Better Unrelated Machine Scheduling for Weighted Completion Time via Random Offsets from Non-Uniform Distributions

In this paper we consider the classic scheduling problem of minimizing total weighted completion time on unrelated machines when jobs have release times, i.e, $R | r_{ij} | \sum_j w_j C_j$ using the three-field notation. For this problem, a 2-approximation is known based on a novel convex programming (J. ACM 2001 by Skutella). It has been a long standing open problem if one can improve upon this 2-approximation (Open Problem 8 in J. of Sched. 1999 by Schuurman and Woeginger). We answer this question in the affirmative by giving a 1.8786-approximation. We achieve this via a surprisingly simple linear programming, but a novel rounding algorithm and analysis. A key ingredient of our algorithm is the use of random offsets sampled from non-uniform distributions. We also consider the preemptive version of the problem, i.e, $R | r_{ij},pmtn | \sum_j w_j C_j$. We again use the idea of sampling offsets from non-uniform distributions to give the first better than 2-approximation for this problem. This improvement also requires use of a configuration LP with variables for each job's complete schedules along with more careful analysis. For both non-preemptive and preemptive versions, we break the approximation barrier of 2 for the first time.Comment: 24 pages. To apper in FOCS 201

Attention-Mediated Neural and Behavioral Oscillation and Their Relationship to Dispositional Mindfulness

Over the past decades, there has been growing interest in mindfulness-based interventions (MBIs) and their association with attention. Preliminary research suggests that self-regulation of attention may mediate clinical benefits of MBIs. Although daily practice of mindfulness exercise and participation in MBI can produce noticeable intra-personal improvement in mindfulness skills over time, it is of greater theoretical significance to assess inter-individual differences in mindfulness and demonstrate the degree to which various levels of dispositional mindfulness relate to outcomes of interest. Furthermore, there are recent developments focusing on systematic temporal fluctuations in the brain waves and subsequent behavioral performance, which has been proposed as an underlying mechanism of attention. Thus, the current study aimed at evaluating the temporal pattern of behavioral performance and concurrent EEG data in visual cueing tasks via time-frequency techniques and investigating whether behavioral and neural parameters of selective attention in visual cueing tasks are associated with levels of dispositional mindfulness.To address these research questions, three experiments were conducted wherein participants completed an endogenous cueing task (n = 44, Experiment 1), an exogenous cueing task (n = 42, Experiment 2), or an endogenous cueing task with concurrent EEG recordings (n = 27). Additionally, participants from Experiment 1 and Experiment 2 completed self-report questionnaires including Mindfulness Attention Awareness Scale (MAAS), Five Facets Mindfulness Questionnaire (FFMQ), and Adult ADHD Self-Report Scale (ASRS). The results from the endogenous attention task suggest significant oscillatory activities at Delta, Theta, and Beta frequency bands for discrimination accuracy and at Delta, Alpha, and Beta frequency bands for reaction time. Likewise, behavioral data from the exogenous attention task indicates significant increases in evoked Theta and Alpha power in discrimination accuracy and reaction time, respectively. EEG data also support significant power spectral suppression in frontocentral electrodes in Delta and Theta bands when participants’ covert attention shifted either to the left or right target location compared with no-target condition. Moreover, we observed a positive correlation between FFMQ subscale scores and evoked power suggesting that levels of dispositional mindfulness are associated with spatial visual attention. Clinical implications and limitations of the current study will be further discussed

SELFISHMIGRATE: A Scalable Algorithm for Non-clairvoyantly Scheduling Heterogeneous Processors

We consider the classical problem of minimizing the total weighted flow-time for unrelated machines in the online \emph{non-clairvoyant} setting. In this problem, a set of jobs $J$ arrive over time to be scheduled on a set of $M$ machines. Each job $j$ has processing length $p_j$, weight $w_j$, and is processed at a rate of $\ell_{ij}$ when scheduled on machine $i$. The online scheduler knows the values of $w_j$ and $\ell_{ij}$ upon arrival of the job, but is not aware of the quantity $p_j$. We present the {\em first} online algorithm that is {\em scalable} ((1+\eps)-speed $O(\frac{1}{\epsilon^2})$-competitive for any constant \eps > 0) for the total weighted flow-time objective. No non-trivial results were known for this setting, except for the most basic case of identical machines. Our result resolves a major open problem in online scheduling theory. Moreover, we also show that no job needs more than a logarithmic number of migrations. We further extend our result and give a scalable algorithm for the objective of minimizing total weighted flow-time plus energy cost for the case of unrelated machines and obtain a scalable algorithm. The key algorithmic idea is to let jobs migrate selfishly until they converge to an equilibrium. Towards this end, we define a game where each job's utility which is closely tied to the instantaneous increase in the objective the job is responsible for, and each machine declares a policy that assigns priorities to jobs based on when they migrate to it, and the execution speeds. This has a spirit similar to coordination mechanisms that attempt to achieve near optimum welfare in the presence of selfish agents (jobs). To the best our knowledge, this is the first work that demonstrates the usefulness of ideas from coordination mechanisms and Nash equilibria for designing and analyzing online algorithms