On Learning Submodular Functions A Preliminary Draft

Submodular functions are a central concept in combinatorial optimization. The wide collection of optimization problems involving submodular functions encompasses many important combinatorial problems, such as Min-Cut and Max-Cut in graphs, various plant location problems, etc. In the operations research literature, many heuristics, exact algorithms, and approximation algorithms have been developed for minimizing and maximizing submodular functions, under various side constraints. Motivating by the extensive literature on submodular function optimization, we investigate a fundamental question concerning the structure of submodular functions. We consider the problem of learning a submodular function on ground set of size n. The problem is stated as follows: Can we make a polynomial number of (value) queries to a submodular function, and then approximate the submodular function on any other set to within some factor? We answer this question by proving the following results. First, we show that one cannot find a learning algorithm that achieves a factor better than Ω( √ n/ log n), even if the queries are chosen adaptively, and even if f is monotone. Moreover, we show that any such algorithm must satisfy Ω(n/ log n) if the queries are chosen non-adaptively, even if f is monotone, and this result is tight. That is, there exists an algorithm using non-adaptive queries which can approximate a monotone submodular function to within a factor O(n/ log n). For monotone functions, we prove that there exists a function using O(n) space which approximates a monotone submodular function to within a factor √ n. Finally, we show a connection between our problem and an unresolved question in convex geometry.