229 research outputs found
Randomized Composable Core-sets for Distributed Submodular Maximization
An effective technique for solving optimization problems over massive data
sets is to partition the data into smaller pieces, solve the problem on each
piece and compute a representative solution from it, and finally obtain a
solution inside the union of the representative solutions for all pieces. This
technique can be captured via the concept of {\em composable core-sets}, and
has been recently applied to solve diversity maximization problems as well as
several clustering problems. However, for coverage and submodular maximization
problems, impossibility bounds are known for this technique \cite{IMMM14}. In
this paper, we focus on efficient construction of a randomized variant of
composable core-sets where the above idea is applied on a {\em random
clustering} of the data. We employ this technique for the coverage, monotone
and non-monotone submodular maximization problems. Our results significantly
improve upon the hardness results for non-randomized core-sets, and imply
improved results for submodular maximization in a distributed and streaming
settings.
In summary, we show that a simple greedy algorithm results in a
-approximate randomized composable core-set for submodular maximization
under a cardinality constraint. This is in contrast to a known impossibility result for (non-randomized) composable core-set. Our
result also extends to non-monotone submodular functions, and leads to the
first 2-round MapReduce-based constant-factor approximation algorithm with
total communication complexity for either monotone or non-monotone
functions. Finally, using an improved analysis technique and a new algorithm
, we present an improved -approximation algorithm
for monotone submodular maximization, which is in turn the first
MapReduce-based algorithm beating factor in a constant number of rounds
Estimation of surface turbulent heat fluxes via variational assimilation of sequences of land surface temperatures from Geostationary Operational Environmental Satellites
Recently, a number of studies have focused on estimating surface turbulent heat fluxes via assimilation of sequences of land surface temperature (LST) observations into variational data assimilation (VDA) schemes. Using the full heat diffusion equation as a constraint, the surface energy balance equation can be solved via assimilation of sequences of LST within a VDA framework. However, the VDA methods have been tested only in limited field sites that span only a few climate and land use types. Hence, in this study, combined-source (CS) and dual-source (DS) VDA schemes are tested extensively over six FluxNet sites with different vegetation covers (grassland, cropland, and forest) and climate conditions. The CS model groups the soil and canopy together as a single source and does not consider their different contributions to the total turbulent heat fluxes, while the DS model considers them to be different sources. LST data retrieved from the Geostationary Operational Environmental Satellites are assimilated into these two VDA schemes. Sensible and latent heat flux estimates from the CS and DS models are compared with the corresponding measurements from flux tower stations. The results indicate that the performance of both models at dry, lightly vegetated sites is better than that at wet, densely vegetated sites. Additionally, the DS model outperforms the CS model at all sites, implying that the DS scheme is more reliable and can characterize the underlying physics of the problem better
A PTAS for planar group Steiner tree via spanner bootstrapping and prize collecting
We present the first polynomial-time approximation scheme (PTAS), i.e., (1 + ϵ)-approximation algorithm for any constant ϵ > 0, for the planar group Steiner tree problem (in which each group lies on a boundary of a face). This result improves on the best previous approximation factor of O(logn(loglogn)O(1)). We achieve this result via a novel and powerful technique called spanner bootstrapping, which allows one to bootstrap from a superconstant approximation factor (even superpolynomial in the input size) all the way down to a PTAS. This is in contrast with the popular existing approach for planar PTASs of constructing lightweight spanners in one iteration, which notably requires a constant-factor approximate solution to start from. Spanner bootstrapping removes one of the main barriers for designing PTASs for problems which have no known constant-factor approximation (even on planar graphs), and thus can be used to obtain PTASs for several difficult-to-approximate problems. Our second major contribution required for the planar group Steiner tree PTAS is a spanner construction, which reduces the graph to have total weight within a factor of the optimal solution while approximately preserving the optimal solution. This is particularly challenging because group Steiner tree requires deciding which terminal in each group to connect by the tree, making it much harder than recent previous approaches to construct spanners for planar TSP by Klein [SIAM J. Computing 2008], subset TSP by Klein [STOC 2006], Steiner tree by Borradaile, Klein, and Mathieu [ACM Trans. Algorithms 2009], and Steiner forest by Bateni, Hajiaghayi, and Marx [J. ACM 2011] (and its improvement to an efficient PTAS by Eisenstat, Klein, and Mathieu [SODA 2012]. The main conceptual contribution here is realizing that selecting which terminals may be relevant is essentially a complicated prize-collecting process: we have to carefully weigh the cost and benefits of reaching or avoiding certain terminals in the spanner. Via a sequence of involved prize-collecting procedures, we can construct a spanner that reaches a set of terminals that is sufficient for an almost-optimal solution. Our PTAS for planar group Steiner tree implies the first PTAS for geometric Euclidean group Steiner tree with obstacles, as well as a (2 + ϵ)-approximation algorithm for group TSP with obstacles, improving over the best previous constant-factor approximation algorithms. By contrast, we show that planar group Steiner forest, a slight generalization of planar group Steiner tree, is APX-hard on planar graphs of treewidth 3, even if the groups are pairwise disjoint and every group is a vertex or an edge
Advances on Matroid Secretary Problems: Free Order Model and Laminar Case
The most well-known conjecture in the context of matroid secretary problems
claims the existence of a constant-factor approximation applicable to any
matroid. Whereas this conjecture remains open, modified forms of it were shown
to be true, when assuming that the assignment of weights to the secretaries is
not adversarial but uniformly random (Soto [SODA 2011], Oveis Gharan and
Vondr\'ak [ESA 2011]). However, so far, there was no variant of the matroid
secretary problem with adversarial weight assignment for which a
constant-factor approximation was found. We address this point by presenting a
9-approximation for the \emph{free order model}, a model suggested shortly
after the introduction of the matroid secretary problem, and for which no
constant-factor approximation was known so far. The free order model is a
relaxed version of the original matroid secretary problem, with the only
difference that one can choose the order in which secretaries are interviewed.
