A. POTHEN. The complexity of optimal elimination trees. Technical Report CS-88-13, Pennsylvania State University, U.S.A., 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....[IRV88a, Lei80, SDG92] We then review the results on the vertex ranking problem. The vertex ranking problem was posed in 1988 by Iyer et al. in relation with the applications in VLSI layout and in manufacturing systems [IRV88a] Pothen proved that the vertexranking problem is NP hard in general [BDJ94, Pot88], and hence it is very unlikely that there is a polynomial time algorithm for solving the problem for general graphs 1.1. Backgrounds 5 (a) b) 7 6 6 1 5 4 3 2 1 1 5 4 3 2 1 2 1 1 3 4 4 2 5 1 3 Figure 1.2: a) An optimal vertex ranking of a graph G, and (b) an optimal ....

....number, and is denoted by r c (G) A c vertex ranking of G using r c (G) ranks is called an optimal c vertex ranking of G. The c vertex ranking problem is to find an optimal c vertex ranking of a given graph. The problem is NP hard in general, since the ordinary vertex ranking problem is NP hard [BDJ94, Pot88]. Figure 1.3(a) depicts an optimal 2 vertex ranking of a graph G using three ranks, where vertex names are drawn in circles and ranks next to the circles. The c vertex ranking problem of a graph G is equivalent to finding a c vertex separator tree of G having the minimum height. Consider the ....

A. Pothen, The complexity of optimal elimination trees, Technical Report CS-88-13, Pennsylvania State University, USA, 1988.

....tree height measures the parallel time needed to factor A by Gaussian elimination with unlimited processors. All these parameters depend on the ordering on the rows and columns of A. Unfortunately, determining the orderings that give the optimal values of these parameters is NP complete [2, 9, 25]. Therefore we have to be content with approximations. The main point of this paper is that minimum front size and elimination tree height are intimately related to three other graph parameters, namely treewidth, pathwidth, and the size of separators in subgraphs of the graph, and that all these ....

....We will show that then we can solve the mutual independent set problem (MUS) in polynomial time. The MUS problem is the following: Given a bipartite graph B = P; Q; E) are there sets V 1 P and V 2 Q with j V 1 j=jV 2 j= k, such that no edge joins a vertex in V 1 to a vertex in V 2 Pothen [25] shows that this problem is NP complete. 14 Let B = P; Q; E) be a bipartite graph, with vertex sets P and Q and edge set E. Graph B is a chain graph if the adjacency sets of vertices in P form a chain, that is, if the vertices of P can be ordered so that Adj(v 1 ) Adj(v 2 ) Delta Delta ....

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A. Pothen. The complexity of optimal elimination trees. Technical Report CS--88--16, The Pennsylvania State University, 1988.

....close links to that of finding orderings which minimize the height of the elimination tree, which is important in the context of parallel sparse elimination. As is well known, the problem of finding orderings which result in elimination trees of minimum height for general graphs is NP hard; see [32]. 0 B B B B B B B B B B B B B B B f f f f f f f f f f f f 1 C C C C C C C C C C C C C C C A 0 B B B B B B B B B B B B B B B 1 C C C C C C C C C C C C C C C A Fig. 1.1: Patterns of the matrix A (left) and of L Gamma1 , the inverse of its ....

A. Pothen, The Complexity of Optimal Elimination Trees, Technical Report CS-88-13, Pennsylvania State University, 1988.

.... [17, 26] The vertex ranking problem Given a graph G and a positive integer t, decide whether r (G) t is NP complete even when restricted to cobipartite graphs since Pothen has shown that the equivalent minimum elimination tree height problem remains NP complete on cobipartite graphs [20]. A short proof of the NP completeness of vertex ranking is given in Section 3. Much work has been done in finding optimal rankings of trees. For trees there is a lineartime algorithm finding an optimal vertex ranking [24] For the closely related edge ranking problem on trees a O(n 3 ) ....

....Remark 3 this problem is equivalent to the vertex ranking problem Given a graph G and a positive integer t, decide whether r (G) t when restricted to line graphs. On the other hand, it is a consequence of the NP completeness of the minimum elimination tree height problem shown by Pothen in [20] and the equivalence of this problem with the vertex ranking problem [6, 7] that the latter is NP complete even when restricted to graphs that are the complement of bipartite graphs, the so called cobipartite graphs. For reasons of self containedness, we start with a short proof of the ....

A. POTHEN. The complexity of optimal elimination trees. Technical Report CS-88-13, Pennsylvania State University, U.S.A., 1988.

....edge ranking. 2 1 2 1 3 4 1 1 3 5 4 1 2 3 Figure 1: An optimal edge ranking of a tree. The node ranking and edge ranking problems have been studied by a number of researchers as they find applications in different context [1, 6, 10, 11, 14] Both problems are now known to be NP hard for graphs [12, 9, 7]. Nonetheless, in most applications, the graphs in concern are restricted to trees only. This initiates the study of node ranking and edge ranking of trees. With respect to trees, the node ranking problem seems easier than the edge ranking problem. A linear time algorithm for finding an optimal ....

A. Pothen, The Complexity of Optimal Elimination Trees, Technical Report CS-88-13, The Pennsylvania State University, 1988.

