Submodular Functions Maximization Problems

In this chapter we study fundamental results on maximizing a special class of functions called submodular functions under various combinatorial constraints. The study of submodular functions is motivated both by their many real world applications and by their frequent occurrence in more theoretical fields such as economy and algorithmic game theory. In particular, submodular functions and submodular maximization play a major role in combinatorial optimization as several well known combinatorial functions turn out to be submodular. A few examples of such functions include cuts functions of graphs and hypergraphs, rank functions of matroids and covering functions. We discuss some of these examples further in the following. Let us begin by providing basic notation used throughout the chapter. We then give two definitions of submodular functions and prove that they are equivalent. Let N = {u1, u2, . . . , un} be a ground set of elements. For a set A and an element u ∈ N we denote the union A ∪ {u} by A+ u. Similarly, we denote A \ {u} as A− u. The following is the first definition of submodular functions.