Online Two-Dimensional Load Balancing

In this paper, we consider the problem of assigning 2-dimensional vector jobs to identical machines online so to minimize the maximum load on any dimension of any machine. For arbitrary number of dimensions d, this problem is known as vector scheduling, and recent research has established the optimal competitive ratio as O((log d)/(log log d)) (Im et al. FOCS 2015, Azar et al. SODA 2018). But, these results do not shed light on the situation for small number of dimensions, particularly for d = 2 which is of practical interest. In this case, a trivial analysis shows that the classic list scheduling greedy algorithm has a competitive ratio of 3. We show the following improvements over this baseline in this paper: - We give an improved, and tight, analysis of the list scheduling algorithm establishing a competitive ratio of 8/3 for two dimensions. - If the value of opt is known, we improve the competitive ratio to 9/4 using a variant of the classic best fit algorithm for two dimensions. - For any fixed number of dimensions, we design an algorithm that is provably the best possible against a fractional optimum solution. This algorithm provides a proof of concept that we can simulate the optimal algorithm online up to the integrality gap of the natural LP relaxation of the problem

Scheduling in the Secretary Model

This paper studies online makespan minimization in the secretary model. Jobs, specified by their processing times, are presented in a uniformly random order. The input size n is known in advance. An online algorithm has to non-preemptively assign each job permanently and irrevocably to one of m parallel and identical machines such that the expected time it takes to process them all, the makespan, is minimized. We give two deterministic algorithms. First, a straightforward adaptation of the semi-online strategy Light Load [Albers and Hellwig, 2012] provides a very simple approach retaining its competitive ratio of 1.75. A new and sophisticated algorithm is 1.535-competitive. These competitive ratios are not only obtained in expectation but, in fact, for all but a very tiny fraction of job orders. Classically, online makespan minimization only considers the worst-case order. Here, no competitive ratio below 1.885 for deterministic algorithms and 1.581 using randomization is possible. The best randomized algorithm so far is 1.916-competitive. Our results show that classical worst-case orders are quite rare and pessimistic for many applications. We complement our results by providing first lower bounds. A competitive ratio obtained on nearly all possible job orders must be at least 1.257. This implies a lower bound of 1.043 for both deterministic and randomized algorithms in the general model

On robust online scheduling algorithms

While standard parallel machine scheduling is concerned with good assignments of jobs to machines, we aim to understand how the quality of an assignment is affected if the jobs' processing times are perturbed and therefore turn out to be longer (or shorter) than declared. We focus on online scheduling with perturbations occurring at any time, such as in railway systems when trains are late. For a variety of conditions on the severity of perturbations, we present bounds on the worst case ratio of two makespans. For the first makespan, we let the online algorithm assign jobs to machines, based on the non-perturbed processing times. We compute the makespan by replacing each job's processing time with its perturbed version while still sticking to the computed assignment. The second is an optimal offline solution for the perturbed processing times. The deviation of this ratio from the competitive ratio of the online algorithm tells us about the "price of perturbations”. We analyze this setting for Graham's algorithm, and among other bounds show a competitive ratio of 2 for perturbations decreasing the processing time of a job arbitrarily, and a competitive ratio of less than 2.5 for perturbations doubling the processing time of a job. We complement these results by providing lower bounds for any online algorithm in this setting. Finally, we propose a risk-aware online algorithm tailored for the possible bounded increase of the processing time of one job, and we show that this algorithm can be worse than Graham's algorithm in some case

Randomized algorithms for fully online multiprocessor scheduling with testing

We contribute the first randomized algorithm that is an integration of arbitrarily many deterministic algorithms for the fully online multiprocessor scheduling with testing problem. When there are two machines, we show that with two component algorithms its expected competitive ratio is already strictly smaller than the best proven deterministic competitive ratio lower bound. Such algorithmic results are rarely seen in the literature. Multiprocessor scheduling is one of the first combinatorial optimization problems that have received numerous studies. Recently, several research groups examined its testing variant, in which each job $J_j$ arrives with an upper bound $u_j$ on the processing time and a testing operation of length $t_j$; one can choose to execute $J_j$ for $u_j$ time, or to test $J_j$ for $t_j$ time to obtain the exact processing time $p_j$ followed by immediately executing the job for $p_j$ time. Our target problem is the fully online version, in which the jobs arrive in sequence so that the testing decision needs to be made at the job arrival as well as the designated machine. We propose an expected $(\sqrt{\varphi + 3} + 1) (\approx 3.1490)$-competitive randomized algorithm as a non-uniform probability distribution over arbitrarily many deterministic algorithms, where $\varphi = \frac {\sqrt{5} + 1}2$ is the Golden ratio. When there are two machines, we show that our randomized algorithm based on two deterministic algorithms is already expected $\frac {3 \varphi + 3 \sqrt{13 - 7\varphi}}4 (\approx 2.1839)$-competitive. Besides, we use Yao's principle to prove lower bounds of $1.6682$ and $1.6522$ on the expected competitive ratio for any randomized algorithm at the presence of at least three machines and only two machines, respectively, and prove a lower bound of $2.2117$ on the competitive ratio for any deterministic algorithm when there are only two machines.Comment: 21 pages with 1 plot; an extended abstract to be submitte

On randomized online scheduling

Notable alumni on this page include: Roger J. Miner,line 1https://digitalcommons.nyls.edu/ledger_book_15/1001/thumbnail.jp

On randomized online scheduling

ABSTRACT We study one of the most basic problems in online scheduling. A sequence of jobs has to be scheduled on m identical parallel machines so as to minimize the makespan. Whenever a new job arrives, its processing time is known in advance. The job has to be scheduled immediately on one of the machines without knowledge of any future jobs. In the sixties Graham presented the famous List scheduling algorithm which is (2 \Gamma