Local Load Balancing for Data-Parallel Branch-and-Bound

Branch-and-bound (B&B) is a well-known and general combinatorial optimisation technique that is used especially for NP-hard problems where no special purpose algorithm is known. Because of the high problem complexity, parallel implementations are required for speeding up the computations. Algorithms with high and stable efficiency, i.e., utilisation of the processing elements (PEs), are desired. This can be achieved by load balancing techniques that intervene if the PE utilisation worsens. We use a common version of the sequential B&B formulation as a basis for the parallel algorithm from [8] (see Figure 1). The initialisation is hidden in a function call and is discussed in detail in [1]. The main data structure, the OPEN set, stores single nodes of the search tree. Any search strategy (best-first, depth-first, etc.) can be applied. The dataparallel B&B works quite similarly to the serial algorithm, except that each PE has its own OPEN set. All PEs execute the main loop of the B&B synchronously: select a node from its OPEN set, expand it, insert the evaluated successors back into the OPEN set, and update the incumbent if necessary. After a couple of iterations, some PEs become idle because their OPEN sets are empty or their memory space is exhausted. In both cases, nodes of the local OPEN sets have to be redistributed such that a maximum number of PEs can continue processing. Load balancing is the task of equilibrating the load as evenly as possible. The easier task of load sharing is to supply each PE at least with some load. For MIMD computers, many approaches have already been studied [4]. Recently, parallel B&B has been investigated for SIMD computers. A global approach is to map busy to idle PEs in a one-to-one-correspondence and to transfer nodes by global communication. This approach has been applied to the IDA* algorithm in [6, 7]. Static or dynamic trigger conditions indicate when load balancing is reasonable (see [3] also).