40 research outputs found
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
Scheduling Algorithms for Procrastinators
This paper presents scheduling algorithms for procrastinators, where the
speed that a procrastinator executes a job increases as the due date
approaches. We give optimal off-line scheduling policies for linearly
increasing speed functions. We then explain the computational/numerical issues
involved in implementing this policy. We next explore the online setting,
showing that there exist adversaries that force any online scheduling policy to
miss due dates. This impossibility result motivates the problem of minimizing
the maximum interval stretch of any job; the interval stretch of a job is the
job's flow time divided by the job's due date minus release time. We show that
several common scheduling strategies, including the "hit-the-highest-nail"
strategy beloved by procrastinators, have arbitrarily large maximum interval
stretch. Then we give the "thrashing" scheduling policy and show that it is a
\Theta(1) approximation algorithm for the maximum interval stretch.Comment: 12 pages, 3 figure
Unconstrained and Constrained Fault-Tolerant Resource Allocation
First, we study the Unconstrained Fault-Tolerant Resource Allocation (UFTRA)
problem (a.k.a. FTFA problem in \cite{shihongftfa}). In the problem, we are
given a set of sites equipped with an unconstrained number of facilities as
resources, and a set of clients with set as corresponding
connection requirements, where every facility belonging to the same site has an
identical opening (operating) cost and every client-facility pair has a
connection cost. The objective is to allocate facilities from sites to satisfy
at a minimum total cost. Next, we introduce the Constrained
Fault-Tolerant Resource Allocation (CFTRA) problem. It differs from UFTRA in
that the number of resources available at each site is limited by .
Both problems are practical extensions of the classical Fault-Tolerant Facility
Location (FTFL) problem \cite{Jain00FTFL}. For instance, their solutions
provide optimal resource allocation (w.r.t. enterprises) and leasing (w.r.t.
clients) strategies for the contemporary cloud platforms.
In this paper, we consider the metric version of the problems. For UFTRA with
uniform , we present a star-greedy algorithm. The algorithm
achieves the approximation ratio of 1.5186 after combining with the cost
scaling and greedy augmentation techniques similar to
\cite{Charikar051.7281.853,Mahdian021.52}, which significantly improves the
result of \cite{shihongftfa} using a phase-greedy algorithm. We also study the
capacitated extension of UFTRA and give a factor of 2.89. For CFTRA with
uniform , we slightly modify the algorithm to achieve
1.5186-approximation. For a more general version of CFTRA, we show that it is
reducible to FTFL using linear programming
Approximation algorithms for fault tolerant facility allocation
Given nf sites, each equipped with one facility, and n c cities, fault tolerant facility location (FTFL) [K. Jain and V. V. Vazirani, APPROX '00: Proceedings of the Third International Workshop on Approximation Algorithms for Combinatorial Optimization, Spinger, New York, 2000, pp. 177-183] requires computing a minimum-cost connection scheme such that each city connects to a specified number of facilities. When each city connects to exactly one facility, FTFL becomes the classical uncapacitated facility location problem (UFL) that is well-known NP hard. The current best solution to FTFL admits an approximation ratio 1.7245 due to Byrka, Srinivasan, and Swamy applying the dependent rounding technique announced recently [Proceedings of IPCO, 2010, pp. 244-257], which improves the ratio 2.076 obtained by Swamy and Shmoys based on LP rounding [ACM Trans. Algorithms, 4 (2008), pp. 1-27]. In this paper, we study a variant of the FTFL problem, namely, fault tolerant facility allocation (FTFA), as another generalization of UFL by allowing each site to hold multiple facilities and show that we can obtain better solutions for this problem. We first give two algorithms with 1.81 and 1.61 approximation ratios in time complexity O(mRlogm) and O(Rn3), respectively, where R is the maximum number of facilities required by any city, m = nfnc, and n = max{ nf, nc}. Instead of applying the dual-fitting technique that reduces the dual problem's solution to fit the original problem as used in the literature [K. Jain et al., Journal of the ACM, 50 (2003), pp. 795-824; K. Jain, M. Mahdian, and A. Saberi, STOC'02: Proceedings of the 34th Annual ACM Symposium on the Theory of Computing, New York, 2002, pp. 731-740; A. Saberi et al., Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques, Springer, New York, 2001, pp. 127-137], we propose a method called inverse dual-fitting that alters the original problem to fit the dual solution and show that this method is more effective for obtaining solutions of multifactor approximation. We show that applying inverse dual-fitting and factor-revealing techniques our second algorithm is also (1.11,1.78)- And (1,2)-approximation simultaneously. These results can be further used to achieve solutions of 1.52-approximation to FTFA and 4-approximation to the fault tolerant k-facility allocation problem in which the total number of facilities is bounded by k. These are currently the best bifactor and single-factor approximation ratios for the problems concerned. ©2013 Society for Industrial and Applied Mathematics.Hong Shen and Shihong X
Molecular mechanisms by which HERV-K Gag interferes with HIV-1 Gag assembly and particle infectivity
A 5-approximation for capacitated facility location
In this paper, we propose and analyze a local search algorithm for the capacitated facility location problem. Our algorithm is a modification of the algorithm proposed by Zhang et al. [7] and improves the approximation ratio from 5.83 to 5. We achieve this by modifying the close, open and multi operations. The idea of taking linear combinations of inequalities used in Aggarwal et al [1] is crucial in achieving this result. The example proposed by Zhang et al. also shows that our analysis is tight