A Tight Approximation for Co-flow Scheduling for Minimizing Total Weighted Completion Time

Co-flows model a modern scheduling setting that is commonly found in a variety of applications in distributed and cloud computing. In co-flow scheduling, there are $m$ input ports and $m$ output ports. Each co-flow $j \in J$ can be represented by a bipartite graph between the input and output ports, where each edge $(i,o)$ with demand $d_{i,o}^j$ means that $d_{i,o}^j$ units of packets must be delivered from port $i$ to port $o$. To complete co-flow $j$, we must satisfy all of its demands. Due to capacity constraints, a port can only transmit (or receive) one unit of data in unit time. A feasible schedule at each time $t$ must therefore be a bipartite matching. We consider co-flow scheduling and seek to optimize the popular objective of total weighted completion time. Our main result is a $(2+\epsilon)$-approximation for this problem, which is essentially tight, as the problem is hard to approximate within a factor of $(2 - \epsilon)$. This improves upon the previous best known 4-approximation. Further, our result holds even when jobs have release times without any loss in the approximation guarantee. The key idea of our approach is to construct a continuous-time schedule using a configuration linear program and interpret each job's completion time therein as the job's deadline. The continuous-time schedule serves as a witness schedule meeting the discovered deadlines, which allows us to reduce the problem to a deadline-constrained scheduling problem. * This result is flawed; see the first page for the details.