115 research outputs found
Submodular Optimization with Submodular Cover and Submodular Knapsack Constraints
We investigate two new optimization problems -- minimizing a submodular
function subject to a submodular lower bound constraint (submodular cover) and
maximizing a submodular function subject to a submodular upper bound constraint
(submodular knapsack). We are motivated by a number of real-world applications
in machine learning including sensor placement and data subset selection, which
require maximizing a certain submodular function (like coverage or diversity)
while simultaneously minimizing another (like cooperative cost). These problems
are often posed as minimizing the difference between submodular functions [14,
35] which is in the worst case inapproximable. We show, however, that by
phrasing these problems as constrained optimization, which is more natural for
many applications, we achieve a number of bounded approximation guarantees. We
also show that both these problems are closely related and an approximation
algorithm solving one can be used to obtain an approximation guarantee for the
other. We provide hardness results for both problems thus showing that our
approximation factors are tight up to log-factors. Finally, we empirically
demonstrate the performance and good scalability properties of our algorithms.Comment: 23 pages. A short version of this appeared in Advances of NIPS-201
Maximizing a Submodular Function with Bounded Curvature under an Unknown Knapsack Constraint
This paper studies the problem of maximizing a monotone submodular function
under an unknown knapsack constraint. A solution to this problem is a policy
that decides which item to pack next based on the past packing history. The
robustness factor of a policy is the worst case ratio of the solution obtained
by following the policy and an optimal solution that knows the knapsack
capacity. We develop an algorithm with a robustness factor that is decreasing
in the curvature of the submodular function. For the extreme cases
corresponding to a modular objective, it matches a previously known and best
possible robustness factor of . For the other extreme case of it
yields a robustness factor of improving over the best previously
known robustness factor of
Maximizing a Submodular Function with Bounded Curvature Under an Unknown Knapsack Constraint
This paper studies the problem of maximizing a monotone submodular function under an unknown knapsack constraint. A solution to this problem is a policy that decides which item to pack next based on the past packing history. The robustness factor of a policy is the worst case ratio of the solution obtained by following the policy and an optimal solution that knows the knapsack capacity. We develop an algorithm with a robustness factor that is decreasing in the curvature c of the submodular function. For the extreme cases c = 0 corresponding to a modular objective, it matches a previously known and best possible robustness factor of 1/2. For the other extreme case of c = 1 it yields a robustness factor of ? 0.35 improving over the best previously known robustness factor of ? 0.06
Test Score Algorithms for Budgeted Stochastic Utility Maximization
Motivated by recent developments in designing algorithms based on individual
item scores for solving utility maximization problems, we study the framework
of using test scores, defined as a statistic of observed individual item
performance data, for solving the budgeted stochastic utility maximization
problem. We extend an existing scoring mechanism, namely the replication test
scores, to incorporate heterogeneous item costs as well as item values. We show
that a natural greedy algorithm that selects items solely based on their
replication test scores outputs solutions within a constant factor of the
optimum for a broad class of utility functions. Our algorithms and
approximation guarantees assume that test scores are noisy estimates of certain
expected values with respect to marginal distributions of individual item
values, thus making our algorithms practical and extending previous work that
assumes noiseless estimates. Moreover, we show how our algorithm can be adapted
to the setting where items arrive in a streaming fashion while maintaining the
same approximation guarantee. We present numerical results, using synthetic
data and data sets from the Academia.StackExchange Q&A forum, which show that
our test score algorithm can achieve competitiveness, and in some cases better
performance than a benchmark algorithm that requires access to a value oracle
to evaluate function values
Randomized Strategies for Robust Combinatorial Optimization
In this paper, we study the following robust optimization problem. Given an
independence system and candidate objective functions, we choose an independent
set, and then an adversary chooses one objective function, knowing our choice.
Our goal is to find a randomized strategy (i.e., a probability distribution
over the independent sets) that maximizes the expected objective value. To
solve the problem, we propose two types of schemes for designing approximation
algorithms. One scheme is for the case when objective functions are linear. It
first finds an approximately optimal aggregated strategy and then retrieves a
desired solution with little loss of the objective value. The approximation
ratio depends on a relaxation of an independence system polytope. As
applications, we provide approximation algorithms for a knapsack constraint or
a matroid intersection by developing appropriate relaxations and retrievals.
The other scheme is based on the multiplicative weights update method. A key
technique is to introduce a new concept called -reductions for
objective functions with parameters . We show that our scheme
outputs a nearly -approximate solution if there exists an
-approximation algorithm for a subproblem defined by
-reductions. This improves approximation ratio in previous
results. Using our result, we provide approximation algorithms when the
objective functions are submodular or correspond to the cardinality robustness
for the knapsack problem
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