New bounds for truthful scheduling on two unrelated selfish machines

We consider the minimum makespan problem for $n$ tasks and two unrelated parallel selfish machines. Let $R_n$ be the best approximation ratio of randomized monotone scale-free algorithms. This class contains the most efficient algorithms known for truthful scheduling on two machines. We propose a new $Min-Max$ formulation for $R_n$, as well as upper and lower bounds on $R_n$ based on this formulation. For the lower bound, we exploit pointwise approximations of cumulative distribution functions (CDFs). For the upper bound, we construct randomized algorithms using distributions with piecewise rational CDFs. Our method improves upon the existing bounds on $R_n$ for small $n$. In particular, we obtain almost tight bounds for $n=2$ showing that $|R_2-1.505996|<10^{-6}$.Comment: 28 pages, 3 tables, 1 figure. Theory Comput Syst (2019

Parallel machine scheduling with release dates, due dates and family setup times

In manufacturing, there is a fundamental conflict between efficient production and delivery performance. Maximizing machine utilization by batching similar jobs may lead to poor delivery performance. Minimizing customers' dissatisfaction may lead to an inefficient use of the machines. In this paper, we consider the problem of scheduling n independent jobs with release dates, due dates, and family setup times on m parallel machines. The objective is to minimize the maximum lateness of any job. We present a branch-and-bound algorithm to solve this problem. This algorithm exploits the fact that an optimal schedule is contained in a specific subset of all feasible schedules. For lower bounding purposes, we see setup times as setup jobs with release dates, due dates and processing times. We present two lower bounds for the problem with setup jobs, one of which proceeds by allowing preemption

Optimal on-line flow time with resource augmentation

AbstractWe study the problem of scheduling n jobs that arrive over time. We consider a non-preemptive setting on a single machine. The goal is to minimize the total flow time. We use extra resource competitive analysis: an optimal off-line algorithm which schedules jobs on a single machine is compared to a more powerful on-line algorithm that has ℓ machines. We design an algorithm of competitive ratio 1+2min(Δ1/ℓ,n1/ℓ), where Δ is the maximum ratio between two job sizes, and provide a lower bound which shows that the algorithm is optimal up to a constant factor for any constant ℓ. The algorithm works for a hard version of the problem where the sizes of the smallest and the largest jobs are not known in advance, only Δ and n are known. This gives a trade-off between the resource augmentation and the competitive ratio.We also consider scheduling on parallel identical machines. In this case the optimal off-line algorithm has m machines and the on-line algorithm has ℓm machines. We give a lower bound for this case. Next, we give lower bounds for algorithms using resource augmentation on the speed. Finally, we consider scheduling with hard deadlines, and scheduling so as to minimize the total completion time

New results on flow time with resource augmentation

We study the problem of scheduling $n$ jobs that arrive over time. We consider a non-preemptive setting on a single machine. The goal is to minimize the total flow time. We use extra resource competitive analysis: an optimal off-line algorithm which schedules jobs on a single machine is compared to a more powerful on-line algorithm that has $l$ machines. We design an algorithm of competitive ratio O(min(Delta^{1/l,n^{1/l)), where $Delta$ is the maximum ratio between two job sizes, and provide a lower bound which shows that the algorithm is optimal up to a constant factor for any constant $l$. The algorithm works for a hard version of the problem where the sizes of the smallest and the largest jobs are not known in advance, only $Delta$ is known. This gives a trade-off between the resource augmentation and the competitive ratio. We also consider scheduling on parallel identical machines. In this case the optimal off-line algorithm has $m$ machines and the on-line algorithm has $lm$ machines. We give a lower bound for this case. Next, we give lower bounds for algorithms using resource augmentation on the speed. Finally, we consider scheduling with hard deadlines

Rolling horizon and fix-and-relax heuristcs for the parallel machines lot-sizing and scheduling problem with sequence dependent set-up costs

none4In this paper we develop newrolling-horizon and fix-and-relax heuristics for the identical parallel machine lot-sizing and scheduling problem with sequence-dependent set-up costs. Unlike previous papers, our procedures are based on a compact formulation relying on the hypotheses of identical machines. This feature makes our approach suitable for large-scale applications (with hundreds of machines) arising in the textile and fiberglass industries. Moreover, our procedures are shown to provide a feasible solution for any feasible instance. Comparisons with lower bounds provided by a truncated branch-and-bound show that the gap between the best heuristic solution and the lower bound never exceeds 3%P.BERALDI; G. GHIANI; A. GRIECO; E. GUERRIEROP., Beraldi; Ghiani, Gianpaolo; Grieco, Antonio Domenico; Guerriero, Emanuel

Special cases of online parallel job scheduling

In this paper we consider the online scheduling of jobs, which require processing on a number of machines simultaneously. These jobs are presented to a decision maker one by one, where the next job becomes known as soon as the current job is scheduled. The objective is to minimize the makespan. For the problem with three machines we give a 2.8-competitive algorithm, improving upon the 3-competitive greedy algorithm. For the special case with arbitrary number of machines, where the jobs appear in non-increasing order of machine requirement, we give a 2.4815-competitive algorithm, improving the 2.75-competitive greedy algorithm

Competitive-Ratio Approximation Schemes for Minimizing the Makespan in the Online-List Model

We consider online scheduling on multiple machines for jobs arriving one-by-one with the objective of minimizing the makespan. For any number of identical parallel or uniformly related machines, we provide a competitive-ratio approximation scheme that computes an online algorithm whose competitive ratio is arbitrarily close to the best possible competitive ratio. We also determine this value up to any desired accuracy. This is the first application of competitive-ratio approximation schemes in the online-list model. The result proves the applicability of the concept in different online models. We expect that it fosters further research on other online problems

Stochastic scheduling on unrelated machines

Two important characteristics encountered in many real-world scheduling problems are heterogeneous machines/processors and a certain degree of uncertainty about the actual sizes of jobs. The first characteristic entails machine dependent processing times of jobs and is captured by the classical unrelated machine scheduling model.The second characteristic is adequately addressed by stochastic processing times of jobs as they are studied in classical stochastic scheduling models. While there is an extensive but separate literature for the two scheduling models, we study for the first time a combined model that takes both characteristics into account simultaneously. Here, the processing time of job $j$ on machine $i$ is governed by random variable $P_{ij}$, and its actual realization becomes known only upon job completion. With $w_j$ being the given weight of job $j$, we study the classical objective to minimize the expected total weighted completion time $E[\sum_j w_jC_j]$, where $C_j$ is the completion time of job $j$. By means of a novel time-indexed linear programming relaxation, we compute in polynomial time a scheduling policy with performance guarantee $(3+\Delta)/2+\epsilon$. Here, $\epsilon>0$ is arbitrarily small, and $\Delta$ is an upper bound on the squared coefficient of variation of the processing times. We show that the dependence of the performance guarantee on $\Delta$ is tight, as we obtain a $\Delta/2$ lower bound for the type of policies that we use. When jobs also have individual release dates $r_{ij}$, our bound is $(2+\Delta)+\epsilon$. Via $\Delta=0$, currently best known bounds for deterministic scheduling are contained as a special case