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Register a new userImproved Approximations for Multiprocessor Scheduling Under Uncertainty
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Added by
Jacob Scott
Publication date
2008
fields
Informatics Engineering
and research's language is
English
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This paper presents improved approximation algorithms for the problem of multiprocessor scheduling under uncertainty, or SUU, in which the execution of each job may fail probabilistically. This problem is motivated by the increasing use of distributed computing to handle large, computationally intensive tasks. In the SUU problem we are given n unit-length jobs and m machines, a directed acyclic graph G of precedence constraints among jobs, and unrelated failure probabilities q_{ij} for each job j when executed on machine i for a single timestep. Our goal is to find a schedule that minimizes the expected makespan, which is the expected time at which all jobs complete. Lin and Rajaraman gave the first approximations for this NP-hard problem for the special cases of independent jobs, precedence constraints forming disjoint chains, and precedence constraints forming trees. In this paper, we present asymptotically better approximation algorithms. In particular, we give an O(loglog min(m,n))-approximation for independent jobs (improving on the previously best O(log n)-approximation). We also give an O(log(n+m) loglog min(m,n))-approximation algorithm for precedence constraints that form disjoint chains (improving on the previously best O(log(n)log(m)log(n+m)/loglog(n+m))-approximation by a (log n/loglog n)^2 factor when n = poly(m). Our algorithm for precedence constraints forming chains can also be used as a component for precedence constraints forming trees, yielding a similar improvement over the previously best algorithms for trees.
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We present improved results for approximating maximum-weight independent set ($MaxIS$) in the CONGEST and LOCAL models of distributed computing. Given an input graph, let $n$ and $Delta$ be the number of nodes and maximum degree, respectively, and let $MIS(n,Delta)$ be the the running time of finding a emph{maximal} independent set ($MIS$) in the CONGEST model. Bar-Yehuda et al. [PODC 2017] showed that there is an algorithm in the CONGEST model that finds a $Delta$-approximation for $MaxIS$ in $O(MIS(n,Delta)log W)$ rounds, where $W$ is the maximum weight of a node in the graph, which can be as high as $poly (n)$. Whether their algorithm is deterministic or randomized depends on the $MIS$ algorithm that is used as a black-box. Our main result in this work is a randomized $(poly(loglog n)/epsilon)$-round algorithm that finds, with high probability, a $(1+epsilon)Delta$-approximation for $MaxIS$ in the CONGEST model. That is, by sacrificing only a tiny fraction of the approximation guarantee, we achieve an emph{exponential} speed-up in the running time over the previous best known result. Due to a lower bound of $Omega(sqrt{log n/log log n})$ that was given by Kuhn, Moscibroda and Wattenhofer [JACM, 2016] on the number of rounds for any (possibly randomized) algorithm that finds a maximal independent set (even in the LOCAL model) this result implies that finding a $(1+epsilon)Delta$-approximation for $MaxIS$ is exponentially easier than $MIS$.
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The problem of attaining energy efficiency in distributed systems is of importance, but a general, non-domain-specific theory of energy-minimal scheduling is far from developed. In this paper, we classify the problems of energy-minimal scheduling and present theoretical foundations of the same. We derive results concerning energy-minimal scheduling of independent jobs in a distributed system with functionally similar machines with different working and idle power ratings. The machines considered in our system can have identical as well as different speeds. If the jobs can be divided into arbitrary parts, we show that the minimum-energy schedule can be generated in linear time and give exact scheduling algorithms. For the cases where jobs are non-divisible, we prove that the scheduling problems are NP-hard and also give approximation algorithms for the same along with their bounds.
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