Constrained submodular maximization problems encompass a wide variety of applications, including personalized recommendation, team formation, and revenue maximization via viral marketing. The massive instances occurring in modern day applications can render existing algorithms prohibitively slow, while frequently, those instances are also inherently stochastic. Focusing on these challenges, we revisit the classic problem of maximizing a (possibly non-monotone) submodular function subject to a knapsack constraint. We present a simple randomized greedy algorithm that achieves a $5.83$ approximation and runs in $O(n \log n)$ time, i.e., at least a factor $n$ faster than other state-of-the-art algorithms. The robustness of our approach allows us to further transfer it to a stochastic version of the problem. There, we obtain a 9-approximation to the best adaptive policy, which is the first constant approximation for non-monotone objectives. Experimental evaluation of our algorithms showcases their improved performance on real and synthetic data.

翻译：受约束的子模最大化问题涵盖广泛的应用场景，包括个性化推荐、团队组建及通过病毒式营销实现收益最大化。现代应用中出现的大规模实例可能导致现有算法运行速度过慢，而这类实例往往本身也具有随机性。针对这些挑战，我们重新审视了在背包约束下最大化（可能非单调）子模函数的经典问题。我们提出一种简单的随机贪心算法，该算法可实现5.83的近似比，运行时间为O(n log n)，即比现有最优算法至少快一个数量级n倍。我们方法的鲁棒性使其能进一步推广到该问题的随机版本。在随机版本中，我们获得了针对最优自适应策略的9-近似比，这是非单调目标函数领域的首个常数近似比。实验评估表明，我们的算法在真实数据和合成数据上均展现出更优性能。