Minimum Latency Submodular Cover

Abstract

We study the Minimum Latency Submodular Cover (MLSC) problem, which 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. This generalizes well-studied problems, such as Submodular Ranking [Azar and Gamzu 2011] and the Group Steiner Tree [Garg et al. 2000]. We give a polynomial time O (log 1/ϵ ċ log 2+δ |V|)-approximation algorithm for MLSC, where ϵ > 0 is the smallest non-zero marginal increase of any { f i } m i = 1 and δ > 0 is any constant. We also consider the Latency Covering Steiner Tree (LCST) problem, which is the special case of MLSC where the f i s are multi-coverage functions. This is a common generalization of the Latency Group Steiner Tree [Gupta et al. 2010; Chakrabarty and Swamy 2011] and Generalized Min-sum Set Cover [Azar et al. 2009; Bansal et al. 2010] problems. We obtain an O (log 2 | V |)-approximation algorithm for LCST. Finally, we study a natural stochastic extension of the Submodular Ranking problem and obtain an adaptive algorithm with an O (log 1/ϵ)-approximation ratio, which is best possible. This result also generalizes some previously studied stochastic optimization problems, such as Stochastic Set Cover [Goemans and Vondrák 2006] and Shared Filter Evaluation [Munagala et al. 2007; Liu et al. 2008].