Robust Submodular Maximization: A Non-Uniform Partitioning Approach
This solves a key open problem in robust optimization for submodular functions, with applications in data summarization and influence maximization, though it is incremental as it builds on prior work.
The paper tackles the problem of robust submodular maximization under a cardinality constraint where up to τ items can be removed, solving an open problem by achieving a constant-factor approximation guarantee for τ = o(k), improving from τ = o(√k). It demonstrates gains in data summarization and influence maximization over existing algorithms.
We study the problem of maximizing a monotone submodular function subject to a cardinality constraint $k$, with the added twist that a number of items $τ$ from the returned set may be removed. We focus on the worst-case setting considered in (Orlin et al., 2016), in which a constant-factor approximation guarantee was given for $τ= o(\sqrt{k})$. In this paper, we solve a key open problem raised therein, presenting a new Partitioned Robust (PRo) submodular maximization algorithm that achieves the same guarantee for more general $τ= o(k)$. Our algorithm constructs partitions consisting of buckets with exponentially increasing sizes, and applies standard submodular optimization subroutines on the buckets in order to construct the robust solution. We numerically demonstrate the performance of PRo in data summarization and influence maximization, demonstrating gains over both the greedy algorithm and the algorithm of (Orlin et al., 2016).