Suppose some objects are hidden in a finite set $S$ of hiding places which must be examined one-by-one. The cost of searching subsets of $S$ is given by a submodular function and the probability that all objects are contained in a subset is given by a supermodular function. We seek an ordering of $S$ that finds all the objects in minimal expected cost. This problem is NP-hard and we give an efficient combinatorial $2$-approximation algorithm, generalizing analogous results in scheduling theory. We also give a new scheduling application $1|prec|\sum w_A h(C_A)$, where a set of jobs must be ordered subject to precedence constraints to minimize the weighted sum of some concave function $h$ of the completion times of {\em subsets} of jobs. We go on to give better approximations for submodular functions with low {\em total curvature} and we give a full solution when the problem is what we call {\em series-parallel decomposable}. Next, we consider a zero-sum game between a cost-maximizing Hider and a cost-minimizing Searcher. We prove that the equilibrium mixed strategies for the Hider are in the base polyhedron of the cost function, suitably scaled, and we solve the game in the series-parallel decomposable case, giving approximately optimal strategies in other cases.
Optimal Algorithms for Continuous Non-monotone Submodular and DR-Submodular Maximization
Stanford University（斯坦福大学）
在本文中，我们研究了极大化超立方体上的连续非单调子模函数的基本问题，包括有和没有坐标方向的限制。这类优化问题在机器学习、经济学和通信系统中有几个应用。我们的主要结果是第一个$\frac{1}{2}$ -近似算法连续次模函数最大化，这个近似因子的$\frac{1}{2}$是最好的算法，只查询目标函数在多项式多点。对于DR-次模最大化的特殊情况，即当次模函数沿所有坐标也是坐标方向凹时，我们提供了一个不同的$\frac{1}{2}$ -近似算法，它在准线性时间内运行。这两个结果都改进了先前的工作[Bian et al，2017，Soma and Yoshida，2017]。我们的第一个算法使用了新的想法，如减少保证近似问题，以分析每个坐标的零和游戏，并结合了这个零和游戏的几何形状，以固定在这个坐标的值。我们的第二个算法利用坐标明智的搜索，以确定一个单调的平衡条件足以得到所需的近似保证，并使用二进制搜索的平衡点狩猎。我们进一步运行实验来验证我们提出的算法在相关机器学习应用中的性能。
Boosting Gradient Ascent for Continuous DR-submodular Maximization
University of Chinese Academy of Sciences（中国科学院大学）；City University of Hong Kong（香港城市大学）；ByteDance（字节跳动）；Institute of Computing Technology, Chinese Academy of Sciences（中国科学院计算技术研究所）
