E. Dahlhaus. Minimal elimination ordering inside a given chordal graph. In Proceedings 23rd International Workshop on Graph-Theoretic Concepts in Computer Science WG'97, pages 132--143. Springer Verlag, Lecture Notes in Computer Science, vol. 1335, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....is NP hard [11] In this extended abstract, we study the problem of finding a minimal triangulation. A minimal triangulation H of a given graph G is a triangulation such that no subgraph of H is a triangulation of G. Several practical algorithms exist for finding minimal triangulations [1] 2] [3], 5] 8] 9] One such classical algorithm, called Lex M [9] is derived from the Lex BFS (lexicographic breadth first search) algorithm [9] for recognizing chordal graphs. Both Lex BFS and Lex M use lexicographic labels of the unprocessed vertices. As processing continues, the remaining labels ....

E. Dahlhaus, Minimal elimination ordering inside a given chordal graph, in Graph Theoretical Concepts in Computer Science - WG '97, R. H. Mohring, ed., Springer Verlag, 1997, pp. 132--143. Lecture Notes in Computer Science 1335.

....is known theoretically as to its quality. It has in fact been analyzed theoretically only to a limited extent, which makes it dicult to gain control over this heuristic in order to improve it yet further, although recent research has been done on algorithms for low ll minimal triangulations [5, 7, 23]. In this paper, we use recent graph theoretical results on minimal triangulation and minimal separation to explain, at least in part, why MD yields such good results. In fact, it turns out that one of the reasons why MD works so well is that the EG algorithm is remarkably robust, in the sense ....

E. Dahlhaus. Minimal elimination ordering inside a given chordal graph. In R. H. Mohring, editor, Graph Theoretical Concepts in Computer Science, pages 132-143. Springer Verlag, 1997. Lecture Notes in Computer Science 1335.

....that removes fill edges from G ff in order to solve this problem. The complexity of their algorithm is O(f(m f ) where f is the number of filled edges in the initial simplicial filled graph G ff , thus the algorithm works fast for elimination orderings resulting in low fill. Dahlhaus [11] later presented an algorithm for solving the same problem with a time complexity evaluated as O(nm) which uses a clique tree representation of the graph as an intermediate structure. The most recent among algorithms solving the Minimal Triangulation Sandwich Problem is presented by Peyton [25] ....

E. Dahlhaus. Minimal elimination ordering inside a given chordal graph. In R. H. Mohring, editor, Graph Theoretical Concepts in Computer Science, pages 132--143. Springer Verlag, 1997. Lecture Notes in Computer Science 1335.

....f) of our algorithm depends on structural properties of the filled graph, and our tests indicate that this worst case bound is usually not met. A preliminary version [2] of this paper was presented at the Fifth Scandinavian Workshop on Algorithm Theory in 1996. Subsequently Dahlhaus presented in [6] an O(ne) algorithm to solve the same problem. These two algorithms thus have the same worst case asymptotic time complexity when the fill is linear in the number of vertices, i.e. when f = Theta(n) while O(f(e f) wins if f = o(n) and O(ne) wins if f = n) Our tests indicate that there are ....

E. Dahlhaus. Minimal elimination ordering inside a given chordal graph. In Graph Theoretical Concepts in Computer Science, pages 132--143. Springer Verlag, 1997. Lecture Notes in Computer Science 1335.

....8 partition defines a metric in an obvious way. One might build up a minimum spanning tree, and one gets a first approximation of a minimal elimination ordering. This approach might also be helpful to solve the sandwich problem to get a minimal elimination inside a given chordal supergraph (see [3, 5]) ....

E. Dahlhaus, Minimal Elimination Ordering inside a Given Chordal Graph, WG 97 (R. Mohring ed.), LLNCS 1335, pp. 132-143.

....But still the problem is an interesting one, because the algorithm in this paper might be helpful as a subprocedure to solve the problem to transform the fill in of a nested dissection ordering of a planar graph to a minimal fill in. For general graphs the problem has been considered in [2] and in [4]. The complexities of the algorithms are not linear. For planar graphs it might be interesting to get a linear time algorithm. In section 2 we will introduce the basic notation. In section 3 we distance levels (denoted by L i ) leed to a first approximation of a minimal elimination ordering ....

....sorting. We order the vertices v of G in first priority with respect to c(v) and in second priority with respect to 00 . This ordering is a minimal elimination ordering due to previous discussions and can be determined in linear time. Q.E.D. 6 Conclusions One could ask as in in [2] or [4] whether one could combine the nested dissection algorithm for planar graphs (see [10] and [1] with a planar minimal elimination algorithm, such that we get the performance of the nested dissection algorithm affecting the number of fill in edges and an inclusion minimal fill in. We would like to ....

E. Dahlhaus, Minimal Elimination Ordering inside a Given Chordal Graph, WG 97 (R. Mohring ed.), LLNCS 1335, pp. 132-143.

....partition defines a metric in an obvious way. One might build up a minimum spanning tree, and one gets a first approximation of a minimal elimination ordering. This approach might also be helpful to solve the sandwich problem to get a minimal elimination inside a given chordal supergraph (see [3, 5]) ....

E. Dahlhaus, Minimal Elimination Ordering inside a Given Chordal Graph, WG 97 (R. Mohring ed.), LLNCS 1335, pp. 132-143.

....the subset relation (Minimal Elimination Ordering (MEO) This problem can be solved in O(nm) time [15] Unfortunately, a minimal fill in can have a size that is far from the size of a fill in of minimum cardinality. This is shown by the following example. A preliminary version appeared in WG 97 [6], partially supported by ESPRIT Long Term Research Project Nr. 20244 (ALCOM IT) 1 Figure 1: A graph with a small minimum fill in and a large minimal fill in The vertex set of G consists of a vertex set V = X [ fxg [ fyg and an edge set fxvjv 2 Xg [ fyvjv 2 Xg (see figure 1) Numbering x first ....

E. Dahlhaus, Minimal Elimination Ordering inside a Given Chordal Graph, GraphTheoretic Concepts in Computer Science, LLNCS 1335 (1997), pp. 132-143.

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E. Dahlhaus. Minimal elimination ordering inside a given chordal graph. In Proceedings 23rd International Workshop on Graph-Theoretic Concepts in Computer Science WG'97, pages 132--143. Springer Verlag, Lecture Notes in Computer Science, vol. 1335, 1997.

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E. Dahlhaus. Minimal elimination ordering inside a given chordal graph. In R. H. Mohring, editor, Graph Theoretical Concepts in Computer Science - WG '97, LNCS 1335, pages 132-143. Springer Verlag, 1997. 17

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E. Dahlhaus, Minimal elimination ordering inside a given chordal graph, in Graph Theoretical Concepts in Computer Science - WG '97, 13 R. H. Mohring, ed., Springer Verlag, 1997, pp. 132-143. Lecture Notes in Computer Science 1335.

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