The Buffer Minimization Problem for Scheduling Flow Jobs with Conflicts

TL;DR

This paper investigates online buffer minimization for scheduling conflicting processors under the flow model, where workloads arrive continuously and processing occurs at unit rate subject to conflict constraints. It provides tight bounds for all graphs on four vertices, as well as tight results for complete graphs and complete bipartite graphs, and nearly tight bounds for complete k-partite graphs, with corresponding translations to the original model. The authors introduce and analyze simple yet effective algorithms (Greedy, PrioGreedy, PrioCenter) and invariants that govern total delay, establishing exact competitive ratios such as $H_n-1/2$ for complete graphs, $3/2$ for complete bipartite graphs, and $4/3$ for certain four-vertex graphs in the flow model. For the K3+e graph, the flow model achieves a $4/3$-competitive bound, while the original model shows a gap with bounds between $13/6$ and $9/4$, highlighting the impact of model choice on performance. Overall, the work reveals how graph structure and flow-based arrivals influence online buffering strategies, with implications for resource-sharing in multiprocessor systems.

Abstract

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.

Figures (1)

• Figure 1: Graphs with four vertices. The numbers $(R_f; R_o)$ represent the competitive ratios in the flow model ($R_f$) and the original model ($R_o$), respectively.

Theorems & Definitions (27)

• definition 1
• definition 2
• lemma 1
• definition 3
• lemma 2: Lemma 4.1 10.1007/3-540-48224-5_70, Original model
• theorem 1
• theorem 2
• theorem 3
• lemma 3: Lemma 4.2 10.1007/3-540-48224-5_70, Original model
• theorem 4