
A Note on Monotone Submodular Maximization with Cardinality Constraint
We show that for the cardinality constrained monotone submodular maximiz...
read it

Nearly LinearTime, Parallelizable Algorithms for NonMonotone Submodular Maximization
We study parallelizable algorithms for maximization of a submodular func...
read it

On the Coordinator's Rule for Fast Paxos
Fast Paxos is an algorithm for consensus that works by a succession of r...
read it

Submodular Maximization subject to a Knapsack Constraint: Combinatorial Algorithms with Nearoptimal Adaptive Complexity
The growing need to deal with massive instances motivates the design of ...
read it

Stochastic Submodular Cover with Limited Adaptivity
In the submodular cover problem, we are given a nonnegative monotone su...
read it

Nonmonotone Submodular Maximization in Exponentially Fewer Iterations
In this paper we consider parallelization for applications whose objecti...
read it

Submodular Optimization in the MapReduce Model
Submodular optimization has received significant attention in both pract...
read it
A polynomial lower bound on adaptive complexity of submodular maximization
In largedata applications, it is desirable to design algorithms with a high degree of parallelization. In the context of submodular optimization, adaptive complexity has become a widelyused measure of an algorithm's "sequentiality". Algorithms in the adaptive model proceed in rounds, and can issue polynomially many queries to a function f in each round. The queries in each round must be independent, produced by a computation that depends only on query results obtained in previous rounds. In this work, we examine two fundamental variants of submodular maximization in the adaptive complexity model: cardinalityconstrained monotone maximization, and unconstrained nonmonotone maximization. Our main result is that an rround algorithm for cardinalityconstrained monotone maximization cannot achieve a factor better than 1  1/e  Ω(min{1/r, log^2 n/r^3}), for any r < n^c (where c>0 is some constant). This is the first result showing that the number of rounds must blow up polynomially large as we approach the optimal factor of 11/e. For the unconstrained nonmonotone maximization problem, we show a positive result: For every instance, and every δ>0, either we obtain a (1/2δ)approximation in 1 round, or a (1/2+Ω(δ^2))approximation in O(1/δ^2) rounds. In particular (in contrast to the cardinalityconstrained case), there cannot be an instance where (i) it is impossible to achieve a factor better than 1/2 regardless of the number of rounds, and (ii) it takes r rounds to achieve a factor of 1/2O(1/r).
READ FULL TEXT
Comments
There are no comments yet.