Robust monotone submodular function maximization

• Published in: Mathematical programming 2018-11, Vol.172 (1-2), p.505-537
• Main Authors: Orlin, James B., Schulz, Andreas S., Udwani, Rajan
• Format: Article
• Language: English
• Subjects: Algorithms; Approximation; Calculus of Variations and Optimal Control; Optimization; Combinatorics; Full Length Paper; Greedy algorithms; Mathematical analysis; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Mathematics of Computing; Maximization; Numerical Analysis; Queries; Robustness; Theoretical
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-018-1320-2

We consider a robust formulation, introduced by Krause et al. (J Artif Intell Res 42:427–486, 2011 ), of the classical cardinality constrained monotone submodular function maximization problem, and give the first constant factor approximation results. The robustness considered is w.r.t. adversarial removal of up to τ elements from the chosen set. For the fundamental case of τ = 1 , we give a deterministic ( 1 - 1 / e ) - 1 / Θ ( m ) approximation algorithm, where m is an input parameter and number of queries scale as O ( n m + 1 ) . In the process, we develop a deterministic ( 1 - 1 / e ) - 1 / Θ ( m ) approximate greedy algorithm for bi-objective maximization of (two) monotone submodular functions. Generalizing the ideas and using a result from Chekuri et al. (in: FOCS 10, IEEE, pp 575–584, 2010 ), we show a randomized ( 1 - 1 / e ) - ϵ approximation for constant τ and ϵ ≤ 1 Ω ~ ( τ ) , making O ( n 1 / ϵ 3 ) queries. Further, for τ ≪ k , we give a fast and practical 0.387 algorithm. Finally, we also give a black box result result for the much more general setting of robust maximization subject to an Independence System.