....PAP T and keeps the Cholesky factor L of PAP T sparse. It is however not difficult to show that these two requirements might be in conflict with each other. To make matters worse, both the problem of minimizing fill and the problem of finding the lowest possible etree are known to be NP hard [21, 26]. There are methods for finding orderings that give low etrees and few fill edges. Nested dissection is a method for ordering G that was developed to reduce fill [5, 6] and has also been shown to produce low etrees [13] Another approach is first to compute a fill reducing ordering P and then to ....

A. Pothen, The complexity of optimal elimination trees, Tech. Report CS-88-13, Pennsylvania State University, 1988.

....between our work and these results is that we start with a chordal completion of a graph, and focus our efforts on finding parallel elimination orders with linear fill. 1. 4 Related work: Height Ignoring fill, computing an elimination ordering for a given graph with minimum height is NP hard [30], and remains so even if an additive error in the estimate of the height is allowed [5] Pan and Reif give one of the first analyses of the parallel height of nested dissection orderings as well as how nested dissection can be used for solving the shortest path problem in graphs [28, 27] ....

A. Pothen. The complexity of optimal elimination trees. Technical Report CS-88-16, Department of Computer Science, The Pennsylvania State University, 1988.

....see [23] 26] 30] and [31] The techniques proposed in these papers provide as a side effect a decrease in the overlap between closures of the graph vertices. Note that the problem of finding orderings which result in elimination trees of minimum height for general graphs is NP hard; see [35]. 0 B B B B B B B B B B B B B B B f f f f f f f f f f f f 1 C C C C C C C C C C C C C C C A 0 B B B B B B B B B B B B B B B 1 C C C C C C C C C C C C C C C A Fig. 1.1: Patterns of the matrix A (left) and of L Gamma1 , the inverse of its ....

A. Pothen, The Complexity of Optimal Elimination Trees, Technical Report CS-88-13, Pennsylvania State University, 1988.

....of one vertex from each of the connected components constitutes an independent set, and hence can be eliminated in parallel. The height of a graph is the minimum number of parallel elimination steps needed to eliminate all of the vertices of the graph. Finding the height of a graph is NP hard [24], but minimum height orders may be found for specific classes of graphs. Aspvall and Heggernes [5] present an algorithm that finds elimination orders with minimum height for interval graphs, but the orders produced have not been analyzed in terms of fill. A family of chordal graphs for which any ....

A. Pothen. The complexity of optimal elimination trees. Technical Report CS-88-16, Department of Computer Science, The Pennsylvania State University, 1988.

....filled graphs and elimination trees. A perfect elimination ordering is an ordering that results in no fill in edges. Finding orderings that minimize the fill in size and finding orderings that minimize the elimination tree height are NP hard problems, as shown by Yannakakis in [26] and Pothen in [20]. In this paper, we assume that the graphs we are working with are connected. The results can easily be extended to include disconnected graphs. A separator S ae V is a set of vertices whose removal from G disconnects G into two or more connected components. A separator S is minimal if no proper ....

A. Pothen, The complexity of optimal elimination trees, Tech. Report CS-88-13, Pennsylvania State University, 1988.

....between our work and these results is that we start with a chordal completion of a graph, and focus our efforts on finding parallel elimination orders with linear fill. 1. 4 Related work: Height Ignoring fill, computing an elimination order for a given graph with minimum height is NP hard [33], and remains so even if an additive error in the estimate of the height is allowed [5] Pan and Reif give one of the first analyses of the parallel height of nested dissection orders and show how nested dissection can be used for solving the shortest path problem in graphs [31, 30] Bodlaender et ....

A. Pothen. The complexity of optimal elimination trees. Technical Report CS-88-16, Department of Computer Science, The Pennsylvania State University, University Park, PA, 1988.

....form Ax = b using parallel Cholesky factorization where A is an nn matrix. Various algorithms for doing this exist (Liu [9] and Hafsteinsson [4] They all have in common that their speed depends on the height of the elimination tree of A. However it is known that this problem is NP hard (Pothen [12]) The other important parameter for Cholesky factorization is the amount of fill in the Cholesky factor L. It is also known that minimizing this quantity is NP hard (Yannakakis [13] Various heuristics for achieving a low elimination tree have been proposed. They are centered around two ....

A. Pothen, The complexity of optimal elimination trees, Tech. Report CS-88-13, Pennsylvania State University, 1988.

....log 2 n if O(n) fill is permitted. Hence the question: For what classes of graphs does increasing fill not lead to increased parallelism in sparse Cholesky factorization The problem of computing a vertex ordering that leads to a shortest elimination tree is NP complete for an arbitrary graph [19]. If G is chordal, then Liu [15] has shown that a scheme due to Jess and Kees that recursively eliminates a maximum independent subset of the simplicial vertices computes a shortest elimination tree of G over all PEOs of G. If G is a P 4 free chordal graph, then it has the property that in every ....

A. Pothen, The complexity of optimal elimination trees, Tech. Report CS-88-16, Computer Science, Pennsylvania State University, 1988.

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A. POTHEN. The complexity of optimal elimination trees. Technical Report CS-88-13, Pennsylvania State University, U.S.A., 1988.

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A. Pothen. \The complexity of optimal elimination trees". Technical Report CS-88-16, Department of Computer Science, The Pennsylvania State University, University Park, PA (1988).

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