On the Application of the Minimum Degree Algorithm to Finite Element Systems

A supernodal formulation of vertex colouring with applications in course timetabling

“…As is well known, the problem of finding the minimum cardinality of a clique partition,χ(G), is N P-Hard in general graphs and as hard to This relation has been studied previously in the context of pivoting in matrix factorisation, in connection with mass elimination (George & McIntyre, 1978), supervariables (Duff & Reid, 1983), and prototype vertices (Eisenstat et al, 1984). It is easy to observe the indistinguishability relation is reflexive, symmetric, and transitive.…”

“…This can also be viewed as the supernodal integer programming formulation of graph colouring, where supernode of George and McIntyre (1978) is the complete subset of vertices of a graph where each two vertices have the same neighbours outside of the subset. It remains to be seen, if the transformation could be used in conjuction with other formulations of multicolouring.…”

“…These seven encodings of graph colouring, often together with the corresponding integer programming formulations, are surveyed in Section 2. In Section 3, we first review the concept of a "supernode", a complete subset of vertices of a graph, where each two vertices have the same neighbourhoods outside of the subset; this concept has been described many times previously (George & McIntyre, 1978;Duff & Reid, 1983;Eisenstat, Elman, Schultz, & Sherman, 1984). See Figure 1 for a simple illustration.…”

Review of literature on the finite-element solution of the equations of two-dimensional surface-water flow in the horizontal plane

“…George and Liu (1978) incorporate a modified form of the Gibbs-Poole-Stockmeyer algorithm with the reverse Cuthill-McKee strategy to form a technique that successfully reduces bandwidth in finiteelement networks with appendages and (or) holes. George and Mcintyre ( 1978) describe another nodal renumbering scheme based on a similar minimum-degree concept and the grouping of nodes into cliques (that is, sets of nodes, all of which are adjacent to one another). King ( 1970) developed an algorithm based on the concept of a "minimum front growth" criterion.…”

Combinatorial Problems in Solving Linear Systems

“…The methods used to achieve this goal are described in [65,85,86]. The survey by George and Liu [88] lists, inter alia, the following improvements and algorithmic follow-ups: mass eliminations [90], where it is shown that, in case of finite-element problems, after a minimum degree vertex is eliminated a subset of adjacent vertices can be eliminated next, together at the same time; indistinguishable nodes [87], where it is shown that two adjacent nodes having the same adjacency can be merged and treated as one; incomplete degree update [75], where it is shown that if the adjacency set of a vertex becomes a subset of the adjacency set of another one, then the degree of the first vertex does not need to be updated before the second one has been eliminated; element absorption [66], where based on a compact representation of elimination graphs, redundant structures (cliques being subsets of other cliques) are detected and removed; multiple elimination [134], where it was shown that once a vertex v is eliminated, if there is a vertex with the same degree that is not adjacent to the eliminated vertex, then that vertex can be eliminated before updating the degree of the vertices in adj (v), that is the degree updates can be postponed; external degree [134], where instead of the true degree of a vertex, the number of adjacent and indistinguishable nodes is used as a selection criteria. Some further improvements include the use of compressed graphs [11], where the indistinguishable nodes are detected even before the elimination process and the graph is reduced, and the extensions of the concept of the external degree [44,94].…”