The Container Selection Problem
We introduce and study a network resource management problem that is a special case of non-metric k-median, naturally arising in cross platform scheduling and cloud computing. In the continuous d-dimensional container selection problem, we are given a set C of input points in d-dimensional Euclidean space, for some d >= 2, and a budget k. An input point p can be assigned to a "container point" c only if c dominates p in every dimension. The assignment cost is then equal to the L1-norm of the container point. The goal is to find k container points in the d-dimensional space, such that the total assignment cost for all input points is minimized. The discrete variant of the problem has one key distinction, namely, the container points must be chosen from a given set F of points.
For the continuous version, we obtain a polynomial time approximation scheme for any fixed dimension d>= 2. On the negative side, we show that the problem is NP-hard for any d>=3. We further show that the discrete version is significantly harder, as it is NP-hard to approximate without violating the budget k in any dimension d>=3. Thus, we focus on obtaining bi-approximation algorithms. For d=2, the bi-approximation guarantee is (1+epsilon,3), i.e., for any epsilon>0, our scheme outputs a solution of size 3k and cost at most (1+epsilon) times the optimum. For fixed d>2, we present a (1+epsilon,O((1/epsilon)log k)) bi-approximation algorithm.
non-metric k-median
geometric hitting set
approximation algorithms
cloud computing
cross platform scheduling.
416-434
Regular Paper
Viswanath
Nagarajan
Viswanath Nagarajan
Kanthi K.
Sarpatwar
Kanthi K. Sarpatwar
Baruch
Schieber
Baruch Schieber
Hadas
Shachnai
Hadas Shachnai
Joel L.
Wolf
Joel L. Wolf
10.4230/LIPIcs.APPROX-RANDOM.2015.416
Marcel R. Ackermann, Johannes Blömer, and Christian Sohler. Clustering for metric and nonmetric distance measures. ACM Transactions on Algorithms (TALG), 6(4):59, 2010.
Amazon EC2. In URL: http://aws.amazon.com/ec2/.
http://aws.amazon.com/ec2/
Hyung-Chan An, Aditya Bhaskara, Chandra Chekuri, Shalmoli Gupta, Vivek Madan, and Ola Svensson. Centrality of trees for capacitated k-center. In IPCO, pages 52-63, 2014.
Sanjeev Arora, Prabhakar Raghavan, and Satish Rao. Approximation schemes for euclidean k-medians and related problems. In STOC, pages 106-113, 1998.
Hervé Brönnimann and Michael T. Goodrich. Almost optimal set covers in finite vc-dimension. Discrete & Computational Geometry, 14(4):463-479, 1995.
Jaroslaw Byrka, Thomas Pensyl, Bartosz Rybicki, Aravind Srinivasan, and Khoa Trinh. An Improved Approximation for k-median, and Positive Correlation in Budgeted Optimization. In SODA, 2015.
Tomás Feder and Daniel H. Greene. Optimal algorithms for approximate clustering. In STOC, pages 434-444, 1988.
David Haussler and Emo Welzl. ε-Nets and Simplex Range Queries. Discrete &Computational Geometry, 2:127-151, 1987.
Benjamin Hindman, Andy Konwinski, Matei Zaharia, Ali Ghodsi, Anthony Joseph, Scott Shenker, and Ion Stoica. Mesos: A Platform for Fine-Grained Resource Sharing in the Data Center. In NSDI, 2011.
A. K. Jain, M. N. Murty, and P. J. Flynn. Data clustering: A review. ACM Comput. Surv., 31(3), September 1999.
Shi Li and Ola Svensson. Approximating k-median via pseudo-approximation. In STOC, pages 901-910, 2013.
Jyh-Han Lin and Jeffrey Scott Vitter. epsilon-approximations with minimum packing constraint violation (extended abstract). In STOC, pages 771-782, 1992.
Private cloud. In URL: http://wikipedia.org/wiki/Cloud_computing#Private_cloud.
http://wikipedia.org/wiki/Cloud_computing#Private_cloud
Evangelia Pyrga and Saurabh Ray. New existence proofs epsilon-nets. In SOCG, pages 199-207, 2008.
Baruch Schieber. Computing a Minimum Weight k-Link Path in Graphs with the Concave Monge Property. Journal of Algorithms, 29(2):204-222, 1998.
V. Vavilapalli, A. Murthy, C. Douglis, A. Agarwal, M. Konar, R. Evans, T. Graves, J. Lowe, H. Shah, S. Seth, B. Saha, C. Curino, O. O'Malley, S. Radia, B. Reed, and E. Baldeschwiele. Apache Hadoop YARN: Yet Another Resource Negotiator. In SoCC, 2013.
Joel Wolf, Zubair Nabi, Viswanath Nagarajan, Robert Saccone, Rohit Wagle, Kirsten Hildrum, Edward Ping, and Kanthi Sarpatwar. The X-Flex Cross-Platform Scheduler: Who’s The Fairest Of Them All? In Middleware, 2014.
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