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Co-flows model a modern scheduling setting that is commonly found in avariety of applications in distributed and cloud computing. In co-flowscheduling, there are $m$ input ports and $m$ output ports. Each co-flow $j \inJ$ 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 ofpackets 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 canonly transmit (or receive) one unit of data in unit time. A feasible scheduleat each time $t$ must therefore be a bipartite matching.  We consider co-flow scheduling and seek to optimize the popular objective oftotal weighted completion time. Our main result is a$(2+\epsilon)$-approximation for this problem, which is essentially tight, asthe problem is hard to approximate within a factor of $(2 - \epsilon)$. Thisimproves upon the previous best known 4-approximation. Further, our resultholds even when jobs have release times without any loss in the approximationguarantee. The key idea of our approach is to construct a continuous-timeschedule using a configuration linear program and interpret each job'scompletion time therein as the job's deadline. The continuous-time scheduleserves as a witness schedule meeting the discovered deadlines, which allows usto reduce the problem to a deadline-constrained scheduling problem.  * This result is flawed; see the first page for the details.

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