Decomposition by clique separators

Chordal graphs play a central role in techniques for exploiting sparsityin large semidefinite optimization problems and in related convexoptimization problems involving sparse positive semidefinite matrices.Chordal graph properties are also fundamental to several classicalresults in combinatorial optimization, linear algebra, statistics,signal processing, machine learning, and nonlinear optimization. Thissurvey covers the theory and applications of chordal graphs, with anemphasis on algorithms developed in the literature on sparse Choleskyfactorization. These algorithms are formulated as recursions on eliminationtrees, supernodal elimination trees, or clique trees associatedwith the graph. The best known example is the multifrontal Choleskyfactorization algorithm, but similar algorithms can be formulated fora variety of related problems, including the computation of the partialinverse of a sparse positive definite matrix, positive semidefinite andEuclidean distance matrix completion problems, and the evaluation ofgradients and Hessians of logarithmic barriers for cones of sparse positivesemidefinite matrices and their dual cones. The purpose of thesurvey is to show how these techniques can be applied in algorithmsfor sparse semidefinite optimization, and to point out the connectionswith related topics outside semidefinite optimization, such as probabilisticnetworks, matrix completion problems, and partial separabilityin nonlinear optimization.

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Chordal Graphs and Semidefinite Optimization

http://www.ii.uib.no/~pinar/MinTriSurvey.pdf

Minimal triangulations of graphs: A survey

We define two types of bipartite graphs, chordal bipartite graphs and perfect elimination bipartite graphs, and prove theorems analogous to those of Dirac and Rose for chordal graphs (rigid circuit graphs, triangulated graphs). Our results are applicable to Gaussian elimination on sparse matrices where a sequence of pivots preserving zeros is sought. Our work removes the constraint imposed by Haskins and Rose that pivots must be along the main diagonal.

Perfect Elimination and Chordal Bipartite Graphs

We study the parameterized complexity of three NP-hard graph completion problems.
The minimum fill-in problem asks if a graph can be triangulated by adding at most k edges. We develop O(ck m) and O(k2mn+f(k)) algorithms for this problem on a graph with n vertices and m edges. Here f(k) is exponential in k and the constants hidden by the big-O notation are small and do not depend on k. In particular, this implies that the problem is fixed-parameter tractable (FPT).
The proper interval graph completion problem, motivated by molecular biology, asks if a graph can be made proper interval by adding no more than k edges. We show that the problem is FPT by providing a simple search-tree-based algorithm that solves it in O(ck m)-time. Similarly, we show that the parameterized version of the strongly chordal graph completion problem is FPT by giving an O(ck m log n)-time algorithm for it.
All of our algorithms can actually enumerate all possible k-completions within the same time bounds.

Tractability of Parameterized Completion Problems on Chordal, Strongly Chordal, and Proper Interval Graphs

Minimal triangulation of a graph and optimal pivoting order in a sparse matrix

This paper is concerned with eigenvalue approaches for the pole-zero calculations of the transfer function of a linear timeinvariant network. Combinatorial and sparse-matrix algorithms are fully used to increase numerical accuracy and computational speed in the two-sets-of-eigenvalues approach. Some matrix decompositionreduction algorithms are presented to simplify and stabilize the numerical eigenvalue-finding procedures.

Tatsuo Ohtsuki

Papers

Minimal triangulation of a graph and optimal pivoting order in a sparse matrix

A matrix decomposition-reduction procedure for the pole-zero calculation of transfer functions

Pole-Zero Calculation of Transfer Functions