Minimum latency submodular cover

We study the submodular ranking problem in the presence of metric costs. The input to the minimum latency submodular cover (mlsc ) problem consists of a metric (v,d) with source r?v and m monotone submodular functions f 1, f 2, .., f m : 2 v ?[0,1]. The goal is to find a path originating at r that minimizes the total cover time of all functions; the cover time of function f i is the smallest value t such that f i has value one for the vertices visited within distance t along the path. This generalizes many previously studied problems, such as submodular ranking [1] when the metric is uniform, and group steiner tree [14] when m?=?1 and f 1 is a coverage function. We give a polynomial time o(\log \frac{1}{\epsilon } \cdot \log^{2+\delta} |v|)o(log1? ·log 2+d |v|)o(\log \frac{1}{\epsilon } \cdot \log^{2+\delta} |v|)-approximation algorithm for mlsc, where e?>?0 is the smallest non-zero marginal increase of any \{f_i\}_{i=1}^m{f i } m i=1 \{f_i\}_{i=1}^m and d?>?0 is any constant. This result is enabled by a simpler analysis of the submodular ranking algorithm from [1].we also consider the stochastic submodular ranking problem where elements v have random instantiations, and obtain an adaptive algorithm with an o(log1/ e) approximation ratio, which is best possible. This result also generalizes several previously studied stochastic problems, eg. Adaptive set cover [15] and shared filter evaluation [24,23].