Furthermore, we consider the classical matroid secretary problem for the
special case of laminar matroids. Only recently, a constant-factor
approximation has been found for this case, using a clever but rather involved
method and analysis (Im and Wang, [SODA 2011]) that leads to a
16000/3-approximation. This is arguably the most involved special case of the
matroid secretary problem for which a constant-factor approximation is known.
We present a considerably simpler and stronger -approximation, based on reducing the problem to a matroid secretary
problem on a partition matroid
On Complexity of 1-Center in Various Metrics
We consider the classic 1-center problem: Given a set P of n points in a
metric space find the point in P that minimizes the maximum distance to the
other points of P. We study the complexity of this problem in d-dimensional
-metrics and in edit and Ulam metrics over strings of length d. Our
results for the 1-center problem may be classified based on d as follows.
Small d: We provide the first linear-time algorithm for 1-center
problem in fixed-dimensional metrics. On the other hand, assuming the
hitting set conjecture (HSC), we show that when , no
subquadratic algorithm can solve 1-center problem in any of the
-metrics, or in edit or Ulam metrics.
Large d. When , we extend our conditional lower bound
to rule out sub quartic algorithms for 1-center problem in edit metric
(assuming Quantified SETH). On the other hand, we give a
-approximation for 1-center in Ulam metric with running time
.
We also strengthen some of the above lower bounds by allowing approximations
or by reducing the dimension d, but only against a weaker class of algorithms
which list all requisite solutions. Moreover, we extend one of our hardness
results to rule out subquartic algorithms for the well-studied 1-median problem
in the edit metric, where given a set of n strings each of length n, the goal
is to find a string in the set that minimizes the sum of the edit distances to
the rest of the strings in the set
Pricing in Social Networks with Negative Externalities
We study the problems of pricing an indivisible product to consumers who are
embedded in a given social network. The goal is to maximize the revenue of the
seller. We assume impatient consumers who buy the product as soon as the seller
posts a price not greater than their values of the product. The product's value
for a consumer is determined by two factors: a fixed consumer-specified
intrinsic value and a variable externality that is exerted from the consumer's
neighbors in a linear way. We study the scenario of negative externalities,
which captures many interesting situations, but is much less understood in
comparison with its positive externality counterpart. We assume complete
information about the network, consumers' intrinsic values, and the negative
externalities. The maximum revenue is in general achieved by iterative pricing,
which offers impatient consumers a sequence of prices over time.
We prove that it is NP-hard to find an optimal iterative pricing, even for
unweighted tree networks with uniform intrinsic values. Complementary to the
hardness result, we design a 2-approximation algorithm for finding iterative
pricing in general weighted networks with (possibly) nonuniform intrinsic
values. We show that, as an approximation to optimal iterative pricing, single
pricing can work rather well for many interesting cases, but theoretically it
can behave arbitrarily bad
Covering problems in edge- and node-weighted graphs
This paper discusses the graph covering problem in which a set of edges in an
edge- and node-weighted graph is chosen to satisfy some covering constraints
while minimizing the sum of the weights. In this problem, because of the large
integrality gap of a natural linear programming (LP) relaxation, LP rounding
algorithms based on the relaxation yield poor performance. Here we propose a
stronger LP relaxation for the graph covering problem. The proposed relaxation
is applied to designing primal-dual algorithms for two fundamental graph
covering problems: the prize-collecting edge dominating set problem and the
multicut problem in trees. Our algorithms are an exact polynomial-time
algorithm for the former problem, and a 2-approximation algorithm for the
latter problem, respectively. These results match the currently known best
results for purely edge-weighted graphs.Comment: To appear in SWAT 201
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