization for weakly triangulated graphs, and use this to interpret the known properties of weakly triangulated graphs as an extension of the corresponding properties of triangulated graphs. Our new ..."
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Cited by 10 (7 self)
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We extend Dirac's characterization by the minimal separators of a triangulated graph to a new characterization for weakly triangulated graphs, and use this to interpret the known properties of weakly triangulated graphs as an extension of the corresponding properties of triangulated graphs. Our new insight, applied to a result of Rose on triangulated graphs, enables us to bound the number of minimal separators of a weakly triangulated graph.
A vertex incremental approach for maintaining chordality
 Discrete Mathematics
, 2006
"... For a chordal graph G = (V, E), we study the problem of whether a new vertex u � ∈ V and a given set of edges between u and vertices in V can be added to G so that the resulting graph remains chordal. We show how to resolve this efficiently, and at the same time, if the answer is no, specify a maxim ..."
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Cited by 10 (5 self)
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For a chordal graph G = (V, E), we study the problem of whether a new vertex u � ∈ V and a given set of edges between u and vertices in V can be added to G so that the resulting graph remains chordal. We show how to resolve this efficiently, and at the same time, if the answer is no, specify a maximal subset of the proposed edges that can be added along with u, or conversely, a minimal set of extra edges that can be added in addition to the given set, so that the resulting graph is chordal. In order to do this, we give a new characterization of chordal graphs and, for each potential new edge uv, a characterization of the set of edges incident to u that also must be added to G along with uv. We propose a data structure that can compute and add each such set in O(n) time. Based on these results, we present an algorithm that computes both a minimal triangulation and a maximal chordal subgraph of an arbitrary input graph in O(nm) time, using a totally new vertex incremental approach. In contrast to previous algorithms, our process is online in that each new vertex is added without reconsidering any choice made at previous steps, and without requiring any knowledge of the vertices that might be added subsequently. 1
Computing and exploiting treedecompositions for solving constraint networks
 In Proceedings of CP
, 2005
"... Abstract. Methods exploiting treedecompositions seem to provide the best approach for solving constraint networks w.r.t. the theoretical time complexity. However, they have not shown a real practical interest yet. In this paper, we study several methods for computing a rough optimal treedecomposit ..."
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Cited by 7 (5 self)
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Abstract. Methods exploiting treedecompositions seem to provide the best approach for solving constraint networks w.r.t. the theoretical time complexity. However, they have not shown a real practical interest yet. In this paper, we study several methods for computing a rough optimal treedecomposition and assess their relevance for solving CSPs. 1
Towards Improving Phylogeny Reconstruction with CombinatorialBased Constraints on an Underlying Family of Graphs
"... We address the issue of improving phylogenetic data by modifying a dissimilarity matrix so that it will be closer to an additive matrix (the matrix corresponding to a phylogeny), in the case where the thresholds are known to be too low. Our approach uses the thresholds defined by the input matrix t ..."
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Cited by 4 (2 self)
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We address the issue of improving phylogenetic data by modifying a dissimilarity matrix so that it will be closer to an additive matrix (the matrix corresponding to a phylogeny), in the case where the thresholds are known to be too low. Our approach uses the thresholds defined by the input matrix to define a family of pairwise inclusive undirected graphs. We run an experimental analysis on a previous algorithm, which leads us to a combinatorial study of all the successions of subgraphs on four vertices which an additive matrix can span, thereby extracting some interesting rules on the additive family of graphs.
Decomposition By Clique Minimal Separators
 Res. Rep. 97213
, 1999
"... Clique decomposition consists in repeatedly finding a clique, the removal of which leaves the graph disconnected, and copying this clique into each of the connected components thus defined. This was implemented efficiently by Tarjan in 1985, and is an important tool for a Divide and Conquer appr ..."
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Cited by 2 (2 self)
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Clique decomposition consists in repeatedly finding a clique, the removal of which leaves the graph disconnected, and copying this clique into each of the connected components thus defined. This was implemented efficiently by Tarjan in 1985, and is an important tool for a Divide and Conquer approach to many hard graph problems, as it splits the graph into smaller subgraphs, while preserving holes and antiholes. We define a new sort of decomposition, which uses at each step only clique separators that are also minimal separators of the graph, and has the additional property of being unique. We characterize the atoms obtained and calculate their number, and show the coherence of this decomposition, as any clique minimal separator chosen in some subgraph is a clique minimal separator in its own right for the original input graph. We propose an extension of Tarjan's algorithm which computes the unique decomposition while retaining the complexity. Keywords: Clique minimal se...
Computing and exploiting treedecomposition for (Max)CSP
, 2005
"... Methods exploiting the treedecomposition notion seem to provide the best approach for solving constraint networks w.r.t. the theoretical time complexity. Nevertheless, they have not shown a real practical interest yet. So, in this paper, we first study several methods for computing an approximate o ..."
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Cited by 2 (1 self)
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Methods exploiting the treedecomposition notion seem to provide the best approach for solving constraint networks w.r.t. the theoretical time complexity. Nevertheless, they have not shown a real practical interest yet. So, in this paper, we first study several methods for computing an approximate optimal treedecomposition before assessing their relevance for solving CSPs. Then, we propose and compare several strategies to achieve the best depthfirst traversal of the associated cluster tree w.r.t. CSP solving. These strategies concern the choice of the root cluster (i.e. the first visited cluster) and the order according to which we visit the sons of a given cluster. Finally, we propose a new decomposition strategy and heuristics which both rely on probabilistic criteria and which significantly improve the runtime. 1