1Google Research · 2Dept. of Computer, Control and Management Engineering, Sapienza University of Rome · 3EPFL
Consistency is an important property in dynamic submodular maximization and entails maintaining a near-optimal solution at all times, making only a small number of adjustments to the solution in each step. Prior work has explored this question for the insertion-only case, where the algorithm faces a stream of
n insertions, and has established lower and upper bounds for the cardinality-constrained version of the problem. We consider this question in the fully dynamic setting, where the stream of operations may contain both insertions and deletions. We develop a general framework for designing algorithms for this setting, and instantiate it to obtain the first constant-factor approximations with sublinear consistency. For cardinality constraints, we propose a
21−O(ε) approximation that is
O(ε21) consistent. For rank-
k matroid constraints, we construct a
41−O(ε) approximation to the dynamic optimum that is
O(ε2logk) consistent.