4,009 research outputs found
LP-Based Algorithms for Capacitated Facility Location
Linear programming has played a key role in the study of algorithms for
combinatorial optimization problems. In the field of approximation algorithms,
this is well illustrated by the uncapacitated facility location problem. A
variety of algorithmic methodologies, such as LP-rounding and primal-dual
method, have been applied to and evolved from algorithms for this problem.
Unfortunately, this collection of powerful algorithmic techniques had not yet
been applicable to the more general capacitated facility location problem. In
fact, all of the known algorithms with good performance guarantees were based
on a single technique, local search, and no linear programming relaxation was
known to efficiently approximate the problem.
In this paper, we present a linear programming relaxation with constant
integrality gap for capacitated facility location. We demonstrate that the
fundamental theories of multi-commodity flows and matchings provide key
insights that lead to the strong relaxation. Our algorithmic proof of
integrality gap is obtained by finally accessing the rich toolbox of LP-based
methodologies: we present a constant factor approximation algorithm based on
LP-rounding.Comment: 25 pages, 6 figures; minor revision
A new approximation algorithm for the multilevel facility location problem
In this paper we propose a new integer programming formulation for the multi-level facility location problem and a novel 3-approximation algorithm based on LP rounding. The linear program we are using has a polynomial number of variables and constraints, being thus more efficient than the one commonly used in the approximation algorithms for this type of problems
Improved approximation algorithm for k-level UFL with penalties, a simplistic view on randomizing the scaling parameter
The state of the art in approximation algorithms for facility location
problems are complicated combinations of various techniques. In particular, the
currently best 1.488-approximation algorithm for the uncapacitated facility
location (UFL) problem by Shi Li is presented as a result of a non-trivial
randomization of a certain scaling parameter in the LP-rounding algorithm by
Chudak and Shmoys combined with a primal-dual algorithm of Jain et al. In this
paper we first give a simple interpretation of this randomization process in
terms of solving an aux- iliary (factor revealing) LP. Then, armed with this
simple view point, Abstract. we exercise the randomization on a more
complicated algorithm for the k-level version of the problem with penalties in
which the planner has the option to pay a penalty instead of connecting chosen
clients, which results in an improved approximation algorithm
Approximation algorithms for clustering and facility location problems
In this thesis we design and analyze algorithms for various facility location and clustering problems. The problems we study are NP-Hard and therefore, assuming P is not equal NP, there do not exist polynomial time algorithms to solve them optimally. One approach to cope with the intractability of these problems is to design approximation algorithms which run in polynomial-time and output a near-optimal solution for all instances of the problem. However these algorithms do not always work well in practice. Often heuristics with no explicit approximation guarantee perform quite well. To bridge this gap between theory and practice, and to design algorithms that are tuned for instances arising in practice, there is an increasing emphasis on beyond worst-case analysis. In this thesis we consider both these approaches.
In the first part we design worst case approximation algorithms for Uniform Submodular Facility Location (USFL), and Capacitated k-center (CapKCenter) problems. USFL is a generalization of the well-known Uncapacitated Facility Location problem. In USFL the cost of opening a facility is a submodular function of the clients assigned to it (the function is identical for all facilities). We show that a natural greedy algorithm (which gives constant factor approximation for Uncapacitated Facility Location and other facility location problems) has a lower bound of log(n), where n is the number of clients. We present an O(log^2 k) approximation algorithm where k is the number of facilities. The algorithm is based on rounding a convex relaxation. We further consider several special cases of the problem and give improved approximation bounds for them. The CapKCenter problem is an extension of the well-known k-center problem: each facility has a maximum capacity on the number of clients that can be assigned to it. We obtain a 9-approximation for this problem via a linear programming (LP) rounding procedure. Our result, combined with previously known lower bounds, almost settles the integrality gap for a natural LP relaxation.
In the second part we consider several well-known clustering problems like k-center, k-median, k-means and their corresponding outlier variants. We use beyond worst-case analysis due to the practical relevance of these problems. In particular we show that when the input instances are 2-perturbation resilient (i.e. the optimal solution does not change when the distances change by a multiplicative factor of 2), the LP integrality gap for k-center (and also asymmetric k-center) is 1. We further introduce a model of perturbation resilience for clustering with outliers. Under this new model, we show that previous results (including our LP integrality result) known for clustering under perturbation resilience also extend for clustering with outliers. This leads to a dynamic programming based heuristic for k-means with outliers (k-means-outlier) which gives an optimal solution when the instance is 2-perturbation resilient. We propose two more algorithms for k-means-outlier — a sampling based algorithm which gives an O(1) approximation when the optimal clusters are not “too small”, and an LP rounding algorithm which gives an O(1) approximation at the expense of violating the number of clusters and outliers by a small constant. We empirically study our proposed algorithms on several clustering datasets
Approximation algorithms for stochastic and risk-averse optimization
We present improved approximation algorithms in stochastic optimization. We
prove that the multi-stage stochastic versions of covering integer programs
(such as set cover and vertex cover) admit essentially the same approximation
algorithms as their standard (non-stochastic) counterparts; this improves upon
work of Swamy \& Shmoys which shows an approximability that depends
multiplicatively on the number of stages. We also present approximation
algorithms for facility location and some of its variants in the -stage
recourse model, improving on previous approximation guarantees. We give a
-approximation algorithm in the standard polynomial-scenario model and
an algorithm with an expected per-scenario -approximation guarantee,
which is applicable to the more general black-box distribution model.Comment: Extension of a SODA'07 paper. To appear in SIAM J. Discrete Mat
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