Linear Query Approximation Algorithms for Non-monotone Submodular Maximization under Knapsack Constraint

This work, for the first time, introduces two constant factor approximation algorithms with linear query complexity for non-monotone submodular maximization over a ground set of size n subject to a knapsack constraint, mathsfDLA and mathsfRLA. mathsfDLA is a deterministic algorithm that provides an approximation factor of 6+ε while mathsfRLA is a randomized algorithm with an approximation factor of 4+ε. Both run in O(n log(1/ε)/ε) query complexity. The key idea to obtain a constant approximation ratio with linear query lies in: (1) dividing the ground set into two appropriate subsets to find the near-optimal solution over these subsets with linear queries, and (2) combining a threshold greedy with properties of two disjoint sets or a random selection process to improve solution quality. In addition to the theoretical analysis, we have evaluated our proposed solutions with three applications: Revenue Maximization, Image Summarization, and Maximum Weighted Cut, showing that our algorithms not only return comparative results to state-of-the-art algorithms but also require significantly fewer queries.