2024/09/22 by Niv Buchbinder, Buchbinder, Niv, Moran Feldman +1 · 2 citations
Computer Science · Engineering · #68R05 (Primary) 68Q87 #68W40 (Secondary) #Complexity and Algorithms in Graphs #Data Structures and Algorithms (cs.DS) #Discrete Mathematics (cs.DM) #F.2.2 #FOS: Computer and information sciences #G.2.1 #Imbalanced Data Classification Techniques #Infrastructure Maintenance and Monitoring
paper · pdf · doi:10.48550/arxiv.2409.14325
openalex publication_date 2024/09/22 · openalex created_date 2024/10/25 · openalex updated_date 2026/07/28
Maximization of submodular functions under various constraints is a fundamental problem that has been studied extensively. A powerful technique that has emerged and has been shown to be extremely effective for such problems is the following. First, a continues 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 the value of 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.