59,108 research outputs found
Stability of Influence Maximization
The present article serves as an erratum to our paper of the same title,
which was presented and published in the KDD 2014 conference. In that article,
we claimed falsely that the objective function defined in Section 1.4 is
non-monotone submodular. We are deeply indebted to Debmalya Mandal, Jean
Pouget-Abadie and Yaron Singer for bringing to our attention a counter-example
to that claim.
Subsequent to becoming aware of the counter-example, we have shown that the
objective function is in fact NP-hard to approximate to within a factor of
for any .
In an attempt to fix the record, the present article combines the problem
motivation, models, and experimental results sections from the original
incorrect article with the new hardness result. We would like readers to only
cite and use this version (which will remain an unpublished note) instead of
the incorrect conference version.Comment: Erratum of Paper "Stability of Influence Maximization" which was
presented and published in the KDD1
Locally Adaptive Optimization: Adaptive Seeding for Monotone Submodular Functions
The Adaptive Seeding problem is an algorithmic challenge motivated by
influence maximization in social networks: One seeks to select among certain
accessible nodes in a network, and then select, adaptively, among neighbors of
those nodes as they become accessible in order to maximize a global objective
function. More generally, adaptive seeding is a stochastic optimization
framework where the choices in the first stage affect the realizations in the
second stage, over which we aim to optimize.
Our main result is a -approximation for the adaptive seeding
problem for any monotone submodular function. While adaptive policies are often
approximated via non-adaptive policies, our algorithm is based on a novel
method we call \emph{locally-adaptive} policies. These policies combine a
non-adaptive global structure, with local adaptive optimizations. This method
enables the -approximation for general monotone submodular functions
and circumvents some of the impossibilities associated with non-adaptive
policies.
We also introduce a fundamental problem in submodular optimization that may
be of independent interest: given a ground set of elements where every element
appears with some small probability, find a set of expected size at most
that has the highest expected value over the realization of the elements. We
show a surprising result: there are classes of monotone submodular functions
(including coverage) that can be approximated almost optimally as the
probability vanishes. For general monotone submodular functions we show via a
reduction from \textsc{Planted-Clique} that approximations for this problem are
not likely to be obtainable. This optimization problem is an important tool for
adaptive seeding via non-adaptive policies, and its hardness motivates the
introduction of \emph{locally-adaptive} policies we use in the main result
Non-monotone Submodular Maximization with Nearly Optimal Adaptivity and Query Complexity
Submodular maximization is a general optimization problem with a wide range
of applications in machine learning (e.g., active learning, clustering, and
feature selection). In large-scale optimization, the parallel running time of
an algorithm is governed by its adaptivity, which measures the number of
sequential rounds needed if the algorithm can execute polynomially-many
independent oracle queries in parallel. While low adaptivity is ideal, it is
not sufficient for an algorithm to be efficient in practice---there are many
applications of distributed submodular optimization where the number of
function evaluations becomes prohibitively expensive. Motivated by these
applications, we study the adaptivity and query complexity of submodular
maximization. In this paper, we give the first constant-factor approximation
algorithm for maximizing a non-monotone submodular function subject to a
cardinality constraint that runs in adaptive rounds and makes
oracle queries in expectation. In our empirical study, we use
three real-world applications to compare our algorithm with several benchmarks
for non-monotone submodular maximization. The results demonstrate that our
algorithm finds competitive solutions using significantly fewer rounds and
queries.Comment: 12 pages, 8 figure
Optimal policy design for the sugar tax
Healthy nutrition promotions and regulations have long been regarded as a
tool for increasing social welfare. One of the avenues taken in the past decade
is sugar consumption regulation by introducing a sugar tax. Such a tax
increases the price of extensive sugar containment in products such as soft
drinks. In this article we consider a typical problem of optimal regulatory
policy design, where the task is to determine the sugar tax rate maximizing the
social welfare. We model the problem as a sequential game represented by the
three-level mathematical program. On the upper level, the government decides
upon the tax rate. On the middle level, producers decide on the product
pricing. On the lower level, consumers decide upon their preferences towards
the products. While the general problem is computationally intractable, the
problem with a few product types is polynomially solvable, even for an
arbitrary number of heterogeneous consumers. This paper presents a simple,
intuitive and easily implementable framework for computing optimal sugar tax in
a market with a few products. This resembles the reality as the soft drinks,
for instance, are typically categorized in either regular or no-sugar drinks,
e.g. Coca-Cola and Coca-Cola Zero. We illustrate the algorithm using an example
based on the real data and draw conclusions for a specific local market
- …