A Parallel Algorithm for Minimum Cost Submodular Cover

08/10/2021

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by   Yingli Ran, et al.

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In a minimum cost submodular cover problem (MinSMC), given a monotone non-decreasing submodular function f 2^V →ℤ^+, a cost function c: V→ℝ^+, an integer k≤ f(V), the goal is to find a subset A⊆ V with the minimum cost such that f(A)≥ k. MinSMC has a lot of applications in machine learning and data mining. In this paper, we design a parallel algorithm for MinSMC which obtains a solution with approximation ratio at most H(min{Δ,k})/1-5ε with probability 1-3ε in O(log mlog nlog^2 mn/ε^4) rounds, where Δ=max_v∈ Vf(v), H(·) is the Hamornic number, n=f(V), m=|V| and ε is a constant in (0,1/5). This is the first paper obtaining a parallel algorithm for the weighted version of the MinSMC problem with an approximation ratio arbitrarily close to H(min{Δ,k}).