37,340 research outputs found
Large-Scale Distributed Algorithms for Facility Location with Outliers
This paper presents fast, distributed, O(1)-approximation algorithms for metric facility location problems with outliers in the Congested Clique model, Massively Parallel Computation (MPC) model, and in the k-machine model. The paper considers Robust Facility Location and Facility Location with Penalties, two versions of the facility location problem with outliers proposed by Charikar et al. (SODA 2001). The paper also considers two alternatives for specifying the input: the input metric can be provided explicitly (as an n x n matrix distributed among the machines) or implicitly as the shortest path metric of a given edge-weighted graph. The results in the paper are:
- Implicit metric: For both problems, O(1)-approximation algorithms running in O(poly(log n)) rounds in the Congested Clique and the MPC model and O(1)-approximation algorithms running in O~(n/k) rounds in the k-machine model.
- Explicit metric: For both problems, O(1)-approximation algorithms running in O(log log log n) rounds in the Congested Clique and the MPC model and O(1)-approximation algorithms running in O~(n/k) rounds in the k-machine model.
Our main contribution is to show the existence of Mettu-Plaxton-style O(1)-approximation algorithms for both Facility Location with outlier problems. As shown in our previous work (Berns et al., ICALP 2012, Bandyapadhyay et al., ICDCN 2018) Mettu-Plaxton style algorithms are more easily amenable to being implemented efficiently in distributed and large-scale models of computation
Scalable Facility Location for Massive Graphs on Pregel-like Systems
We propose a new scalable algorithm for facility location. Facility location
is a classic problem, where the goal is to select a subset of facilities to
open, from a set of candidate facilities F , in order to serve a set of clients
C. The objective is to minimize the total cost of opening facilities plus the
cost of serving each client from the facility it is assigned to. In this work,
we are interested in the graph setting, where the cost of serving a client from
a facility is represented by the shortest-path distance on the graph. This
setting allows to model natural problems arising in the Web and in social media
applications. It also allows to leverage the inherent sparsity of such graphs,
as the input is much smaller than the full pairwise distances between all
vertices.
To obtain truly scalable performance, we design a parallel algorithm that
operates on clusters of shared-nothing machines. In particular, we target
modern Pregel-like architectures, and we implement our algorithm on Apache
Giraph. Our solution makes use of a recent result to build sketches for massive
graphs, and of a fast parallel algorithm to find maximal independent sets, as
building blocks. In so doing, we show how these problems can be solved on a
Pregel-like architecture, and we investigate the properties of these
algorithms. Extensive experimental results show that our algorithm scales
gracefully to graphs with billions of edges, while obtaining values of the
objective function that are competitive with a state-of-the-art sequential
algorithm
Nearly Linear-Work Algorithms for Mixed Packing/Covering and Facility-Location Linear Programs
We describe the first nearly linear-time approximation algorithms for
explicitly given mixed packing/covering linear programs, and for (non-metric)
fractional facility location. We also describe the first parallel algorithms
requiring only near-linear total work and finishing in polylog time. The
algorithms compute -approximate solutions in time (and work)
, where is the number of non-zeros in the constraint
matrix. For facility location, is the number of eligible client/facility
pairs
Optimistic Concurrency Control for Distributed Unsupervised Learning
Research on distributed machine learning algorithms has focused primarily on
one of two extremes - algorithms that obey strict concurrency constraints or
algorithms that obey few or no such constraints. We consider an intermediate
alternative in which algorithms optimistically assume that conflicts are
unlikely and if conflicts do arise a conflict-resolution protocol is invoked.
We view this "optimistic concurrency control" paradigm as particularly
appropriate for large-scale machine learning algorithms, particularly in the
unsupervised setting. We demonstrate our approach in three problem areas:
clustering, feature learning and online facility location. We evaluate our
methods via large-scale experiments in a cluster computing environment.Comment: 25 pages, 5 figure
Super-Fast Distributed Algorithms for Metric Facility Location
This paper presents a distributed O(1)-approximation algorithm, with
expected- running time, in the model for
the metric facility location problem on a size- clique network. Though
metric facility location has been considered by a number of researchers in
low-diameter settings, this is the first sub-logarithmic-round algorithm for
the problem that yields an O(1)-approximation in the setting of non-uniform
facility opening costs. In order to obtain this result, our paper makes three
main technical contributions. First, we show a new lower bound for metric
facility location, extending the lower bound of B\u{a}doiu et al. (ICALP 2005)
that applies only to the special case of uniform facility opening costs. Next,
we demonstrate a reduction of the distributed metric facility location problem
to the problem of computing an O(1)-ruling set of an appropriate spanning
subgraph. Finally, we present a sub-logarithmic-round (in expectation)
algorithm for computing a 2-ruling set in a spanning subgraph of a clique. Our
algorithm accomplishes this by using a combination of randomized and
deterministic sparsification.Comment: 15 pages, 2 figures. This is the full version of a paper that
appeared in ICALP 201
Separable Concave Optimization Approximately Equals Piecewise-Linear Optimization
We study the problem of minimizing a nonnegative separable concave function
over a compact feasible set. We approximate this problem to within a factor of
1+epsilon by a piecewise-linear minimization problem over the same feasible
set. Our main result is that when the feasible set is a polyhedron, the number
of resulting pieces is polynomial in the input size of the polyhedron and
linear in 1/epsilon. For many practical concave cost problems, the resulting
piecewise-linear cost problem can be formulated as a well-studied discrete
optimization problem. As a result, a variety of polynomial-time exact
algorithms, approximation algorithms, and polynomial-time heuristics for
discrete optimization problems immediately yield fully polynomial-time
approximation schemes, approximation algorithms, and polynomial-time heuristics
for the corresponding concave cost problems.
We illustrate our approach on two problems. For the concave cost
multicommodity flow problem, we devise a new heuristic and study its
performance using computational experiments. We are able to approximately solve
significantly larger test instances than previously possible, and obtain
solutions on average within 4.27% of optimality. For the concave cost facility
location problem, we obtain a new 1.4991+epsilon approximation algorithm.Comment: Full pape
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