The problem of maximizing a monotone submodular function subject to a knapsack constraint admits a tight $(1-e^{-1})$-approximation: exhaustively enumerate over all subsets of size at most three and extend each using the greedy heuristic [Sviridenko, 2004]. We prove it suffices to enumerate only over all subsets of size at most two and still retain a tight $(1-e^{-1})$-approximation. This improves the running time from $O(n^5)$ to $O(n^4)$ queries. The result is achieved via a refined analysis of the greedy heuristic.

中文翻译：

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受背包约束约束的次模最大化的更快紧逼近

使最大单调子模函数受背包约束约束的问题允许紧密的$（1-e ^ {-1}）$逼近：穷举最多枚举大小为3的所有子集，并使用贪婪启发式方法来扩展每个子集[Sviridenko ，2004]。我们证明仅枚举最多两个大小的所有子集就足够了，并且仍然保持严格的$（1-e ^ {-1}）$近似值。这将运行时间从$ O（n ^ 5）$缩短为$ O（n ^ 4）$查询。通过对贪婪启发式算法的精细分析来获得结果。