Greedy Maximization of Submodular Functions

Traditional optimization techniques often rely upon functions that are convex or at least locally convex. Such diverse methods as gradient descent, loopy belief propagation, and linear programming all rely upon convex functions. However, many natural functions are not convex, yet optimizing over them is both possible and necessary. The class of submodular functions is particularly well-behaved and applicable. Submodularity may be thought of as a discrete analogue of convexity for set-functions (that is, functions defined on sets). Minimizing convex functions is generally easy, while maximizing them is generally computationally hard. The same is true of submodular functions, which admit strongly polynomial-time minimization algorithms but which cannot generally be maximized efficiently. An example of this phenomenon, which we explore in some detail below, is the well-known difference between the problems of finding a minimum cut and finding a maximum cut in a graph. Minimum cuts can be found by any efficient maximum-flow algorithm, whereas finding a maximum cut is generally NP-hard. In this paper, we describe two “greedy” approaches to the problem of submodular maximization. As we will show below, maximizing a submodular function is provably hard in a strong sense; nevertheless, simple greedy algorithms provide approximations to optimal solutions in many cases of practical significance. We first restrict ourselves to the problem of maximizing a monotone submodular function subject to a natural constraint. Here, a näıve greedy algorithm finds a good approximation but is inefficient. We present a recent improvement of the greedy algorithm with a much better asymptotic running time due to Badanidiyuru and Vondrák [1]. We then consider general submodular maximization. A näıve greedy algorithm is ineffective in this case, but we will present a new algorithm due to Buchbinder et al. [2] that combines two greedy algorithms to achieve a (1/2)-approximation guarantee. We end by considering open problems in this area and briefly surveying some recent trends in the field.