2,291 research outputs found
Differentially Private Decomposable Submodular Maximization
We study the problem of differentially private constrained maximization of
decomposable submodular functions. A submodular function is decomposable if it
takes the form of a sum of submodular functions. The special case of maximizing
a monotone, decomposable submodular function under cardinality constraints is
known as the Combinatorial Public Projects (CPP) problem [Papadimitriou et al.,
2008]. Previous work by Gupta et al. [2010] gave a differentially private
algorithm for the CPP problem. We extend this work by designing differentially
private algorithms for both monotone and non-monotone decomposable submodular
maximization under general matroid constraints, with competitive utility
guarantees. We complement our theoretical bounds with experiments demonstrating
empirical performance, which improves over the differentially private
algorithms for the general case of submodular maximization and is close to the
performance of non-private algorithms
Streaming Algorithms for Submodular Function Maximization
We consider the problem of maximizing a nonnegative submodular set function
subject to a -matchoid
constraint in the single-pass streaming setting. Previous work in this context
has considered streaming algorithms for modular functions and monotone
submodular functions. The main result is for submodular functions that are {\em
non-monotone}. We describe deterministic and randomized algorithms that obtain
a -approximation using -space, where is
an upper bound on the cardinality of the desired set. The model assumes value
oracle access to and membership oracles for the matroids defining the
-matchoid constraint.Comment: 29 pages, 7 figures, extended abstract to appear in ICALP 201
Multi-objective Evolutionary Algorithms are Still Good: Maximizing Monotone Approximately Submodular Minus Modular Functions
As evolutionary algorithms (EAs) are general-purpose optimization algorithms,
recent theoretical studies have tried to analyze their performance for solving
general problem classes, with the goal of providing a general theoretical
explanation of the behavior of EAs. Particularly, a simple multi-objective EA,
i.e., GSEMO, has been shown to be able to achieve good polynomial-time
approximation guarantees for submodular optimization, where the objective
function is only required to satisfy some properties but without explicit
formulation. Submodular optimization has wide applications in diverse areas,
and previous studies have considered the cases where the objective functions
are monotone submodular, monotone non-submodular, or non-monotone submodular.
To complement this line of research, this paper studies the problem class of
maximizing monotone approximately submodular minus modular functions (i.e.,
) with a size constraint, where is a non-negative monotone
approximately submodular function and is a non-negative modular function,
resulting in the objective function being non-monotone non-submodular. We
prove that the GSEMO can achieve the best-known polynomial-time approximation
guarantee. Empirical studies on the applications of Bayesian experimental
design and directed vertex cover show the excellent performance of the GSEMO
Constrained Non-Monotone Submodular Maximization: Offline and Secretary Algorithms
Constrained submodular maximization problems have long been studied, with
near-optimal results known under a variety of constraints when the submodular
function is monotone. The case of non-monotone submodular maximization is less
understood: the first approximation algorithms even for the unconstrainted
setting were given by Feige et al. (FOCS '07). More recently, Lee et al. (STOC
'09, APPROX '09) show how to approximately maximize non-monotone submodular
functions when the constraints are given by the intersection of p matroid
constraints; their algorithm is based on local-search procedures that consider
p-swaps, and hence the running time may be n^Omega(p), implying their algorithm
is polynomial-time only for constantly many matroids. In this paper, we give
algorithms that work for p-independence systems (which generalize constraints
given by the intersection of p matroids), where the running time is poly(n,p).
Our algorithm essentially reduces the non-monotone maximization problem to
multiple runs of the greedy algorithm previously used in the monotone case.
Our idea of using existing algorithms for monotone functions to solve the
non-monotone case also works for maximizing a submodular function with respect
to a knapsack constraint: we get a simple greedy-based constant-factor
approximation for this problem.
With these simpler algorithms, we are able to adapt our approach to
constrained non-monotone submodular maximization to the (online) secretary
setting, where elements arrive one at a time in random order, and the algorithm
must make irrevocable decisions about whether or not to select each element as
it arrives. We give constant approximations in this secretary setting when the
algorithm is constrained subject to a uniform matroid or a partition matroid,
and give an O(log k) approximation when it is constrained by a general matroid
of rank k.Comment: In the Proceedings of WINE 201
Online Non-Monotone DR-submodular Maximization
In this paper, we study fundamental problems of maximizing DR-submodular
continuous functions that have real-world applications in the domain of machine
learning, economics, operations research and communication systems. It captures
a subclass of non-convex optimization that provides both theoretical and
practical guarantees. Here, we focus on minimizing regret for online arriving
non-monotone DR-submodular functions over different types of convex sets:
hypercube, down-closed and general convex sets.
First, we present an online algorithm that achieves a -approximation
ratio with the regret of for maximizing DR-submodular functions
over any down-closed convex set. Note that, the approximation ratio of
matches the best-known guarantee for the offline version of the problem.
Moreover, when the convex set is the hypercube, we propose a tight
1/2-approximation algorithm with regret bound of . Next, we give
an online algorithm that achieves an approximation guarantee (depending on the
search space) for the problem of maximizing non-monotone continuous
DR-submodular functions over a \emph{general} convex set (not necessarily
down-closed). To best of our knowledge, no prior algorithm with approximation
guarantee was known for non-monotone DR-submodular maximization in the online
setting. Finally we run experiments to verify the performance of our algorithms
on problems arising in machine learning domain with the real-world datasets
A -approximation algorithm for monotone submodular maximization over a -exchange system
We consider the problem of maximizing a monotone submodular function in a
-exchange system. These systems, introduced by Feldman et al., generalize
the matroid k-parity problem in a wide class of matroids and capture many other
combinatorial optimization problems. Feldman et al. show that a simple
non-oblivious local search algorithm attains a approximation ratio
for the problem of linear maximization in a -exchange system. Here, we
extend this approach to the case of monotone submodular objective functions. We
give a deterministic, non-oblivious local search algorithm that attains an
approximation ratio of for the problem of maximizing a monotone
submodular function in a -exchange system
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