Constrained Submodular Maximization via New Bounds for DR-Submodular Functions.
CoRR(2023)
摘要
Submodular maximization under various constraints is a fundamental problem
studied continuously, in both computer science and operations research, since
the late $1970$'s. A central technique in this field is to approximately
optimize the multilinear extension of the submodular objective, and then round
the solution. The use of this technique requires a solver able to approximately
maximize multilinear extensions. Following a long line of work, Buchbinder and
Feldman (2019) described such a solver guaranteeing $0.385$-approximation for
down-closed constraints, while Oveis Gharan and Vondr\'ak (2011) showed that no
solver can guarantee better than $0.478$-approximation. In this paper, we
present a solver guaranteeing $0.401$-approximation, which significantly
reduces the gap between the best known solver and the inapproximability result.
The design and analysis of our solver are based on a novel bound that we prove
for DR-submodular functions. This bound improves over a previous bound due to
Feldman et al. (2011) that is used by essentially all state-of-the-art results
for constrained maximization of general submodular/DR-submodular functions.
Hence, we believe that our new bound is likely to find many additional
applications in related problems, and to be a key component for further
improvement.
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关键词
new bounds,dr-submodular
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