Resource augmentation in load balancing

We consider load-balancing in the following setting. The on-line algorithm is allowed to use $n$ machines, whereas the optimal off-line algorithm is limited to $m$ machines, for some fixed $m < n$. We show that while the greedy algorithm has a competitive ratio which decays linearly in the inverse of $n/m$, the best on-line algorithm has a ratio which decays exponentially in $n/m$. Specifically, we give an algorithm with competitive ratio of 1+2^{- frac{n{m (1- o (1)), and a lower bound of 1+ e^{ - frac{n{m (1+ o(1)) on the competitive ratio of any randomized algorithm. We also consider the preemptive case. We show an on-line algorithm with a competitive ratio of 1+ e^{ - frac{n{m (1+ o(1)). We show that the algorithm is optimal by proving a matching lower bound. We also consider the non-preemptive model with temporary tasks. We prove that for $n=m+1$, the greedy algorithm is optimal. (It is not optimal for permanent tasks.

Minimizing the maximum starting time on-line

We study the scheduling problem of minimizing the maximum starting time on-line. The goal is to minimize the last time that a job starts. We show that while the greedy algorithm has a competitive ratio of $Theta(log m)$, we can give a constant competitive algorithm for this problem. We also show that the greedy algorithm is optimal for resource augmentation in the sense that it requires 2m-1 machines to have a competitive ratio of 1, whereas no algorithm can achieve this with 2m-1 machines

O(1/ε)O(1/\varepsilon) is the answer in online weighted throughput maximization

We study a fundamental online scheduling problem where jobs with processing times, weights, and deadlines arrive online over time at their release dates. The task is to preemptively schedule these jobs on a single or multiple (possibly unrelated) machines with the objective to maximize the weighted throughput, the total weight of jobs that complete before their deadline. To overcome known lower bounds for the competitive analysis, we assume that each job arrives with some slack $\varepsilon > 0$; that is, the time window for processing job $j$ on any machine $i$ on which it can be executed has length at least $(1+\varepsilon)$ times $j$'s processing time on machine $i$. Our contribution is a best possible online algorithm for weighted throughput maximization on unrelated machines: Our algorithm is $O\big(\frac1\varepsilon\big)$-competitive, which matches the lower bound for unweighted throughput maximization on a single machine. Even for a single machine, it was not known whether the problem with weighted jobs is "harder" than the problem with unweighted jobs. Thus, we answer this question and close weighted throughput maximization on a single machine with a best possible competitive ratio $\Theta\big(\frac1\varepsilon\big)$. While we focus on non-migratory schedules, our algorithm achieves the same (up to constants) performance guarantee when compared to an optimal migratory schedule

The Power of Migration for Online Slack Scheduling

We investigate the power of migration in online scheduling for parallel identical machines. Our objective is to maximize the total processing time of accepted jobs. Once we decide to accept a job, we have to complete it before its deadline d that satisfies d >= (1+epsilon)p + r, where p is the processing time, r the submission time and the slack epsilon > 0 a system parameter. Typically, the hard case arises for small slack epsilon << 1, i.e. for near-tight deadlines. Without migration, a greedy acceptance policy is known to be an optimal deterministic online algorithm with a competitive factor of (1+epsilon)/epsilon (DasGupta and Palis, APPROX 2000). Our first contribution is to show that migrations do not improve the competitive ratio of the greedy acceptance policy, i.e. the competitive ratio remains (1+epsilon)/epsilon for any number of machines. Our main contribution is a deterministic online algorithm with almost tight competitive ratio on any number of machines. For a single machine, the competitive factor matches the optimal bound of (1+epsilon)/epsilon of the greedy acceptance policy. The competitive ratio improves with an increasing number of machines. It approaches (1+epsilon) ln((1+epsilon)/epsilon) as the number of machines converges to infinity. This is an exponential improvement over the greedy acceptance policy for small epsilon. Moreover, we show a matching lower bound on the competitive ratio for deterministic algorithms on any number of machines

How to whack moles ଁ

Abstract In the classical whack-a-mole game moles that pop up at certain locations must be whacked by means of a hammer before they go under ground again. The goal is to maximize the number of moles whacked. This problem can be formulated as an online optimization problem: requests (moles) appear over time at points in a metric space and must be served (whacked) by a server (hammer) before their deadlines (i.e., before they disappear). An online algorithm learns each request only at its release time and must base its decisions on incomplete information. We study the online whack-a-mole problem (WHAM) on the real line and on the uniform metric space. While on the line no deterministic algorithm can achieve a constant competitive ratio, we provide competitive algorithms for the uniform metric space. Our online investigations are complemented by complexity results for the offline problem