Submodular functions are a fundamental object of study in combinatorial optimization, economics, machine learning, etc. and exhibit a rich combinatorial structure. Many subclasses of submodular functions have also been well studied and these subclasses widely vary in their complexity. Our motivation is to understand the relative complexity of these classes of functions. Towards this, we conside...

We investigate two new optimization problems — minimizing a submodular function subject to a submodular lower bound constraint (submodular cover) and maximizing a submodular function subject to a submodular upper bound constraint (submodular knapsack). We are motivated by a number of real-world applications in machine learning including sensor placement and data subset selection, which require ...

We consider two types of grouping among users in utility systems. The first type of grouping is from Chen et al. (2014), where each user belongs to a group of users having social ties with it. For this type of utility system, each user’s strategy maximizes its social group utility function, giving rise to the notion of social-aware Nash equilibrium. We prove that, for a valid utility system, if...

The main problem considered is submodular set cover, the problem of minimizing a linear function under a nondecreasing submodular constraint, which generalizes both wellknown set cover and minimum matroid base problems. The problem is NP-hard, and two natural greedy heuristics are introduced along with analysis of their performance. As applications of these heuristics we consider various specia...

We are motivated by an application to extract a representative subset of machine learning training data and by the poor empirical performance we observe of the popular minimum norm algorithm. In fact, for our application, minimum norm can have a running time of about O(n) (O(n) oracle calls). We therefore propose a fast approximate method to minimize arbitrary submodular functions. For a large ...

Influence maximization is the problem of selecting k nodes in a social network to maximize their influence spread. The problem has been extensively studied but most works focus on the submodular influence diffusion models. In this paper, motivated by empirical evidences, we explore influence maximization in the nonsubmodular regime. In particular, we study the general threshold model in which a...

Integer-valued elements of an integral submodular flow polyhedron Q are investigated which decreasingly minimal (dec-min) in the sense that their largest component is as small possible, within this, second and so on. As a main result, we prove set dec-min another arising from by intersecting face with box. Based on this description, develop strongly polynomial algorithm for computing not only i...