Efficient Approximation Algorithms for the Bounded Flexible Scheduling Problem in Clouds

Clouds, such as Amazon Infrastructure-as-a-Service (IaaS) clouds and EMC Hybrid Cloud, impose growing requirements of resource-efficiency scheduling. The bounded flexible scheduling (BFS) problem is one of the problems proposed to meet such requirements. In BFS, we are given a set of identical machines and a set of jobs, each of which is with a value, a workload, a deadline and a parallelism degree, i.e., the maximum number of machines on which the job can execute concurrently. The problem is to compute an assignment of the given jobs to the machines, such that the total value of the jobs successfully completed by their deadlines is maximized. This paper presents a factor- <inline-formula><tex-math notation="LaTeX">$\frac{C-k}{C}$</tex-math><alternatives> <inline-graphic xlink:href="shen-ieq1-2731843.gif"/></alternatives></inline-formula> approximation algorithm for BFS, where <inline-formula><tex-math notation="LaTeX">$k$</tex-math><alternatives> <inline-graphic xlink:href="shen-ieq2-2731843.gif"/></alternatives></inline-formula> is the maximum parallelism degree and <inline-formula><tex-math notation="LaTeX">$C$</tex-math><alternatives> <inline-graphic xlink:href="shen-ieq3-2731843.gif"/></alternatives></inline-formula> is the capacity of the system (i.e., the number of machines). Since <inline-formula><tex-math notation="LaTeX">$C\gg k$</tex-math><alternatives> <inline-graphic xlink:href="shen-ieq4-2731843.gif"/></alternatives></inline-formula> in BFS, our result significantly improves the known best approximation ratio of <inline-formula><tex-math notation="LaTeX"> $(\frac{C-k}{2C-k})(1-\epsilon)$</tex-math><alternatives><inline-graphic xlink:href="shen-ieq5-2731843.gif"/> </alternatives></inline-formula> for tight deadlines <xref ref-type="bibr" rid="ref17">[17]</xref> , and <inline-formula><tex-math notation="LaTeX">$\frac{C-k}{C}\cdot \frac{s-1}{s}$</tex-math><alternatives> <inline-graphic xlink:href="shen-ieq6-2731843.gif"/></alternatives></inline-formula> for loose deadlines <xref ref-type="bibr" rid="ref18">[18]</xref> on a slackness ratio <inline-formula><tex-math notation="LaTeX">$s\geq 1$</tex-math><alternatives> <inline-graphic xlink:href="shen-ieq7-2731843.gif"/></alternatives></inline-formula> that is the maximum ratio between a job’s earliest actual finish time and its deadline. We first propose <italic>feasibility condition</italic> to determine whether an instance of BFS is feasible, i.e., whether there exists a scheduling according to which all jobs can finish before their deadlines, which is the key to achieve the ratio improvement of our algorithm. To prove the correctness of the feasibility condition, we give a simple linear program (LP) for a weaker version of BFS, and show that it is with an integral polyhedron and hence the version of BFS is polynomial-time solvable. Then we present a greedy algorithm and its equivalent primal-dual algorithm for the complementary problem of BFS. Both algorithms have an approximation ratio of <inline-formula><tex-math notation="LaTeX">$\frac{C-k}{C}$</tex-math><alternatives> <inline-graphic xlink:href="shen-ieq8-2731843.gif"/></alternatives></inline-formula>, and time complexity <inline-formula><tex-math notation="LaTeX">$O(n^{2}+nT)$</tex-math><alternatives> <inline-graphic xlink:href="shen-ieq9-2731843.gif"/></alternatives></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$n$</tex-math><alternatives><inline-graphic xlink:href="shen-ieq10-2731843.gif"/> </alternatives></inline-formula> is the number of jobs and <inline-formula><tex-math notation="LaTeX">$T$</tex-math> <alternatives><inline-graphic xlink:href="shen-ieq11-2731843.gif"/></alternatives></inline-formula> is the number of time slots. As a by-product, we show that the BFS admits a polynomial-time approximation scheme (PTAS) when <inline-formula><tex-math notation="LaTeX">$T$</tex-math><alternatives> <inline-graphic xlink:href="shen-ieq12-2731843.gif"/></alternatives></inline-formula> is fixed.