Submodular Functions Maximization Problems

This chapter focuses on maximizing a special class of functions called submodular functions under various combinatorial constraints. It deals with algorithms maximizing submodular functions subject to combinatorial constraints. The chapter discusses basic discrete algorithms for maximizing submodular functions subject to various constraints. It explores maximization subject to a cardinality constraint and also discusses the unconstrained submodular maximization problem. The chapter presents a discrete algorithm for the submodular welfare problem and describes continuous methods that are based on extensions of submodular functions to the n-dimensional cube. It explains the basic properties of these extensions and analyses basic algorithms obtaining fractional solutions which approximately maximize these extensions. The chapter also describes a method for obtaining inapproximability results for submodular maximization problems which is based on similar techniques. Many results for submodular maximization problems are based on a continuous relaxation of these problems known as the multilinear relaxation.