The Buffer Minimization Problem for Scheduling Flow Jobs with Conflicts

We consider the online buffer minimization in multiprocessor systems with conflicts problem (in short, the buffer minimization problem) in the recently introduced flow model. In an online fashion, workloads arrive on some of the $n$ processors and are stored in an input buffer. Processors can run and reduce these workloads, but conflicts between pairs of processors restrict simultaneous task execution. Conflicts are represented by a graph, where vertices correspond to processors and edges indicate conflicting pairs. An online algorithm must decide which processors are run at a time; so provide a valid schedule respecting the conflict constraints. The objective is to minimize the maximal workload observed across all processors during the schedule. Unlike the original model, where workloads arrive as discrete blocks at specific time points, the flow model assumes workloads arrive continuously over intervals or not at all. We present tight bounds for all graphs with four vertices (except the path, which has been solved previously) and for the families of general complete graphs and complete bipartite graphs. We also recover almost tight bounds for complete $k$-partite graphs. For the original model, we narrow the gap for the graph consisting of a triangle and an additional edge to a fourth vertex.