Perfect Gaussian Elimination and Chordality By Shashank Rao

 Contents  Problem statement  Construction of graph  R-W problem leading to graph problem  Bipartite graphs  Perfect elimination bipartite graphs  Simplicial edges  Solution  Chordality  Special properties

 Problem Statement  When performing Gaussian elimination on sparse matrices an arbitrary choice of pivots may result in the unfortunate side effect of turning zeros to non zero values, which may possibly even lead to dense matrix.

 Construction Of Graph  An instance of the problem can be represented by a {0, 1} matrix M where Mi,j = 1 denotes that the original matrix contains a non-zero value at element (i, j).  We can construct the bipartite graph G[M] with vertices corresponding to the rows and columns in M where vertices i and j are adjacent iff Mi,j is nonzero

 Cont…  For example the matrix (a) has the corresponding bipartite graph (b)

 Graph Problem To show that a bipartite graph is a perfect elimination bipartite graph. If it is a perfect elimination bipartite graph then how to find its perfect edge elimination scheme.

 Bipartite Graphs A bipartite graph is an undirected graph whose vertices can be partitioned into disjoint sets X and Y such that every edge has one vertex in X and the other in Y.

 Perfect Elimination Bipartite Graphs

 Simplicial Edges An edge xy is called simplicial if Adj (x) U Adj(y) induces a complete bipartite subgraph. For the following bipartite graph (b) the simplicial edges are represented in bold lines.

Example: For example the following graph has the perfect elimination ordering [x1y1,x2y2,x3y3,x4y4]

 Solution We repeatedly find a simplicial edge of a bipartite graph and delete all the edges adjacent to it.If we succeed in eliminating all edges, then the bipartite graph was perfect elimination. Such procedure can be implemented to run in a polynomial time.

 Chordal Bipartite graphs A bipartite graph is called chordal bipartite if every cycle of length greater than 4 has a cord.

 Special Properties Every chordal bipartite graph is a perfect elimination bipartite graph. Chordal bipartite graphs give a characterization of the matrices in an important class of linear programming problems for which the greedy heuristic approach gives an optimum solution.

 Cont.. It is found that removal of simplicial edge from a chordal bipartite graph produces a chordal bipartite graph. As consequence occurrence of arithmetic zeros will not terminate perfect Gaussian elimination on sparse matrices having associated a chordal bipartite graph.

 References M. C. Golumbic, A note o n perfect Gaussian elimination Fall2014/Lecture4.pdfhttp:// Fall2014/Lecture4.pdf 5. urnals/DAM_BomhoffManthey_Bisimplicial.pdf

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