Before revision:  We propose a new concept, adaptive DR-submodularity , and show our Adaptive-IMMA is the problem that maximizing an adaptive monotone and DR-submodular function under the  knapsack constraint.
After revision:  We propose a new concept, adaptive DR-submodularity , and show our Adaptive-IMMA is the problem that maximizing an adaptive monotone and DR-submodular function under the expected knapsack constraint.