Robust monotone submodular function maximization

Abstract

Abstract 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

$$\tau $$

τ

elements from the chosen set. For the fundamental case of

$$\tau =1$$

τ = 1

, we give a deterministic

$$(1-1/e)-1/\varTheta (m)$$

( 1 - 1 / e ) - 1 / Θ ( m )

approximation algorithm, where m is an input parameter and number of queries scale as

$$O(n^{m+1})$$

O (

n

m + 1

)

. In the process, we develop a deterministic

$$(1-1/e)-1/\varTheta (m)$$

( 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

$$(1-1/e)-\epsilon $$

( 1 - 1 / e ) - ϵ

approximation for constant

$$\tau $$

τ

and

$$\epsilon \le \frac{1}{\tilde{\varOmega }(\tau )}$$

ϵ ≤

1

Ω ~

( τ )

, making

$$O(n^{1/\epsilon ^3})$$

O (

n

1 /

ϵ 3

)

queries. Further, for

$$\tau \ll \sqrt{k}$$

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.