Streaming Algorithms for Non-Submodular Maximization on the Integer Lattice

Many practical problems emphasize the importance of not only knowing whether an element is selected but also deciding to what extent it is selected, which imposes a challenge on submodule optimization. In this study, we consider the monotone, nondecreasing, and non-submodular maximization on the integer lattice with a cardinality constraint. We first design a two-pass streaming algorithm by refining the estimation interval of the optimal value. For each element, the algorithm not only decides whether to save the element but also gives the number of reservations. Then, we introduce the binary search as a subroutine to reduce the time complexity. Next, we obtain a one-pass streaming algorithm by dynamically updating the estimation interval of optimal value. Finally, we improve the memory complexity of this algorithm.

Automated summary

In this study, we address the challenge of submodule optimization by proposing two streaming algorithms for the maximization problem on the integer lattice with a cardinality constraint. These algorithms not only determine whether to select an element but also quantify the extent of its selection, and improve both time and memory complexity.