Polyhedral aspects of Submodularity, Convexity and Concavity

The seminal work by Edmonds [9] and Lovász [39] shows the strong connection between submodular functions and convex functions. Submodular functions have tight modular lower bounds, and a subdifferential structure [16] in a manner akin to convex functions. They also admit polynomial time algorithms for minimization and satisfy the Fenchel duality theorem [18] and the Discrete Seperation Theorem [15], both of which are fundamental characteristics of convex functions. Submodular functions also show signs similar to concavity. Submodular function maximization, though NP hard, admits constant factor approximation guarantees. Concave functions composed with modular functions are submodular, and they also show the diminishing returns property. In this manuscript, we try to provide a more complete picture on the relationship between submodularity with convexity and concavity, by extending many of the results connecting submodularity with convexity [39, 15, 18, 9, 16] to the concave aspects of submodular functions. We first show the existence of the superdifferentials and efficiently computable tight modular upper bounds of a submodular function. While we show that it is hard to characterize this polyhedron, we obtain inner and outer bounds on the superdifferential along with certain specific and useful supergradients. We then investigate forms of concave extensions of submodular functions and show interesting relationships to submodular maximization. We next show connections between optimality conditions over the superdifferentials and submodular maximization, and show how forms of approximate optimality conditions translate into approximation factors for maximization. We end this paper by studying versions of the discrete seperation theorem and the Fenchel duality theorem when seen from the concave point of view. In every case, we relate our results to the existing results from the convex point of view, thereby improving the analysis of the relationship between submodularity, convexity, and concavity.