Scheduling to Approximate Minimization Objectives on Identical Machines

This paper considers scheduling on identical machines. The scheduling objective considered in this paper generalizes most scheduling minimization problems. In the problem, there are n jobs and each job j is associated with a monotonically increasing function g_j. The goal is to design a schedule that minimizes sum_{j in [n]} g_{j}(C_j) where C_j is the completion time of job j in the schedule. An O(1)-approximation is known for the single machine case. On multiple machines, this paper shows that if the scheduler is required to be either non-migratory or non-preemptive then any algorithm has an unbounded approximation ratio. Using preemption and migration, this paper gives a O(log log nP)-approximation on multiple machines, the first result on multiple machines. These results imply the first non-trivial positive results for several special cases of the problem considered, such as throughput minimization and tardiness. Natural linear programs known for the problem have a poor integrality gap. The results are obtained by strengthening a natural linear program for the problem with a set of covering inequalities we call job cover inequalities. This linear program is rounded to an integral solution by building on quasi-uniform sampling and rounding techniques

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 $n$ jobs, all released at time $0$, where job $j$ has size $p_j$ and an associated arbitrary non-decreasing cost function $f_j$ of its completion time. The goal is to find a schedule on $m$ machines with minimum total cost. We give an $O(1)$ approximation for the problem, improving upon the previous $O(\log \log nP)$ bound ($P$ 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 $\Omega(\log\log P)$ 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 $\Omega(\log \log P)$ loss in previous approaches. In addition to the scheduling problem above, we use this approach to obtain $O(1)$ or inverse Ackermann type bounds for several basic capacitated covering problems

Identification of input-output LPV models

This chapter presents an overview of the available methods for identifying input-output LPV models both in discrete time and continuous time with the main focus on noise modeling issues. First, a least-squares approach and an instrumental variable method are presented for dealing with LPV-ARX models. Then, a refined instrumental variable approach is discussed to address more sophisticated noise models like Box-Jenkins in the LPV context. This latter approach is also introduced in continuous time and efficient solutions are proposed for both the problem of time-derivative approximation and the issue of continuous-time modeling of the noise