  We study the correlated stochastic knapsack problem of a submodular target
function, with optional additional constraints. We utilize the multilinear
extension of submodular function, and bundle it with an adaptation of the
relaxed linear constraints from Ma [Mathematics of Operations Research, Volume
43(3), 2018] on correlated stochastic knapsack problem. The relaxation is then
solved by the stochastic continuous greedy algorithm, and rounded by a novel
method to fit the contention resolution scheme (Feldman et al. [FOCS 2011]). We
obtain a pseudo-polynomial time $(1 - 1/\sqrt{e})/2 \simeq 0.1967$
approximation algorithm with or without those additional constraints,
eliminating the need of a key assumption and improving on the $(1 -
1/\sqrt[4]{e})/2 \simeq 0.1106$ approximation by Fukunaga et al. [AAAI 2019].
