### Abstract

Deshpande et al. presented a k(lnR+1) approximation bound for Stochastic Submodular Cover, where k is the state set size, R is the maximum utility of a single item, and the utility function is integer-valued. This bound is similar to the (ln Q= + 1) bound given by Golovin and Krause, whose analysis was recently found to have an error. Here Q ≥ R is the goal utility and Q is the minimum gap between Q and any attainable utility Q0 < Q. We revisit the proof of the k(lnR+1) bound of Deshpande et al., fill in the details of the proof of a key lemma, and prove two bounds for real-valued utility functions: k(lnR1 +1) and (lnRE +1). Here R1 equals the maximum ratio between the largest increase in utility attainable from a single item, and the smallest non-zero increase attainable from that same item (in the same state). The quantity RE equals the maximum ratio between the largest expected increase in utility from a single item, and the smallest non-zero expected increase in utility from that same item. Our bounds apply only to the stochastic setting with independent states.

Original language | English (US) |
---|---|

Pages (from-to) | 265-279 |

Number of pages | 15 |

Journal | Journal of Artificial Intelligence Research |

Volume | 63 |

State | Published - Oct 1 2018 |

### ASJC Scopus subject areas

- Artificial Intelligence

### Cite this

*Journal of Artificial Intelligence Research*,

*63*, 265-279.

**Revisiting the approximation bound for stochastic submodular cover.** / Hellerstein, Lisa; Kletenik, Devorah.

Research output: Contribution to journal › Article

*Journal of Artificial Intelligence Research*, vol. 63, pp. 265-279.

}

TY - JOUR

T1 - Revisiting the approximation bound for stochastic submodular cover

AU - Hellerstein, Lisa

AU - Kletenik, Devorah

PY - 2018/10/1

Y1 - 2018/10/1

N2 - Deshpande et al. presented a k(lnR+1) approximation bound for Stochastic Submodular Cover, where k is the state set size, R is the maximum utility of a single item, and the utility function is integer-valued. This bound is similar to the (ln Q= + 1) bound given by Golovin and Krause, whose analysis was recently found to have an error. Here Q ≥ R is the goal utility and Q is the minimum gap between Q and any attainable utility Q0 < Q. We revisit the proof of the k(lnR+1) bound of Deshpande et al., fill in the details of the proof of a key lemma, and prove two bounds for real-valued utility functions: k(lnR1 +1) and (lnRE +1). Here R1 equals the maximum ratio between the largest increase in utility attainable from a single item, and the smallest non-zero increase attainable from that same item (in the same state). The quantity RE equals the maximum ratio between the largest expected increase in utility from a single item, and the smallest non-zero expected increase in utility from that same item. Our bounds apply only to the stochastic setting with independent states.

AB - Deshpande et al. presented a k(lnR+1) approximation bound for Stochastic Submodular Cover, where k is the state set size, R is the maximum utility of a single item, and the utility function is integer-valued. This bound is similar to the (ln Q= + 1) bound given by Golovin and Krause, whose analysis was recently found to have an error. Here Q ≥ R is the goal utility and Q is the minimum gap between Q and any attainable utility Q0 < Q. We revisit the proof of the k(lnR+1) bound of Deshpande et al., fill in the details of the proof of a key lemma, and prove two bounds for real-valued utility functions: k(lnR1 +1) and (lnRE +1). Here R1 equals the maximum ratio between the largest increase in utility attainable from a single item, and the smallest non-zero increase attainable from that same item (in the same state). The quantity RE equals the maximum ratio between the largest expected increase in utility from a single item, and the smallest non-zero expected increase in utility from that same item. Our bounds apply only to the stochastic setting with independent states.

UR - http://www.scopus.com/inward/record.url?scp=85055880835&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85055880835&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:85055880835

VL - 63

SP - 265

EP - 279

JO - Journal of Artificial Intelligence Research

JF - Journal of Artificial Intelligence Research

SN - 1076-9757

ER -