Submodular Functions: Optimization and Approximation

Submodular functions are discrete analogue of convex functions, arising in various fields of applied mathematics including game theory, information theory, and queueing theory. This survey aims at providing an overview on fundamental properties of submodular functions and recent algorithmic developments of their optimization and approximation. For submodular function minimization, the ellipsoid method had long been the only polynomial algorithm until combinatorial strongly polynomial algorithms appeared a decade ago. On the other hand, for submodular function maximization, which is NP-hard and known to refuse any polynomial algorithms, constant factor approximation algorithms have been developed with applications to combinatorial auction, machine learning, and social networks. In addition, an efficient method has been developed for approximating submoduar functions everywhere, which leads to a generic framework of designing approximation algorithms for combinatorial optimization problems with submodular costs. In some specific cases, however, one can devise better approximation algorithms. Mathematics Subject Classification (2010). Primary 90C27; Secondary 68W25.