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

A capacity scaling algorithm for the constrained maximum flow problem

Includes bibliographical references (p. 19-20).Supported by the Air Force Office of Scientific Research. AFORS-88-0088 Supported by grants from UPS and Prime Computer Corporation.Ravindra K. Ahuja, I.I.T. Kanpur and James B. Orlin

Risk Aversion in Finite Markov Decision Processes Using Total Cost Criteria and Average Value at Risk

In this paper we present an algorithm to compute risk averse policies in Markov Decision Processes (MDP) when the total cost criterion is used together with the average value at risk (AVaR) metric. Risk averse policies are needed when large deviations from the expected behavior may have detrimental effects, and conventional MDP algorithms usually ignore this aspect. We provide conditions for the structure of the underlying MDP ensuring that approximations for the exact problem can be derived and solved efficiently. Our findings are novel inasmuch as average value at risk has not previously been considered in association with the total cost criterion. Our method is demonstrated in a rapid deployment scenario, whereby a robot is tasked with the objective of reaching a target location within a temporal deadline where increased speed is associated with increased probability of failure. We demonstrate that the proposed algorithm not only produces a risk averse policy reducing the probability of exceeding the expected temporal deadline, but also provides the statistical distribution of costs, thus offering a valuable analysis tool

Learning Deployable Navigation Policies at Kilometer Scale from a Single Traversal

Model-free reinforcement learning has recently been shown to be effective at learning navigation policies from complex image input. However, these algorithms tend to require large amounts of interaction with the environment, which can be prohibitively costly to obtain on robots in the real world. We present an approach for efficiently learning goal-directed navigation policies on a mobile robot, from only a single coverage traversal of recorded data. The navigation agent learns an effective policy over a diverse action space in a large heterogeneous environment consisting of more than 2km of travel, through buildings and outdoor regions that collectively exhibit large variations in visual appearance, self-similarity, and connectivity. We compare pretrained visual encoders that enable precomputation of visual embeddings to achieve a throughput of tens of thousands of transitions per second at training time on a commodity desktop computer, allowing agents to learn from millions of trajectories of experience in a matter of hours. We propose multiple forms of computationally efficient stochastic augmentation to enable the learned policy to generalise beyond these precomputed embeddings, and demonstrate successful deployment of the learned policy on the real robot without fine tuning, despite environmental appearance differences at test time. The dataset and code required to reproduce these results and apply the technique to other datasets and robots is made publicly available at rl-navigation.github.io/deployable

Analysis, preliminary design and simulation systems for control-structure interaction problems

Software aspects of control-structure interaction (CSI) analysis are discussed. The following subject areas are covered: (1) implementation of a partitioned algorithm for simulation of large CSI problems; (2) second-order discrete Kalman filtering equations for CSI simulations; and (3) parallel computations and control of adaptive structures

Differential energy saving algorithms in a distributed router architecture

A distributed multistage software router (MSSR) is composed by several interconnected software routers running on personal computers (PCs). The MSSR architecture overcomes scalability and performance issues of single software router by providing parallel forwarding paths. Like many networking devices, a MSSR must be sized for peak traffic load, which implies energy inefficiency at low loads. Thus, we focus on energy saving schemes to improve the router energy efficiency by dynamically adapting the MSSR architecture to the currently offered load. We first introduce an optimal energy saving algorithm defined as a mixed integer linear programming (MILP) optimization model. Then, heuristic solutions, named differential algorithms are discussed. While the optimal approach provides higher energy savings, the heuristics avoid the complete MSSR reconfiguration, thus reducing forwarding delays and minimizing service interruption. The performance evaluation shows that the proposed heuristic algorithms, that gracefully modifies the internal MSSR configuration, preserve the load proportional energy demand characteristics of the optimal algorithm, with a minimal loss of efficiency, largely compensated by algorithm simplicity

Lecture notes on ridge regression

The linear regression model cannot be fitted to high-dimensional data, as the high-dimensionality brings about empirical non-identifiability. Penalized regression overcomes this non-identifiability by augmentation of the loss function by a penalty (i.e. a function of regression coefficients). The ridge penalty is the sum of squared regression coefficients, giving rise to ridge regression. Here many aspect of ridge regression are reviewed e.g. moments, mean squared error, its equivalence to constrained estimation, and its relation to Bayesian regression. Finally, its behaviour and use are illustrated in simulation and on omics data. Subsequently, ridge regression is generalized to allow for a more general penalty. The ridge penalization framework is then translated to logistic regression and its properties are shown to carry over. To contrast ridge penalized estimation, the final chapter introduces its lasso counterpart