Maximum-weight markings in marked graphs: Algorithms and interpretations based on the simplex method

The problem of determining a maximum-weight marking in a marked graph is mathematically dual to the transshipment problem of operations research. The special structure of the transshipment problem facilitates efficient implementation of the simplex method of linear programming, for solving such problems. In this paper, we first show that the maximum-weight marking problem possesses as much structure as its dual, and then present an implementation of simplex for this problem in terms of marked graph concepts and operations. The pivoting operation in the simplex method is shown to correspond to the subgraph firing operation in marked graphs. A diakoptic reachability theorem is also proved. The formulations presented in this paper cover both live- and nonlive-marked graphs with or without capacity constraints.