7 research outputs found
Geometry of Scheduling on Multiple Machines
We consider the following general scheduling problem: there are m identical machines and n jobs all released at time 0. Each job j has a processing time pj, and an arbitrary non-decreasing function fj that specifies the cost incurred for j, for each possible completion time. The goal is to find a preemptive migratory schedule of minimum cost. This models several natural objectives such as weighted norm of completion time, weighted tardiness and much more.
We give the first O(1) approximation algorithm for this problem, improving upon the O(loglognP) bound due to Moseley (2019). To do this, we first view the job-cover inequalities of Moseley geometrically, to reduce the problem to that of covering demands on a line by rectangular and triangular capacity profiles. Due to the non-uniform capacities of triangles, directly using quasi-uniform sampling loses a O(loglogP) factor, so a second idea is to adapt it to our setting to only lose an O(1) factor. Our ideas for covering points with non-uniform capacity profiles (which have not been studied before) may be of independent int
Non-uniform Geometric Set Cover and Scheduling on Multiple Machines
We consider the following general scheduling problem studied recently by
Moseley. There are jobs, all released at time , where job has size
and an associated arbitrary non-decreasing cost function of its
completion time. The goal is to find a schedule on machines with minimum
total cost. We give an approximation for the problem, improving upon the
previous bound ( is the maximum to minimum size ratio),
and resolving the open question of Moseley.
We first note that the scheduling problem can be reduced to a clean geometric
set cover problem where points on a line with arbitrary demands, must be
covered by a minimum cost collection of given intervals with non-uniform
capacity profiles. Unfortunately, current techniques for such problems based on
knapsack cover inequalities and low union complexity, completely lose the
geometric structure in the non-uniform capacity profiles and incur at least an
loss.
To this end, we consider general covering problems with non-uniform
capacities, and give a new method to handle capacities in a way that completely
preserves their geometric structure. This allows us to use sophisticated
geometric ideas in a black-box way to avoid the loss in
previous approaches. In addition to the scheduling problem above, we use this
approach to obtain or inverse Ackermann type bounds for several basic
capacitated covering problems
Non-uniform geometric set cover and scheduling on multiple machines
We consider the following general scheduling problem studied recently by Moseley [27]. There are n jobs, all released at time 0, where job j has size pj and an associated arbitrary non-decreasing cost function fj of its completion time.