Scheduling Mutual Exclusion Accesses in Equal-Length Jobs

A fundamental problem in parallel and distributed processing is the partial serialization that is imposed due to the need for mutually exclusive access to common resources. In this article, we investigate the problem of optimally scheduling (in terms of makespan) a set of jobs, where each job consists of the same number L of unit-duration tasks, and each task either accesses exclusively one resource from a given set of resources or accesses a fully shareable resource. We develop and establish the optimality of a fast polynomial-time algorithm to find a schedule with the shortest makespan for any number of jobs and for any number of resources for the case of L = 2. In the notation commonly used for job-shop scheduling problems, this result means that the problem J |d_ij=1, n_j =2|C_max is polynomially solvable, adding to the polynomial solutions known for the problems J2 | n_j ≤ 2 | C_max and J2 | d_ij = 1 | C_max (whereas other closely related versions such as J2 | n_j ≤ 3 | C_max, J2 | d_ij ∈ { 1,2} | C_max, J3 | n_j ≤ 2 | C_max, J3 | d_ij=1 | C_max, and J |d_ij=1, n_j ≤ 3| C_max are all known to be NP-complete). For the general case L > 2 (i.e., for the job-shop problem J |d_ij=1, n_j =L> 2| C_max), we present a competitive heuristic and provide experimental comparisons with other heuristic versions and, when possible, with the ideal integer linear programming formulation.