Publication

Combinatorics and Algorithms for Augmenting Graphs

Dominique de Werra
2015

Abstract

The notion of augmenting graphs generalizes Berge’s idea of augmenting chains, which was used by Edmonds in his celebrated solution of the maximum matching problem. This problem is a special case of the more general maximum independent set (MIS) problem. Recently, the augmenting graph approach has been successfully applied to solve MIS in various other special cases. However, our knowledge of augmenting graphs is still very limited and we do not even know what the minimal infinite classes of augmenting graphs are. In the present paper, we find an answer to this question and apply it to extend the area of polynomial-time solvability of the maximum independent set problem.

Official source

https://infoscience.epfl.ch/entities/publication/dcce2c4f-65ee-4fe3-8527-22816b4492a9

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Dominique de Werra
2015

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/dcce2c4f-65ee-4fe3-8527-22816b4492a9

About this result

Related concepts (31)

Matching (graph theory)

In the mathematical discipline of graph theory, a matching or independent edge set in an undirected graph is a set of edges without common vertices. In other words, a subset of the edges is a matching if each vertex appears in at most one edge of that matching. Finding a matching in a bipartite graph can be treated as a network flow problem. Given a graph G = (V, E), a matching M in G is a set of pairwise non-adjacent edges, none of which are loops; that is, no two edges share common vertices.

Maximal independent set

In graph theory, a maximal independent set (MIS) or maximal stable set is an independent set that is not a subset of any other independent set. In other words, there is no vertex outside the independent set that may join it because it is maximal with respect to the independent set property. For example, in the graph P_3, a path with three vertices a, b, and c, and two edges and , the sets {b} and {a, c} are both maximally independent. The set {a} is independent, but is not maximal independent, because it is a subset of the larger independent set {a, c}.

Maximum flow problem

In optimization theory, maximum flow problems involve finding a feasible flow through a flow network that obtains the maximum possible flow rate. The maximum flow problem can be seen as a special case of more complex network flow problems, such as the circulation problem. The maximum value of an s-t flow (i.e., flow from source s to sink t) is equal to the minimum capacity of an s-t cut (i.e., cut severing s from t) in the network, as stated in the max-flow min-cut theorem. The maximum flow problem was first formulated in 1954 by T.

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