Scheduling with Conflicts

We consider the following problem of scheduling with conflicts (SWC): Find a minimum makespan schedule on a constant number of identical machines where conflicting jobs may not be scheduled concurrently. We present the first approximation algorithms for the case in which conflicts between jobs are modeled by general graphs both for unit jobs and jobs with arbitrary processing times. Previous results showed that the problem is weakly NP-hard for two machines and arbitrary processing times, and strong NP-hardness is known to hold in the unit case for at least three machines. Hence, we consider approximation algorithms for general number of machines, and short jobs for two machines. Three different approaches are considered for developing algorithms for SWC. First, SWC with unit jobs can be formulated as a set-cover problem in which the number of elements in each set is bounded by the number of machines. This gives an approximation ratio of Hm − 1 2 = O(log m), where m is the number of machines. The running time of this algorithm is exponential in m. For jobs with arbitrary processing times, we extend this technique by applying scaling. Scaling increases the approximation ratio to O(log maxj pj minj pj · log m) where pj is the processing time of a job j. Second, we analyze the greedy algorithm for arbitrary processing times. We present the first analysis of the approximation ratio of the greedy algorithm, and prove that it equals m+1 2 . We also show the tightness of this analysis, even for unit jobs. Third, we focus on short jobs and two machines. For processing times in the set {1, 2} we introduce an optimal algorithm, that is based on finding a maximum matching in an auxiliary graph. This result is related to finding a maximum bipartite b-matching, for b = 2. When the processing times are from the set {1, 2, 3}, a min-cost matching is used to achieve a 4 3 -approximation ratio is achieved. We also show that the analysis is tight. Chapter