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

International audienceIn this work we propose an efficient branch-and-bound (B&B) algorithm for the permutation flow-shop problem (PFSP) with makespan objective. We present a new node decomposition scheme that combines dynamic branching and lower bound refinement strategies in a computationally efficient way. To alleviate the computational burden of the two-machine bound used in the refinement stage, we propose an online learning-inspired mechanism to predict promising couples of bottleneck machines. The algorithm offers multiple choices for branching and bounding operators and can explore the search tree either sequentially or in parallel on multi-core CPUs. In order to empirically determine the most efficient combination of these components, a series of computational experiments with 600 benchmark instances is performed. A main insight is that the problem size, aswell as interactions between branching and bounding operators substantially modify the trade-off between the computational requirements of a lower bound and the achieved tree size reduction. Moreover, we demonstrate that parallel tree search is a key ingredient for the resolution of largeproblem instances, as strong super-linear speedups can be observed. An overall evaluation using two well-known benchmarks indicates that the proposed approach is superior to previously published B&B algorithms. For the first benchmark we report the exact resolution – within less than20 minutes – of two instances defined by 500 jobs and 20 machines that remained open for more than 25 years, and for the second a total of 89 improved best-known upper bounds, including proofs of optimality for 74 of them

Towards ultra-scale Branch-and-Bound using a high-productivity language

International audienceDue to the highly irregular nature and prohibitive execution times of Branch-and-Bound (B&B) algorithms applied to combinatorial optimization problems (COPs), their parallelization has received these two last decades great attention. Indeed, significant efforts have been made to revisit the parallelization of B&B following the rapid evolution of high-performance computing technologies dealing with their associated scientific and technical challenges. However, these parallelization efforts have always been guided by the performance objective setting aside programming productivity. Nevertheless, this latter is crucial for designing ultra-scale algorithms able to harness modern supercomputers which are increasingly complex, including millions of processing cores and heterogeneous building-block devices. In this paper, we investigate the partitioned global address space (PGAS)-based approach using Chapel for the productivity-aware design and implementation of distributed B&B for solving large COPs. The proposed algorithms are intensively experimented using the Flow-shop scheduling problem as a test-case. The Chapel-based implementation is compared to its MPI+X-based traditionally used counterpart in terms of performance, scalabil-ity, and productivity. The results show that Chapel is much more expressive and up to 7.8× more productive than MPI+Pthreads. In addition, the Chapel-based search presents performance equivalent to MPI+Pthreads for its best results on 1024 cores and reaches up to 84% of the linear speedup. However, there are cases where the built-in load balancing provided by Chapel cannot produce regular load among computer nodes. In such cases, the MPI-based search can be up to 4.2× faster and reaches speedups up to 3× higher than its Chapel-based counterpart. Thorough feedback on the experience is given, pointing out the strengths and limitations of the two opposite approaches (Chapel vs. MPI+X). To the best of our knowledge, the present study is pioneering within the context of exact parallel optimization

A Multi-Core Parallel Branch-and-Bound Algorithm Using Factorial Number System

International audienceMany real-world problems in different industrial and economic fields are permutation combinatorial optimization problems. Solving to optimality large instances of these problems, such as flowshop problem, is a challenge for multi-core computing. This paper proposes a multi-threaded factoradic-based branch-and-bound algorithm to solve permutation combinatorial problems on multi-core processors. The factoradic, called also factorial number system, is a mixed radix numeral system adapted to numbering permutations. In this new parallel algorithm, the B&B is based on a matrix of integers instead of a pool of permutations, and work units exchanged between threads are intervals of factoradics instead of sets of nodes. Compared to a conventional pool-based approach, the obtained results on flowshop instances demonstrate that our new factoradic-based approach, on average, uses about 60 times less memory to store the pool of subproblems, generates about 1.3 times less page faults, waits about 7 times less time to synchronize the access to the pool, requires about 9 times less CPU time to manage this pool, and performs about 30,000 times less context switches

Solving large permutation flow-shop scheduling problems on GPU-accelerated supercomputers

Makespan minimization in permutation flow-shop scheduling is a well-known hard combinatorial optimization problem. Among the 120 standard benchmark instances proposed by E. Taillard in 1993, 23 have remained unsolved for almost three decades. In this paper, we present our attempts to solve these instances to optimality using parallel Branch-and-Bound tree search on the GPU-accelerated Jean Zay supercomputer. We report the exact solution of 11 previously unsolved problem instances and improved upper bounds for 8 instances. The solution of these problems requires both algorithmic improvements and leveraging the computing power of peta-scale high-performance computing platforms. The challenge consists in efficiently performing parallel depth-first traversal of a highly irregular, fine-grained search tree on distributed systems composed of hundreds of massively parallel accelerator devices and multi-core processors. We present and discuss the design and implementation of our permutation-based B&B and experimentally evaluate its parallel performance on up to 384 V100 GPUs (2 million CUDA cores) and 3840 CPU cores. The optimality proof for the largest solved instance requires about 64 CPU-years of computation-using 256 GPUs and over 4 million parallel search agents, the traversal of the search tree is completed in 13 hours, exploring 339 Tera-nodes