A Tight Bound for Stochastic Submodular Cover · arXivDesk
2102.01149Feb 1, 2021This work extends the result of Srinivasan Parthasarathy in his paper arXiv:1803.07639 from the problem of Stochastic Set Cover to that of Stochastic Submodular Cover
A Tight Bound for Stochastic Submodular Cover
Lisa Hellerstein, Devorah Kletenik, Srinivasan Parthasarathy
We show that the Adaptive Greedy algorithm of Golovin and Krause (2011) achieves an approximation bound of (ln(Q/η)+1) for Stochastic Submodular Cover: here Q is the "goal value" and η is the smallest non-zero marginal increase in utility deliverable by an item. (For integer-valued utility functions, we show a bound of H(Q)
Nearby in the stack
, where
H(Q)
is the
Qth
Harmonic number.) Although this bound was claimed by Golovin and Krause in the original version of their paper, the proof was later shown to be incorrect by Nan and Saligrama (2017). The subsequent corrected proof of Golovin and Krause (2017) gives a quadratic bound of
(ln(Q/η)+1)2
. Other previous bounds for the problem are
56(ln(Q/η)+1)
, implied by work of Im et al. (2016) on a related problem, and
k(ln(Q/η)+1)
, due to Deshpande et al. (2016) and Hellerstein and Kletenik (2018), where
k
is the number of states. Our bound generalizes the well-known
(lnm+1)
approximation bound on the greedy algorithm for the classical Set Cover problem, where