2020/06/25 by Santiago, Richard · 3 citations
#Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences
paper · doi:10.48550/arxiv.2006.14312
We consider the following class of submodular k-multiway partitioning problems: (Sub-k-MP) min ∑i=1k f(Si): S1 \uplus S2 \uplus ⋯ \uplus Sk = V and Si ≠ ∅ for all i∈ [k]. Here f is a non-negative submodular function, and \uplus denotes the union of disjoint sets. Hence the goal is to partition V into k non-empty sets S1,S2,…,Sk such that ∑i=1k f(Si) is minimized. These problems were introduced by Zhao et al. partly motivated by applications to network reliability analysis, VLSI design, hypergraph cut, and other partitioning problems. In this work we revisit this class of problems and shed some light onto their hardness of approximation in the value oracle model. We provide new unconditional hardness results for Sub-k-MP in the special settings where the function f is either monotone or symmetric. For symmetric functions we show that given any ε> 0, any algorithm achieving a (2 - ε)-approximation requires exponentially many queries in the value oracle model. For monotone objectives we show that given any ε> 0, any algorithm achieving a (4/3 - ε)-approximation requires exponentially many queries in the value oracle model. We then extend Sub-k-MP to a larger class of partitioning problems, where the functions fi(Si) can be different, and there is a more general partitioning constraint S1 \uplus S2 \uplus ⋯ \uplus Sk ∈ F for some family F ⊆ 2V of feasible sets. We provide a black box reduction that allows us to leverage several existing results from the literature; leading to new approximations for this class of problems.