Abstract
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 : 2V → [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 \(O(\log \frac{1}{\epsilon } \cdot \log^{2+\delta} |V|)\)-approximation algorithm for MLSC, where ε > 0 is the smallest non-zero marginal increase of any \(\{f_i\}_{i=1}^m\) and δ > 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/ ε) 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].
A full version of this extended abstract appears as [21].
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Im, S., Nagarajan, V., van der Zwaan, R. (2012). Minimum Latency Submodular Cover. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds) Automata, Languages, and Programming. ICALP 2012. Lecture Notes in Computer Science, vol 7391. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31594-7_41
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DOI: https://doi.org/10.1007/978-3-642-31594-7_41
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