Complexity Classification of Network Information Flow Problems

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We begin with a definition. A flow network consists of, among other things, a directed graph, G, and we will disallow anti-parallel edges to simplify some of the equation. This is not a serious limitation, as we'll see. We'll distinguish to special vertices in the graph. A source, typically labeled s. This is where whatever is flowing to the network starts from. And a sync typically labeled t, this is where the flow ends up. We call all other vertices internal. To keep our equations a little simpler, we'll assume there are no incoming edges into s, or outgoing edges from t. Associated with every pair of vertices is a capacity which indicates how much flow it is possible to send directly between two vertices. We will assume that these capacity's are always non negative integers. This will make some of the arguments easier and it's not a serious limitation in fact the last algorithm we see will overcome it. Also note that if there is no edge in the graph, then the capacity is defined to