A GPU-based Branch-and-Bound algorithm using Integer–Vector–Matrix data structure

Abstract

Branch-and-Bound (B&B) algorithms are tree-based exploratory methods for solving combinatorial optimization problems exactly to optimality. These problems are often large in size and known to be NP-hard to solve. The construction and exploration of the B&B-tree are performed using four operators: branching, bounding, selection and pruning. Such algorithms are irregular which makes their parallel design and implementation on GPU challenging. Existing GPU-accelerated B&B algorithms perform only a part of the algorithm on the GPU and rely on the transfer of pools of subproblems across the PCI Express bus to the device. To the best of our knowledge, the algorithm presented in this paper is the first GPU-based B&B algorithm that performs all four operators on the device and subsequently avoids the data transfer bottleneck between CPU and GPU. The implementation on GPU is based on the Integer-Vector-Matrix (IVM) data structure which is used instead of a conventional linked-list to store and manage the pool of subproblems. This paper revisits the IVM-based B&B algorithm on the GPU, addressing the irregularity of the algorithm in terms of workload, memory access patterns and control flow. In particular, the focus is put on reducing thread divergence by making a judicious choice for the mapping of threads onto the data. Compared to a GPU-accelerated B&B based on a linked-list, the algorithm presented in this paper solves a set of standard flowshop instances on average 3.3 times faster.

A computationally efficient Branch-and-Bound algorithm for the permutation flow-shop scheduling problem

“…The PFSP has been frequently used as a test-case for parallel B&B algorithms, as the huge amount of generated nodes and the highly irregular structure of the search tree raise multiple challenges in terms of design and implementation on increasingly complex parallel architectures, e. g. grid computing (Mezmaz et al, 2007;Drozdowski et al, 2011;Bendjoudi et al, 2012), multicore CPUs (Mezmaz et al, 2014a;Gmys et al, 2016a), GPUs and many-core devices (Chakroun et al, 2013;Gmys et al, 2016b;Melab et al, 2018), clusters of GPUs (Vu and Derbel, 2016) or FPGAs (Daouri et al, 2015).…”

IVM‐based parallel branch‐and‐bound using hierarchical work stealing on multi‐GPU systems

“…The first version corresponds to the one presented in our earlier work [10,11] and uses a combined PTE + PEB model. We present two versions of the B&B algorithm, both using the same load balancing mechanism.…”

“…In order to enable a B&B-process to explore any arbitrary interval OEB; E an initialization procedure, as described in [10] is necessary. However, it is possible to do the interval splitting between § In the rest of this paper the term IVM designates the data structure as well as, by extension, the exploration process associated with a part of the B&B-tree.…”

“…Our algorithm is based on the Integer-Vector-Matrix (IVM) data structure [9] which allows to implement all four B&B operators on the GPU. For the IVM-based GPU-B&B algorithm presented in [10] our previous work [11] proposes four work stealing (WS) [12] strategies for balancing the irregular workload inside a single GPU. The major contributions of this paper are the following:…”

“…Revisiting the algorithm presented in [10] for fine-grained permutation problems, that is, permutation problems where the cost of evaluating the tree nodes is low compared with the tree search itself. The proposed variant of the algorithm uses the same WS mechanism for load balancing as the original algorithm which uses a second level of parallelism to accelerate the bounding operator.…”