Revisiting Modified Greedy Algorithm for Monotone Submodular Maximization with a Knapsack Constraint
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Monotone submodular maximization with a knapsack constraint is NP-hard. Various approximation algorithms have been devised to address this optimization problem. In this paper, we revisit the widely known modified greedy algorithm. First, we show that this algorithm can achieve an approximation factor of 0.405, which significantly improves the known factor of 0.357 given by Wolsey or (1-1/e)/2≈ 0.316 given by Khuller et al. More importantly, our analysis uncovers a gap in Khuller et al.'s proof for the extensively mentioned approximation factor of (1-1/√(e))≈ 0.393 in the literature to clarify a long time of misunderstanding on this issue. Second, we enhance the modified greedy algorithm to derive a data-dependent upper bound on the optimum. We empirically demonstrate the tightness of our upper bound with a real-world application. The bound enables us to obtain a data-dependent ratio typically much higher than 0.405 between the solution value of the modified greedy algorithm and the optimum. It can also be used to significantly improve the efficiency of algorithms such as branch and bound.
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