11,020 research outputs found
A Local Search Algorithm for the Min-Sum Submodular Cover Problem
We consider the problem of solving the Min-Sum Submodular Cover problem using
local search. The Min-Sum Submodular Cover problem generalizes the NP-complete
Min-Sum Set Cover problem, replacing the input set cover instance with a
monotone submodular set function. A simple greedy algorithm achieves an
approximation factor of 4, which is tight unless P=NP [Streeter and Golovin,
NeurIPS, 2008]. We complement the greedy algorithm with analysis of a local
search algorithm. Building on work of Munagala et al. [ICDT, 2005], we show
that, using simple initialization, a straightforward local search algorithm
achieves a -approximate solution in time
, provided that the monotone submodular set function is
also second-order supermodular. Second-order supermodularity has been shown to
hold for a number of submodular functions of practical interest, including
functions associated with set cover, matching, and facility location. We
present experiments on two special cases of Min-Sum Submodular Cover and find
that the local search algorithm can outperform the greedy algorithm on small
data sets
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
Almost Optimal Streaming Algorithms for Coverage Problems
Maximum coverage and minimum set cover problems --collectively called
coverage problems-- have been studied extensively in streaming models. However,
previous research not only achieve sub-optimal approximation factors and space
complexities, but also study a restricted set arrival model which makes an
explicit or implicit assumption on oracle access to the sets, ignoring the
complexity of reading and storing the whole set at once. In this paper, we
address the above shortcomings, and present algorithms with improved
approximation factor and improved space complexity, and prove that our results
are almost tight. Moreover, unlike most of previous work, our results hold on a
more general edge arrival model. More specifically, we present (almost) optimal
approximation algorithms for maximum coverage and minimum set cover problems in
the streaming model with an (almost) optimal space complexity of
, i.e., the space is {\em independent of the size of the sets or
the size of the ground set of elements}. These results not only improve over
the best known algorithms for the set arrival model, but also are the first
such algorithms for the more powerful {\em edge arrival} model. In order to
achieve the above results, we introduce a new general sketching technique for
coverage functions: This sketching scheme can be applied to convert an
-approximation algorithm for a coverage problem to a
(1-\eps)\alpha-approximation algorithm for the same problem in streaming, or
RAM models. We show the significance of our sketching technique by ruling out
the possibility of solving coverage problems via accessing (as a black box) a
(1 \pm \eps)-approximate oracle (e.g., a sketch function) that estimates the
coverage function on any subfamily of the sets
An Efficient Streaming Algorithm for the Submodular Cover Problem
We initiate the study of the classical Submodular Cover (SC) problem in the
data streaming model which we refer to as the Streaming Submodular Cover (SSC).
We show that any single pass streaming algorithm using sublinear memory in the
size of the stream will fail to provide any non-trivial approximation
guarantees for SSC. Hence, we consider a relaxed version of SSC, where we only
seek to find a partial cover.
We design the first Efficient bicriteria Submodular Cover Streaming
(ESC-Streaming) algorithm for this problem, and provide theoretical guarantees
for its performance supported by numerical evidence. Our algorithm finds
solutions that are competitive with the near-optimal offline greedy algorithm
despite requiring only a single pass over the data stream. In our numerical
experiments, we evaluate the performance of ESC-Streaming on active set
selection and large-scale graph cover problems.Comment: To appear in NIPS'1
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