The LOCAL(A, B) randomized task scheduling algorithm is proposed for fully connected multiprocessors. It combines two given task scheduling algorithms (A, and B) using local neighborhood search to give a hybrid of the two given algorithms. Objective is to show that such type of hybridization can give much better performance results in terms of parallel execution times. Two task scheduling algorithms are selected: DSC (Dominant Sequence Clustering as algorithms A), and CF'F'S (Cluster Pair Priority Scheduling as algorithm B) and a hybrid is created (the LOCAL(DSC, CPPS) or simply the LOCAL task scheduling algorithm). The LOCAL task scheduling algorithm has time complexity 0(broken vertical bar V broken vertical bar broken vertical bar 1E broken vertical bar(vertical bar V broken vertical bar + broken vertical bar E broken vertical bar)), where V is the set of vertices, and P is the set of edges in the task graph. The LOCAL task scheduling algorithm is compared with six other algorithms: CF'F'S, DCCL ((Dynamic Computation. Communication Load), DSC, EZ (Edge Zeroing),. (Linear Clustering), and RDCC (Randomized Dynamic Computation Comm,unicalion). Performance evaluation of the I OC AL task scheduling algorithm shows that it gives up to 80.47 C improvement of NSL (Normalized Schedule Length) over other algorithms.