Abstract:
In this work we present the first practical
.
1
e
−ǫ
.
-approximation algorithm to maximise a general
non-negative submodular function subject to a matroid constraint. Our algorithm is based on
combining the decreasing-threshold procedure of Badanidiyuru and Vondrak (SODA 2014) with a
smoother version of the measured continuous greedy algorithm of Feldman et al. (FOCS 2011).
This enables us to obtain an algorithm that requires O(
nr2
ǫ4
. ¯
d+
¯
d
¯
d
.2
log2 (
n
ǫ
)) value oracle calls, where
n is the cardinality of the ground set, r is the matroid rank, and
¯
d,
¯
d ∈R+ are the absolute values
of the minimum and maximum marginal values that the function f can take i.e.: −
¯
d ≤fS(i) ≤
¯
d,
for all i ∈E and S ⊆E, where E is the ground set. The additional value oracle calls with respect
to the work of Badanidiyuru and Vondrak come from the greater spread in the sampling of the multilinear extension that the possibility of negative marginal values introduce.
