We investigate the problem of computing a minimum set of solutions that approximates within a specified accuracy i¾?the Pareto curve of a multiobjective optimization problem. We show that for a broad class of bi-objective problems (containing many important widely studied problems such as shortest paths, spanning tree, and many others), we can compute in polynomial time an i¾?-Pareto set that contains at most twice as many solutions as the minimum such set. Furthermore we show that the factor of 2 is tight for these problems, i.e., it is NP-hard to do better. We present further results for three or more objectives, as well as for the dual problem of computing a specified number kof solutions which provide a good approximation to the Pareto curve.

Small Approximate Pareto Sets for Bi-objective Shortest Paths and Other Problems

Recent advances in virtualization technology have made it a common practice to consolidate virtual machines(VMs) into a fewer number of servers. An efficient consolidation scheme requires that VMs are packed tightly, yet receive resources commensurate with their demands. However, measurements from production data centers show that the network bandwidth demands of VMs are dynamic, making it difficult to characterize the demands by a fixed value and to apply traditional consolidation schemes. In this work, we formulate the VM consolidation into a Stochastic Bin Packing problem and propose an online packing algorithm by which the number of servers required is within equation of the optimum for any ∈ > 0. The result can be improved to within equation of the optimum in a special case. In addition, we use numerical experiments to evaluate the proposed consolidation algorithm and observe 30% server reduction compared to several benchmark algorithms.

[IEEE IEEE INFOCOM 2011 - IEEE Conference on Computer Communications - Shanghai, China (2011.04.10-2011.04.15)] 2011 Proceedings IEEE INFOCOM - Consolidating virtual machines with dynamic bandwidth demand in data centers

https://pure.tue.nl/ws/files/4278084/592901.pdf

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Consolidating virtual machines with dynamic bandwidth demand in data centers

In the last few years, reinforcement learning (RL), also called adaptive (or approximate) dynamic programming, has emerged as a powerful tool for solving complex sequential decision-making problems in control theory. Although seminal research in this area was performed in the artificial intelligence (AI) community, more recently it has attracted the attention of optimization theorists because of several noteworthy success stories from operations management. It is on large-scale and complex problems of dynamic optimization, in particular the Markov decision problem (MDP) and its variants, that the power of RL becomes more obvious. It has been known for many years that on large-scale MDPs, the curse of dimensionality and the curse of modeling render classical dynamic programming (DP) ineffective. The excitement in RL stems from its direct attack on these curses, which allows it to solve problems that were considered intractable via classical DP in the past. The success of RL is due to its strong mathematical roots in the principles of DP, Monte Carlo simulation, function approximation, and AI. Topics treated in some detail in this survey are temporal differences, Q-learning, semi-MDPs, and stochastic games. Several recent advances in RL, e.g., policy gradients and hierarchical RL, are covered along with references. Pointers to numerous examples of applications are provided. This overview is aimed at uncovering the mathematical roots of this science so that readers gain a clear understanding of the core concepts and are able to use them in their own research. The survey points to more than 100 references from the literature.

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Reinforcement Learning: A Tutorial Survey and Recent Advances

We deal with the problem of minimizing makespan on a single batch processing machine. In this problem, each job has both processing time and size (capacity requirement). The batch processing machine can process a number of jobs simultaneously as long as the total size of these jobs being processed does not exceed the machine capacity. The processing time of a batch is just the processing time of the longest job in the batch. An approximation algorithm with worst-case ratio 3/2 is given for the version where the processing times of large jobs (with sizes greater than 1/2) are not less than those of small jobs (with sizes not greater than 1/2). This result is the best possible unless P = NP. For the general case, we propose an approximation algorithm with worst-case ratio 7/4. A number of heuristics by Uzosy are also analyzed and compared. © 2001 John Wiley & Sons, Inc. Naval Research Logistics 48: 226–240, 2001

Minimizing Makespan on a Single Batch Processing Machine with Nonidentical Job Sizes

This paper investigates two different semi on-line scheduling problems on a two-machine system. In the first case, we assume that all jobs have their processing times in between p and rp

 $(p>0, r\geq1)$
 . In the second case, we assume that the largest processing time is known in advance. We show that one has a best possible algorithm with worst case ratio 4/3 while LS is still the best possible for the other problem with ratio $(r+1)/2$
 which is still $3/2$
 in the worst case $r=2$
 .

Semi on-line scheduling on two identical machines

Strategy-proof approximation mechanisms for an obnoxious facility game on networks

We consider the following rectangle packing problem. Given a set of rectangles, each of which is associated with a profit, we are requested to pack a subset of the rectangles into a bigger rectangle so that the total profit of rectangles packed is maximized. The rectangles may not overlap. This problem is strongly NP-hard even for packing squares with identical profits. We first present a simple (3 + e)-approximation algorithm. Then we consider a restricted version of the problem and show a (2 + e)-approximation algorithm. This restricted problem includes the case where rotation by 90° is allowed (and is possible), and the case of packing squares. We apply a similar technique to the general problem, and get an improved algorithm with a worst-case ratio of at most 5/2 + e. Finally, we devise a (2 + e)-approximation algorithm for the general problem.

Guochuan Zhang

Papers

Minimizing Makespan on a Single Batch Processing Machine with Nonidentical Job Sizes

Semi on-line scheduling on two identical machines

Strategy-proof approximation mechanisms for an obnoxious facility game on networks

Maximizing the Total Profit of Rectangles Packed into a Rectangle

A constrained minimum spanning tree problem