42,941 research outputs found
The Geometry of Scheduling
We consider the following general scheduling problem: The input consists of n
jobs, each with an arbitrary release time, size, and a monotone function
specifying the cost incurred when the job is completed at a particular time.
The objective is to find a preemptive schedule of minimum aggregate cost. This
problem formulation is general enough to include many natural scheduling
objectives, such as weighted flow, weighted tardiness, and sum of flow squared.
Our main result is a randomized polynomial-time algorithm with an approximation
ratio O(log log nP), where P is the maximum job size. We also give an O(1)
approximation in the special case when all jobs have identical release times.
The main idea is to reduce this scheduling problem to a particular geometric
set-cover problem which is then solved using the local ratio technique and
Varadarajan's quasi-uniform sampling technique. This general algorithmic
approach improves the best known approximation ratios by at least an
exponential factor (and much more in some cases) for essentially all of the
nontrivial common special cases of this problem. Our geometric interpretation
of scheduling may be of independent interest.Comment: Conference version in FOCS 201
Covering compact metric spaces greedily
A general greedy approach to construct coverings of compact metric spaces by
metric balls is given and analyzed. The analysis is a continuous version of
Chvatal's analysis of the greedy algorithm for the weighted set cover problem.
The approach is demonstrated in an exemplary manner to construct efficient
coverings of the n-dimensional sphere and n-dimensional Euclidean space to give
short and transparent proofs of several best known bounds obtained from
deterministic constructions in the literature on sphere coverings.Comment: (v2) 10 pages, minor revision, accepted in Acta Math. Hunga
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