Streaming Algorithms for Non-Submodular Functions Maximization with D-Knapsack Constraint on the Integer Lattice.

We study the problem of maximizing a monotone non-submodular function under a d knapsack constraint on the integer lattice. We propose three streaming algorithms to approach this problem. We first design a two-pass min{alpha(1 - epsilon)/2(alpha+1)d, 1 - 1/alpha(w)2(alpha) - epsilon}-approximate algorithm with total memory complexity O(log d beta(-1)/beta epsilon), and total query complexity for each element O(log || B ||(infinity) log d beta(-1)/epsilon). The algorithm relies on a binary search technique to determine the amount of the current elements to be added into the output solution. It also requires to have a good estimate of the optimal value, we use the maximum value of the unit standard vector which can be obtained by reading a round of data to construct a guess set of the optimal value. Then, we modify our algorithm to avoid a repetitive reading of data by dynamically update the maximum value of the unit vector along with the coming elements, and obtain a one-pass streaming algorithm with same approximate ratio. Moreover, we design an improved StreamingKnapsack algorithm to reduce the memory complexity to O(d/epsilon(2)).