Minimum Latency Submodular Cover

  • Sungjin Im
  • Viswanath Nagarajan
  • Ruben van der Zwaan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7391)

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 : 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 \(O(\log \frac{1}{\epsilon } \cdot \log^{2+\delta} |V|)\)-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].

Keywords

Approximation Algorithm Steiner Tree Steiner Tree Problem Cover Time Submodular Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Sungjin Im
    • 1
  • Viswanath Nagarajan
    • 2
  • Ruben van der Zwaan
    • 3
  1. 1.Department of Computer ScienceUniversity of IllinoisUSA
  2. 2.IBM T. J. Watson Research CenterUSA
  3. 3.Maastricht UniversityThe Netherlands

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