Extending the Extension: Deterministic Algorithm for Non-monotone Submodular Maximization

Maximization of submodular functions under various constraints is a fundamental problem that has been extensively studied. A powerful technique that has emerged and has been shown to be extremely effective for such problems is the following. First, a continuous relaxation of the problem is obtained by relaxing the (discrete) set of feasible solutions to a convex body, and extending the discrete submodular function f to a continuous function F known as the multilinear extension. Then, two algorithmic steps are implemented. The first step approximately solves the relaxation by finding a fractional solution within the convex body that approximately maximizes F; and the second step rounds this fractional solution to a feasible integral solution. While this "fractionally solve and then round"approach has been a key technique for resolving many questions in the field, the main drawback of algorithms based on it is that evaluating the multilinear extension may require a number of value oracle queries to f that is exponential in the size of f's ground set. The only known way to tackle this issue is to approximate F via sampling, which makes all algorithms based on this approach inherently randomized and quite slow. In this work, we introduce a new tool, that we refer to as the extended multilinear extension, designed to derandomize submodular maximization algorithms that are based on the successful "solve fractionally and then round"approach. We demonstrate the effectiveness of this new tool on the fundamental problem of maximizing a submodular function subject to a matroid constraint, and show that it allows for a deterministic implementation of both the fractionally solving step and the rounding step of the above approach. As a bonus, we also get a randomized algorithm for the problem with an improved query complexity.