Maximum flow problems and an np-complete variant on edge-labeled graphs

The aim of this chapter is to present an overview of the main results for a well-known optimization problem and an emerging optimization area, as well as introducing a new problem which is related to both of them. The first part of the chapter presents an overview of the main existing results for the classical maximum flow problem. The maximum flow problem is one of the most studied optimization problems in the last decades. Besides its many practical applications, it also arises as a subproblem of several other complex problems (e.g., min cost flow, matching, covering on bipartite graphs). Subsequently, the chapter introduces some problems defined on edge-labeled graphs by reviewing the most relevant results in this field. Edge-labeled graphs are used to model situations where it is crucial to represent qualitative differences (instead of quantitative ones) among different regions of the graph itself. Finally, the maximum flow problem with the minimum number of labels (MF-ML) problem is presented and discussed. The aim is to maximize the network flow as well as the homogeneity of the solution on a capacitated network with logic attributes.