The problem of maximizing a monotone submodular function subject to a knapsack constraint admits a tight (1-e^{-1})-approximation : exhaustively enumerate over all subsets of size at most three and extend each using the greedy heuristic [Sviridenko , 2004 ] . We prove it suffices to enumerate only over all subsets of size at most twoand still retain a tight (1-e^{-1 of the greedy heuristic.
Many algorithms for maximizing a monotone submodular function subject to a knapsack constraint rely on the natural greedy heuristic. We present a novel refined analysis of this greedy heuristic which enables us to: (1) reduce the enumeration in the tight (1-e^{-1})-approximation of [Sviridenko 04 ] from subsets of size three to two; (2) present an improved upper bound of 0.42945 for the classic algorithm which returns the better between a single element and the output of the greedy heuristic.
R
The problem of
Many algorithms for
R
admits a
rely on the natural greedy heuristic. We present a novel refined analysis of this greedy heuristic which enables us to: (1) reduce the enumeration in the
R
: exhaustively enumerate over all subsets of size at most three and extend each using the greedy heuristic
of
R
, 2004
04
R
. We prove it suffices to enumerate only over all
from
R
at most twoand still retain a tight (1-e^{-1
three to two; (2) present an improved upper bound of 0.42945 for the classic algorithm which returns the better between a single element and the output