53 research outputs found
Scheduling MapReduce Jobs under Multi-Round Precedences
We consider non-preemptive scheduling of MapReduce jobs with multiple tasks
in the practical scenario where each job requires several map-reduce rounds. We
seek to minimize the average weighted completion time and consider scheduling
on identical and unrelated parallel processors. For identical processors, we
present LP-based O(1)-approximation algorithms. For unrelated processors, the
approximation ratio naturally depends on the maximum number of rounds of any
job. Since the number of rounds per job in typical MapReduce algorithms is a
small constant, our scheduling algorithms achieve a small approximation ratio
in practice. For the single-round case, we substantially improve on previously
best known approximation guarantees for both identical and unrelated
processors. Moreover, we conduct an experimental analysis and compare the
performance of our algorithms against a fast heuristic and a lower bound on the
optimal solution, thus demonstrating their promising practical performance
Stochastic scheduling on unrelated machines
Two important characteristics encountered in many real-world scheduling problems are heterogeneous machines/processors and a certain degree of uncertainty about the actual sizes of jobs. The first characteristic entails machine dependent processing times of jobs and is captured by the classical unrelated machine scheduling model.The second characteristic is adequately addressed by stochastic processing times of jobs as they are studied in classical stochastic scheduling models. While there is an extensive but separate literature for the two scheduling models, we study for the first time a combined model that takes both characteristics into account simultaneously. Here, the processing time of job on machine is governed by random variable , and its actual realization becomes known only upon job completion. With being the given weight of job , we study the classical objective to minimize the expected total weighted completion time , where is the completion time of job . By means of a novel time-indexed linear programming relaxation, we compute in polynomial time a scheduling policy with performance guarantee . Here, is arbitrarily small, and is an upper bound on the squared coefficient of variation of the processing times. We show that the dependence of the performance guarantee on is tight, as we obtain a lower bound for the type of policies that we use. When jobs also have individual release dates , our bound is . Via , currently best known bounds for deterministic scheduling are contained as a special case
Bounds on the Complexity of Halfspace Intersections when the Bounded Faces have Small Dimension
We study the combinatorial complexity of D-dimensional polyhedra defined as
the intersection of n halfspaces, with the property that the highest dimension
of any bounded face is much smaller than D. We show that, if d is the maximum
dimension of a bounded face, then the number of vertices of the polyhedron is
O(n^d) and the total number of bounded faces of the polyhedron is O(n^d^2). For
inputs in general position the number of bounded faces is O(n^d). For any fixed
d, we show how to compute the set of all vertices, how to determine the maximum
dimension of a bounded face of the polyhedron, and how to compute the set of
bounded faces in polynomial time, by solving a polynomial number of linear
programs
Sharing Supermodular Costs
We study cooperative games with supermodular costs. We show that supermodular costs arise in a variety of situations; in particular, we show that the problem of minimizing a linear function over a supermodular polyhedron—a problem that often arises in combinatorial optimization—has supermodular optimal costs. In addition, we examine the computational complexity of the least core and least core value of supermodular cost cooperative games. We show that the problem of computing the least core value of these games is strongly NP-hard and, in fact, is inapproximable within a factor strictly less than 17/16 unless P = NP. For a particular class of supermodular cost cooperative games that arises from a scheduling problem, we show that the Shapley value—which, in this case, is computable in polynomial time—is in the least core, while computing the least core value is NP-hard.National Science Foundation (U.S.) (DMI-0426686
Decomposition algorithm for the single machine scheduling polytope
Given an -vector of processing times of jobs, the single machine scheduling polytope arises as the convex hull of completion times of jobs when these are scheduled without idle time on a single machine. Given a point (x\ud
in C), Carathéodory's theorem implies that can be written as convex combination of at most vertices of . We show that this convex combination can be computed from and in time O(n2), which is linear in the naive encoding of the output. We obtain this result using essentially two ingredients. First, we build on the fact that the scheduling polytope is a zonotope. Therefore, all of its faces are centrally symmetric. Second, instead of , we consider the polytope of half times and its barycentric subdivision. We show that the subpolytopes of this barycentric subdivison of have a simple, linear description. The final decomposition algorithm is in fact an implementation of an algorithm proposed by Grötschel, Lovász, and Schrijver applied to one of these subpolytopes
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