Total completion time minimization in two-machine job shops with unit time operations

We study the problem of minimizing total completion time in two-machine job shop with unit-time operations. We propose an efficient algorithm for the problem. The algorithm is polynomial with respect to a succinct encoding of the problem instances, where the number of bits necessary to encode a job with k operations is O(log(k+1)). This result answers a long standing open question about the complexity of the problem.