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Combinatorial and algebraic tools for optimal multilevel algorithms
, 2007
"... This dissertation presents combinatorial and algebraic tools that enable the design of the first linear work parallel iterative algorithm for solving linear systems involving Laplacian matrices of planar graphs. The major departure of this work from prior suboptimal and inherently sequential approac ..."
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This dissertation presents combinatorial and algebraic tools that enable the design of the first linear work parallel iterative algorithm for solving linear systems involving Laplacian matrices of planar graphs. The major departure of this work from prior suboptimal and inherently sequential approaches is centered around: (i) the partitioning of planar graphs into fixed size pieces that share small boundaries, by means of a local ”bottomup ” approach that improves the customary ”topdown ” approach of recursive bisection, (ii) the replacement of monolithic global preconditioners by graph approximations that are built as aggregates of miniature preconditioners. In addition, we present extensions to the theory and analysis of Steiner tree preconditioners. We construct more general Steiner graphs that lead to natural linear time solvers for classes of graphs that are known a priori to have certain structural properties. We also present a graphtheoretic approach to classical algebraic multigrid algorithms. We show that their design can be
Vertex Rankings of Chordal Graphs and Weighted Trees
, 2006
"... In this paper we consider the vertex ranking problem of weighted trees. We show that this problem is strongly NPhard. We also give a polynomialtime reduction from the problem of vertex ranking of weighted trees to the vertex ranking of (simple) chordal graphs, which proves that the latter problem ..."
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In this paper we consider the vertex ranking problem of weighted trees. We show that this problem is strongly NPhard. We also give a polynomialtime reduction from the problem of vertex ranking of weighted trees to the vertex ranking of (simple) chordal graphs, which proves that the latter problem is NPhard. In this way we solve an open problem of Aspvall and Heggernes. We use this reduction and the algorithm of Bodlaender et al.'s for vertex ranking of partial $k$trees to give an exact polynomialtime algorithm for vertex ranking of a tree with bounded and integer valued weight functions. This algorithm serves as a procedure in designing a PTAS for weighted vertex ranking problem of trees with bounded weight functions.