Scalable Bicriteria Algorithms for Non-Monotone Submodular Cover

Abstract

This paper proposes the optimization problem Non-Monotone Submodular Cover (SC), which is to minimize the cost required to ensure that a non-monotone submodular benefit function exceeds a given threshold. Two algorithms are presented for SC that both give a ((1 + )(4/ 2 + 1), 1/2(1 − )) bicriteria approximation guarantee to the problem. Both algorithms process the ground set in a stream, one in multiple passes. Further, a (1/2(1 − ), (1 + )(4/ 2 + 1)) bicriteria approximation guarantee is given for the related optimization problem Submodular Knapsack (SK).

Bicriteria Algorithms for Approximately Submodular Cover Under Streaming Model

“…As statement in the introduction, Feige et al [18] showed that there is a close collection between the problems of submodular maximization and submodular cover. Based on the previous work, especially Crawford [19] , we know the following theorem still holds. We refer the reader to the detailed proof in the work of Crawford [19] .…”

“…Based on the previous work, especially Crawford [19] , we know the following theorem still holds. We refer the reader to the detailed proof in the work of Crawford [19] .…”

“…To the best of our knowledge, the memory occupied by the elements is independent of the properties of the cover function during the execution of Algorithm 1. Then utilizing the work of Crawford [19] , we get the following lemma for the cost function. The proof idea is similar with that of Lemma 2, and we present a brief proof.…”