One-pass streaming algorithm for DR-submodular maximization with a knapsack constraint over the integer lattice

Abstract

Due to its broad applications, maximizing a diminishing return submodular function with a knapsack constraint has been extensively studied recently. In the paper, we mainly consider this problem on the integer lattice. Assuming the optimal value is known, we first design a one-pass algorithm and prove that its approximation ratio is (1/3−ɛ). Observing the difficult of actually knowing the optimal value, we design a streaming algorithm with two passes, where in the first round we find the maximum value of the unit vector to estimate the range of the OPT. Furthermore, in order to improve the performance of the algorithm, we design an online algorithm called DynamicMRT to reduce the number of rounds, eventually achieving an approximation ratio (1/3−ɛ), a memory complexity O(KlogK∖ε) and query complexity O(log2K∖ε) per element for the knapsack constraint K.