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Register a new userScheduling Under Power and Energy Constraints
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Added by
Shrisha Rao
Publication date
2016
fields
Informatics Engineering
and research's language is
English
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Given a system model where machines have distinct speeds and power ratings but are otherwise compatible, we consider various problems of scheduling under resource constraints on the system which place the restriction that not all machines can be run at once. These can be power, energy, or makespan constraints on the system. Given such constraints, there are problems with divisible as well as non-divisible jobs. In the setting where there is a constraint on power, we show that the problem of minimizing makespan for a set of divisible jobs is NP-hard by reduction to the knapsack problem. We then show that scheduling to minimize energy with power constraints is also NP-hard. We then consider scheduling with energy and makespan constraints with divisible jobs and show that these can be solved in polynomial time, and the problems with non-divisible jobs are NP-hard. We give exact and approximation algorithms for these problems as required.
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Consider the problem where $n$ jobs, each with a release time, a deadline and a required processing time are to be feasibly scheduled in a single- or multi-processor setting so as to minimize the total energy consumption of the schedule. A processor has two available states: a emph{sleep state} where no energy is consumed but also no processing can take place, and an emph{active state} which consumes energy at a rate of one, and in which jobs can be processed. Transitioning from the active to the sleep does not incur any further energy cost, but transitioning from the sleep to the active state requires $q$ energy units. Jobs may be preempted and (in the multi-processor case) migrated. The single-processor case of the problem is known to be solvable in polynomial time via an involved dynamic program, whereas the only known approximation algorithm for the multi-processor case attains an approximation factor of $3$ and is based on rounding the solution to a linear programming relaxation of the problem. In this work, we present efficient and combinatorial approximation algorithms for both the single- and the multi-processor setting. Before, only an algorithm based on linear programming was known for the multi-processor case. Our algorithms build upon the concept of a emph{skeleton}, a basic (and not necessarily feasible) schedule that captures the fact that some processor(s) must be active at some time point during an interval. Finally, we further demonstrate the power of skeletons by providing an $2$-approximation algorithm for the multiprocessor case, thus improving upon the recent breakthrough $3$-approximation result. Our algorithm is based on a novel rounding scheme of a linear-programming relaxation of the problem which incorporates skeletons.
We study the assignment problem of objects to agents with heterogeneous preferences under distributional constraints. Each agent is associated with a publicly known type and has a private ordinal ranking over objects. We are interested in assigning as many agents as possible. Our first contribution is a generalization of the well-known and widely used serial dictatorship. Our mechanism maintains several desirable properties of serial dictatorship, including strategyproofness, Pareto efficiency, and computational tractability while satisfying the distributional constraints with a small error. We also propose a generalization of the probabilistic serial algorithm, which finds an ordinally efficient and envy-free assignment, and also satisfies the distributional constraints with a small error. We show, however, that no ordinally efficient and envy-free mechanism is also weakly strategyproof. Both of our algorithms assign at least the same number of students as the optimum fractional assignment.
We study the role of interactivity in distributed statistical inference under information constraints, e.g., communication constraints and local differential privacy. We focus on the tasks of goodness-of-fit testing and estimation of discrete distributions. From prior work, these tasks are well understood under noninteractive protocols. Extending these approaches directly for interactive protocols is difficult due to correlations that can build due to interactivity; in fact, gaps can be found in prior claims of tight bounds of distribution estimation using interactive protocols. We propose a new approach to handle this correlation and establish a unified method to establish lower bounds for both tasks. As an application, we obtain optimal bounds for both estimation and testing under local differential privacy and communication constraints. We also provide an example of a natural testing problem where interactivity helps.
We investigate graph problems in the following setting: we are given a graph $G$ and we are required to solve a problem on $G^2$. While we focus mostly on exploring this theme in the distributed CONGEST model, we show new results and surprising connections to the centralized model of computation. In the CONGEST model, it is natural to expect that problems on $G^2$ would be quite difficult to solve efficiently on $G$, due to congestion. However, we show that the picture is both more complicated and more interesting. Specifically, we encounter two phenomena acting in opposing directions: (i) slowdown due to congestion and (ii) speedup due to structural properties of $G^2$. We demonstrate these two phenomena via two fundamental graph problems, namely, Minimum Vertex Cover (MVC) and Minimum Dominating Set (MDS). Among our many contributions, the highlights are the following. - In the CONGEST model, we show an $O(n/epsilon)$-round $(1+epsilon)$-approximation algorithm for MVC on $G^2$, while no $o(n^2)$-round algorithm is known for any better-than-2 approximation for MVC on $G$. - We show a centralized polynomial time $5/3$-approximation algorithm for MVC on $G^2$, whereas a better-than-2 approximation is UGC-hard for $G$. - In contrast, for MDS, in the CONGEST model, we show an $tilde{Omega}(n^2)$ lower bound for a constant approximation factor for MDS on $G^2$, whereas an $Omega(n^2)$ lower bound for MDS on $G$ is known only for exact computation. In addition to these highlighted results, we prove a number of other results in the distributed CONGEST model including an $tilde{Omega}(n^2)$ lower bound for computing an exact solution to MVC on $G^2$, a conditional hardness result for obtaining a $(1+epsilon)$-approximation to MVC on $G^2$, and an $O(log Delta)$-approximation to the MDS problem on $G^2$ in $mbox{poly}log n$ rounds.
Given n jobs with release dates, deadlines and processing times we consider the problem of scheduling them on m parallel machines so as to minimize the total energy consumed. Machines can enter a sleep state and they consume no energy in this state. Each machine requires Q units of energy to awaken from the sleep state and in its active state the machine can process jobs and consumes a unit of energy per unit time. We allow for preemption and migration of jobs and provide the first constant approximation algorithm for this problem.
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