AN IMPROVED PARTIAL SOLUTION TO THE TASK ASSIGNMENT AND MULTIWAY CUT PROBLEMS

The Task Assignment problem consists of assigning tasks to processors in order to minimize the sum of execution times and communication costs. It is equivalent to the Multiway Cut problem, where for a weighted graph with specified vertices S = {s(i),..., s(N)} one has to select a set of edges of minimal weight that separates the s(i)'s. A problem's size can be reduced by Stone's observation that vertices in the same components as s1 in a minimal (s1, S - s1) cut remain so in an optimal multiway cut. Hence the optimal assignment of a task to a processor might sometimes be determined by solving an auxiliary two processor problem. which amounts to a max flow computation. In this paper we show that the same property holds in a different two cut-two processor problem, and at least as many tasks are thus assigned as through Stone's procedure. Computational results show that the improvement is significant although for large problems no reduction is to be expected. The method can be extended to identify tasks whose optimal assignment is restricted to be in a subset of the processors.