The symbolic algorithms for maximum flow in networks

The maximum flow problem is one of the classic combinatorial optimization problems with many applications, such as electrical powers, traffics, communications, computer networks and logistics. The problem is to find a flow of maximum value on a network from a source to a sink. Ordered binary decision diagram (OBDD) is a canonical form to represent and manipulate the Boolean functions efficiently. OBDD-based symbolic algorithms appear to give improved results for large-scale combinatorial optimization problems by searching the nodes and edges implicitly. For the maximum flow problem in networks, we present the symbolic algebraic decision diagram (ADD) formulation and symbolic algorithms. The augmenting-path-based symbolic algorithm is the combination of the Gabowʹs scaling algorithm with the binary decision diagram (BDD)-based symbolic algorithm for the maximum flow in 0–1 networks. The Karzanovʹs algorithm is implemented implicitly, resulting in the preflow-based symbolic algorithm (PSA), in which the vertices on each layer of the layered networks are partitioned by the vertexʹs input preflow and the total capacity of its outgoing edges, and the edges to push or pull the preflow are selected in terms of the priority function. The improved PSA is developed by integrating the heuristic of Träffʹs algorithm and the assistant layered networks into the PSA. The symbolic algorithms do not require explicit enumeration of the nodes and edges, and therefore can handle large-scale networks.