• Optimize data transmission for jobs within a data center.
• Machines can communicate with each other directly
• Uniform machine/capacity.

• \(n\) coflows, each represented as matrix \(D^j = [d^j_{io}]\).
  • \(d^j_{io}\): demand of job \(j\) from input port \(i\) to output port \(o\).
• For every coflow \(j\):
  • \(r_j\): release time
  • \(w_j\): weight
• Minimize: \(\sum_{i=1}^n w_j\cdot C_j\).
  • \(C_j\): completion time. A coflow is finished when all its flows are finished.
• Discrete time.
• At each time slot an input/output port can send/receive at most one unit of flow.

• \(n\) jobs to be scheduled on \(m\) different machines.
• Each job \(j\) consists of \(m\) tasks for each machine. For machine \(i\), there is an task that takes \(d^j_i\) time to finish.
• Different tasks of the same job can take place at the same time.
• A job is finished when all its tasks are finished.
• Minimize weighted completion time.

• Chowdhury and Stoica* first model the coflow scheduling problem.
• Chowdhury, Zhong, and Stoica\({}^{\dagger}\) give effective heuristic to optimize coflow completion time.

Qiu, Stein, and Zhong proved the following for coflow scheduling in SPAA 2015:

In SPAA 2016, Khuller and Purohit* improve upon this via a black box reduction to Concurrent Open Shop Problem and get

• Zero release time (deterministic): 8-approx
• Arbitrary release time (deterministic): 12-approx

In SIGCOMM 2018, Agarwal et al. implemented Sincronia which uses a similar primal-dual algorithm without release time. Evaluation results suggest that it not only admits a practical, near-optimal design but also improves upon state-of-the-art network designs for coflows.

• Optimize data transmission for jobs across data centers.
• Data centers may be directly connected, or connected via multiple routers.
• Network modeled as an directed graph with non-uniform structure.
• Sharing a link within a coflow or between coflows is allowed.
• Spliting and merging a flow is also allowed.

• A weighted directed graph \(G = (V, E)\) representing the network.
  • Weight of an edge is its capacity.
• \(n\) coflows, each represented as a matrix \(D^j = [d^j_{io}]\).
  • \(d^j_{io}\): demand of job \(j\) from node \(i\) to node \(o\).
  • Either a path is given, or there is no limitation on paths.
• For every coflow \(j\):
  • \(r_j\): release time
  • \(w_j\): weight
• Minimize: \(\sum_{i=1}^n w_j\cdot C_j\).
  • \(C_j\): completion time. A coflow is finished when all its flows are finished.
• Discrete time.
• An edge can be shared within a coflow or between coflows, as long as the total bandwidth is within its capacity.

• A schedule would decide at each time slot:
  • Which flows to send.
  • How shall we route each flow.
  • At what rate.
• Solution can be fractional.
  • Naturally required by data transmission.
  • Different from the switch based coflow scheduling problem, where the main difficulty is finding an integral solution.
  • The main difficulty comes from coordinating multiple coflows.

• Single path model:
  • Path for each flow \(f^i_j\) given as \(p^i_j\).
  • Only need to decide the rate for each flow at each time.
  • Edge bandwidths need to be respected.
• Free path model:
  • Path for a flow \(f^i_j\) not specified.
  • Need to find routes for every flow.
  • Data can split and merge at vertices to utilize all possible links and their capacities.
  • Edge bandwidths need to be respected.

• Jahanjou, Kantor, and Rajaraman* give a constant approximation (\(17.53\)) algorithm for the single path model.
• In the same paper*, the authors give a tight \(O\left(\frac{\log |E|}{\log \log |E|}\right)\) algorithm for the case where you need to pick a single path for each demand.
• You and Mosharaf\({}^{\dagger}\) introduce the free path model, and give a heuristic for the unweighted case without theoretical bound.

• A tight \(2\) approximation randomized algorithm for both models when release times and demands are polynomial sized.
• NP-hard to get \((2-\epsilon)\) approximation for any \(\epsilon > 0\).
• A \((2 + \epsilon)\) randomized approximation algorithm when release times and demands are super-polynomial sized.
• Experiments show significant improvements on existing algorithms.

• Single Coflow:
  • Multi commodity flow
  • Can be solved optimally in polynomial time.
• Multiple Coflows:
  • Time indexed LP
  • Stretching to round the solution.