Maximizing Submodular Functions under Submodular Constraints

We consider the problem of maximizing submodular functions under submodular constraints by formulating the problem in two ways: SCSK-C and DIFF-C. Given two submodular functions f and g where f is monotone, the objective of SCSK-C problem is to find a set S of size at most k that maximizes f(S) under the constraint that g(S) <= theta, for a given value of.. The problem of DIFF-C focuses on finding a set S of size at most k such that h(S) = f(S) - g(S) is maximized. It is known that these problems are highly inapproximable and do not admit any constant factor multiplicative approximation algorithms unless NP is easy. Known approximation algorithms involve data-dependent approximation factors that are not efficiently computable. We initiate a study of the design of approximation algorithms where the approximation factors are efficiently computable. For the problem of SCSK-C, we prove that the greedy algorithm produces a solution whose value is at least (1 -1/e)f(OPT) - A, where A is the data-dependent additive error. For the DIFF-C problem, we design an algorithm that uses the SCSK-C greedy algorithm as a subroutine. This algorithm produces a solution whose value is at least (1 - 1/e)h(OPT) - B, where B is also a data-dependent additive error. A salient feature of our approach is that the additive error terms can be computed efficiently, thus enabling us to ascertain the quality of the solutions produced.