Lawrence Berkeley National Laboratory

In many applications of parallel computing, distribution of the data unambiguously implies distribution of work among processors. But there are exceptions where some tasks can be assigned to one of several processors without altering the total volume of communication. In this paper, we study the problem of exploiting this flexibility in assignment of tasks to improve load balance. We first model the problem in terms of network flow and use combinatorial techniques for its solution. Our parametric search algorithms use maximum flow algorithms for probing on a candidate optimal solution value. We describe two algorithms to solve the assignment problem with log W_T and bar P bar probe calls, where W_T and bar P bar, respectively, denote the total workload and number of processors. We also define augmenting paths and cuts for this problem, and show that any algorithm based on augmenting paths can be used to find an optimal solution for the task assignment problem. We then consider a continuous version of the problem, and formulate it as a linearly constrained optimization problem, i.e., min bar Ax bar_infty,; rms.t.; Bx=d. To avoid solving an intractable infty-norm optimization problem, we show that in this case minimizing the 2-norm is sufficient to minimize the infty-norm, which reduces the problem to the well-studied linearly-constrained least squares problem. The continuous version of the problem has the advantage of being easily amenable to parallelization.