Although this bounding mechanism allows the considerable reduction of the exploration time, often only small or moderatelysized instances of COPs can be practically solved. For this reason, over the last decades, parallel computing has been revealed as an attractive way to deal with larger instances of COPs. However, while many contributions have been proposed for parallel B\&B methods using massively parallel processors \cite{ch8:Allen_1997}, networks or clusters of workstations \cite{ch8:Quinn_1990}, and Shared Memory Multiprocessors (SMP) machines \cite{ch8:Casadoa_2008}, very few contributions have been proposed for redesigning B\&B algorithms on Graphical Processing Units (GPUs) \cite{ch8:Carneiro_2011}. For years, the use of GPU accelerators was limited to graphics and video applications. Driven by the demand for high-definition 3D graphics on personal computers, GPUs have evolved into a highly parallel, multithreaded and many-core environment. Their utilization has recently been extended to other application domains such as scientific computing \cite{ch8:Kurzak_2010}.\\
-In this chapter, we rethink the design and implementation of irregular tree-based algorithms such as the B\&B algorithm on top of GPUs. During the execution of the B\&B algorithm, the number of newly generated nodes and the number of not yet explored but promising nodes are variable and depend on the level of the tree being explored and on the best solution found so far. Therefore, due to such unstructured and unpredictable nature of its search tree, designing efficient B\&B on top of GPUs is not straightforward. We investigate two different approaches for designing GPU-based B\&B starting from the parallel models for B\&B identified in \cite{ch8:MelabHDR_2005}. The first one is based on the ``parallel tree exploration'' paradigm. This approach consists of exploring in parallel different subspaces of the tree. The second approach is based on the ``parallel evaluation of bounds'' approach. The two approaches have been applied to the permutation Flowshop Scheduling Problem \index{Flowshop Scheduling Problem} (FSP; see Section~\ref{ch8:BB-FSP}) which is an NP-hard combinatorial optimization problem. The lower bound function used in this work for FSP is the one proposed in~\cite{ch8:Johnson_1954} for two machines and generalized in~\cite{ch8:Lenstra_1978} to more than two machines.\\
+In this chapter, we rethink the design and implementation of irregular tree-based algorithms such as the B\&B algorithm on top of GPUs. During the execution of the B\&B algorithm, the number of newly generated nodes and the number of not yet explored but promising nodes are variable and depend on the level of the tree being explored and on the best solution found so far. Therefore, due to such unstructured and unpredictable nature of its search tree, designing efficient B\&B on top of GPUs is not straightforward. We investigate two different approaches for designing GPU-based B\&B starting from the parallel models for B\&B identified in \cite{ch8:MelabHDR_2005}. The first one is based on the ``parallel tree exploration'' paradigm. This approach consists of exploring in parallel different subspaces of the tree. The second approach is based on the ``parallel evaluation of bounds'' approach. The two approaches have been applied to the permutation Flowshop Scheduling Problem \index{flowshop scheduling problem} (FSP; see Section~\ref{ch8:BB-FSP}) which is an NP-hard combinatorial optimization problem. The lower bound function used in this work for FSP is the one proposed in~\cite{ch8:Johnson_1954} for two machines and generalized in~\cite{ch8:Lenstra_1978} to more than two machines.\\
When rethinking those two parallel models for GPU's architectures, our main focus was on the lower bound function. Indeed, preliminary experiments we carried out on some of Taillard's problem instances \cite{ch8:Taillard_1993} show that computing the lower bounds takes on average between 98\% and 99\% of the total execution time of the B\&B. The GPU-based lower bound's implementation raises mainly two challenges. On the one hand, having in mind that the execution model of GPUs is Single Instruction Multiple Data (SIMD), irregular computations (containing loops and conditional instructions) contained in the lower bound function may lead to a very challenging issue: the thread or branch divergence. This problem drops down the performance and arises when threads of a same warp (the smallest executable unit of parallelism on the GPU) execute different data-dependent instructions. On the other hand, the lower bound computation usually uses large and frequently accessed data structures. Since GPU is a many-core coprocessor device that provides a hierarchy of memories having different sizes and access latencies, the placement and sharing of these data sets become challenging.\\
