A parametric study of matchings and coverings in weighted graphs

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University of Michigan, 1967 - Mathematics - 340 pages
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Given a weighted graph, a matching is a subset of the edges such that no two edges of the subset are incident to the same vertex. A covering is a subset of the edges such that each vertex is incident to at least one edge of the subset. Two problems are to find a maximum weight matching and minimum weight covering. Edmonds has developed an algorithm of algebraic growth to find a maximum matching. A review of this method is presented. A new algorithm for the solution of the minimum covering problem is developed. Under certain conditions, Lagrange multiplier methods can be applied to discrete programming problems. This is called the parametric approach, and is used here to find maximum matchings, subject to the condition that the number of edges in the solution is some fixed number, k. A study of methods for this 'k-cardinality' problem indicates several improvements in Edmonds' matching algorithm. The parametric technique is extended to solve the minimum k-cardinality covering problem. This algorithm is shown to be a generalization of Kruskal's algorithm for a minimum spanning tree. This leads to a discussion of matroid systems and greedy algorithms, and their relationship to the parametric approach. Matchings and coverings in weighted graphs have applications in resource allocation, diagnostics, and large scale system design.

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