Weak submodularity implies localizability: Local search for constrained non-submodular function maximization

Local search algorithms are commonly employed to address a variety of problems in the domain of operations research and combinatorial optimization. Most of the literature on the maximization of constrained monotone non-submodular functions is based on a greedy strategy, and few designs of local search approach are involved. In this paper, we explore the problem of maximizing a monotone non-submodular function under a p-matroid ($p\ge 1$) constraint with local search algorithms. And we indicate that weak submodularity implies localizability of set function optimization which can be used to offer provable approximation guarantees of local search algorithms.