Shared Processor Scheduling of Multiprocessor Jobs

TL;DR

The paper studies maximizing the total weighted overlap for $n$ multiprocessor jobs that can run on private processors and multiple shared processors with costs, capturing subcontracting and divisible-load contexts.A key theoretical advance is the existence of synchronized optimal schedules in the MP mode, together with an LP formulation that optimizes overlaps for a fixed private-completion order.An $\alpha$-approximation algorithm is developed via $\alpha$-private schedules with $\alpha=\frac{1}{2}+\frac{1}{4(m+1)}$, reducible to a strongly polynomial minimum-cost network-flow problem; for $m=1$ this yields a $\frac{5}{8}$-approximation.The paper also analyzes antithetical instances via an LP-based, polynomial-time algorithm and provides a structural framework (processor-descending, sequential schedules) culminating in a synchronization theory that eliminates splits and proves the existence of synchronized optimal schedules.Overall, the work advances both exact and approximation methods for shared-processor scheduling, with applications to supply chains and divisible-load problems, while leaving open fundamental complexity questions.

Abstract

We study shared processor scheduling of $\textit{multiprocessor}$ weighted jobs where each job can be executed on its private processor and simultaneously on possibly $\textit{many}$ processors shared by all jobs in order to reduce their completion times due to processing time overlap. Each of $m$ shared processors may charge different fee but otherwise the processors are identical. The total weighted overlap of all jobs is to be maximized. This problem is key to subcontractor scheduling in extended enterprises and supply chains, and divisible load scheduling in computing. We prove that, quite surprisingly, $\textit{synchronized}$ schedules that complete each job using shared processors at the same time on its private and shared processors include optimal schedules. We show that optimal $α$-$\textit{private}$ schedules that require each job to use its private processor for at least $α=1/2+1/(4(m+1))$ of the time required by the job guarantee more than an $α$ fraction of the total weighted overlap of the optimal schedules. This gives an $α$-approximation algorithm that runs in strongly polynomial time for the problem, and improves the $1/2$-approximation reported recently in the literature to $5/8$-approximation for a single shared processor problem. The computational complexity of the problem, both single and multi-shared processor, remains open. We show however an LP-based optimal algorithm for $\textit{antithetical}$ instances where for any pair of jobs $j$ and $i$, if the processing time of $j$ is smaller than or equal to the processing time of $i$, then the weight of $j$ is greater than or equal to the weight of $i$.

Figures (6)

• Figure 1: The example illustrates that allowing jobs to be executed in MP mode simultaneously on several shared processors ($\mathcal{M}_1$ and $\mathcal{M}_2$) may be beneficial for some problem instances: (a) an optimal (synchronized) schedule for the SP mode; (b) an optimal (synchronized) schedule for the MP mode. In this input instance, the shared processors' fees are $c_1=4$ and $c_2=5$, the jobs' processing times are $p_1=p_2=9$, $p_3=5$ and their weights are $w_1=9$, $w_2=7$, $w_3=5$.
• Figure 2: An optimal schedule for an input instance with $\mathcal{J}=\{1,\ldots,5\}$ and $4$ shared processors $\mathcal{M} _1,\ldots,\mathcal{M} _4$. Here $X_j$ denotes the contribution of a job $j$ to the total weighted overlap of the schedule, $X_j=\sum_{i=1}^m\textup{ovlp}_{\mathcal{S}}(j,\mathcal{M} _i)(w_{j}-c_{i})$
• Figure 3: Transformation performed in Step \ref{['it:s2']} for a segment $I=(L,L')$ with total job executions times in this segment being $a_1^I=18$, $a_2^I=12$, $a_3^I=15$, $a_4^I=9$
• Figure 4: (a) $j$-filling for the case when $j$ is present on the shared processors; (b) changing the order of jobs $j_{i-1}$ and $j_i$ on the shared processors when $p_{j_{i-1}}>p_{j_i}$
• Figure 5: In this example we consider two consecutive segments, which have three and two job pieces, respectively. We consider executing $\xi(\mathcal{S}_i,t_i,\varepsilon_i)$ for $i=0,\ldots,4$, where $\varepsilon_0=\varepsilon$ is positive. All five modifications are doable but note that the job $j_{i,3}$ is synchronized in $\mathcal{S}_4$ but is not synchronized in $\mathcal{S}$. Hence, according to Condition \ref{['eq:trans1:CaseI1']} of Main Step I that handles this modification, this is the maximum $\varepsilon>0$ for which all five modifications are doable.

Theorems & Definitions (31)

• Theorem 1.1
• Lemma 2.1
• proof
• Theorem 2.2
• proof
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Theorem 3.3