Structure Analysis of Some Generalizations of Matchings and Matroids under Algorithmic Aspects of Matchings and Matroids Under Algorithmic Aspects

Combinatorial optimization problems whose underlying structures are matchings or matroids are well-known to be solvable with efficient algorithms. Matroids can even be characterized by a simple greedy algorithm. In the first part of this thesis, some generalizations of matroids which allow the ground set to be partially ordered are considered. In particular, it will be shown that a special type of lattice polyhedra, for which Dietrich and Hoffman recently established a dual greedy algorithm, can be reduced to ordinary polymatroids. Moreover, strong exchange structures, Gauss greedoids and Delta-matroids will be extended from Boolean lattices to general distributive lattices, and the resulting structures will be characterized by certain greedy-type algorithms. While a matching of maximal size can be determined by a polynomial algorithm, the dual problem of finding a vertex cover of minimal size in general graphs is one of the hardest problems in combinatorial optimization. However, in case the graph belongs to the class of K\"onig-Egerv\'ary graphs, a maximum matching can be used to construct a minimum vertex cover. Lovasz and Korach characterized König-Egervary graphs by the exclusion of forbidden subgraphs. In the second part of this dissertation, the structure of König-Egervary graphs and the more general Red/Blue-split graphs will be analyzed. Red/Blue-split graphs have red and blue colored edges and the vertices of which can be split into two stable sets with respect to the red and blue edges, respectively. An algorithm that either determines a feasible partition of the vertices, or returns a red-blue colored subgraph (called ``flower'') characterizing non-Red/Blue-split graphs will be presented. This characterization allows the deduction of Lovasz and Korach's characterizations of König-Egerv\'ary graphs in case the red edges of the flower form a maximum matching. Furthermore, weighted Red/Blue-split graphs which model integrally solvable simple systems are introduced. A simple system is an inequality system where the sum of absolute values in each row of the integral matrix does not exceed the value two. A shortest-path algorithm and the presented Red/Blue-split algorithm will be used to find an integral solution of a simple system. These two algorithms lead to a characterization of weighted Red/Blue-split graphs by forbidden weighted subgraphs