The submodularity of two-stage stochastic maximum-weight independent set problems

In this paper, we extend the maximal independent set problem to two-stage stochastic case: given an independence system associated with one deterministic weight function and a random weight function, the goal is to find two nonoverlapping independent subsets from these two stages with the maximum total weight. In this paper, we study the sub -modularity of three kinds of two-stage independent set problems with max-weight. When the independent set problem is a matroid constraint, we can show its submodularity. How-ever, neither submodular nor supermodular maximization problem can be obtained for the knapsack independent set problem by designing a counterexample. At last, we show that the robust two-stage stochastic maximum-weight uniform matroid problem can be formu-lated as a gamma-submodular problem with cardinality constraint and also give a lower bound for gamma.(c) 2022 Elsevier B.V. All rights reserved.