The chapter is organized into seven main sections. Section \ref{ch8:BB} presents the B\&B algorithm. Section \ref{ch8:Parallel-BB} introduces the different models used to parallelize B\&B algorithms. Section \ref{ch8:BB-FSP} briefly describes the Flowshop Scheduling permutation Problem. In Section~\ref{ch8:approach1}, we describe the GPU-accelerated B\&B based on the parallel tree exploration. In Section~\ref{ch8:approach2}, details about the second approach, the GPU-accelerated B\&B based on the parallel evaluation of lower bounds, are given. In Section \ref{ch8:ThreadDivergence}, the thread divergence issue related to the location of nodes in the B\&B tree and to the control flow instructions within the bounding operator is described. In Section \ref{ch8:DataAccessOpt}, the memory access optimization challenge is addressed and an overview of the GPU memory hierarchy and the used memory access pattern is given. In Section~\ref{ch8:Experiments}, we report experimental results showing the performances of each of two studied approaches compared to a sequential CPU-based execution of the B\&B and demonstrating the efficiency of the proposed optimizations.
-\section{Branch-and-bound \index{Branch-and-bound} algorithm}
+\section{Branch-and-bound \index{branch-and-bound} algorithm}
\label{ch8:BB}
Branch-and-bound algorithms are by far the most widely used methods for exactly solving large scale NP-hard combinatorial optimization problems. Indeed, they allow the finding of the optimal solution of a problem with proof of optimality. \\
Tree-based strategies consist of building and/or exploring the solution tree in parallel by performing operations on several subproblems simultaneously. This coarse-grained type of parallelism affects the general structure of the B\&B algorithm and makes it highly irregular.\\
-The parallel tree exploration \index{parallel tree exploration} model, illustrated in Figure \ref{ch8:parallel_tree}, consists of visiting in parallel different paths of the same tree. The search tree is explored in parallel by performing the branching, selection, bounding, and elimination operators on several subproblems simultaneously.\\
+The parallel tree exploration \index{parallel!tree exploration} model, illustrated in Figure \ref{ch8:parallel_tree}, consists of visiting in parallel different paths of the same tree. The search tree is explored in parallel by performing the branching, selection, bounding, and elimination operators on several subproblems simultaneously.\\
\begin{figure}
\begin{center}
Node-based strategies introduce parallelism when performing the operations on a single problem. For instance, they consist of executing the bounding operation in parallel for each subproblem to accelerate the execution. This type of parallelism has no influence on the general structure of the B\&B algorithm and is particular to the problem being solved.\\
-The parallel evaluation of bounds \index{parallel evaluation of bounds} model, as shown in Figure \ref{ch8:bounds_parallel}, allows the parallelization of the bounding of subproblems generated by the branching operator. This model is used in the case where the bounding operator is performed several times after the branching operator. Compared to the sequential B\&B, the model does not change the order and the number of explored subproblems in the parallel B\&B algorithm.
+The parallel evaluation of bounds \index{parallel!evaluation of bounds} model, as shown in Figure \ref{ch8:bounds_parallel}, allows the parallelization of the bounding of subproblems generated by the branching operator. This model is used in the case where the bounding operator is performed several times after the branching operator. Compared to the sequential B\&B, the model does not change the order and the number of explored subproblems in the parallel B\&B algorithm.
\begin{figure}
\begin{center}
\end{figure}
-\subsection{Lower bound \index{Lower bound} for the flowshop scheduling problem}
+\subsection{Lower bound \index{lower bound} for the flowshop scheduling problem}
The lower bounding technique provides a lower bound (LB) for each subproblem generated by the branching operator. The more the bound is accurate, the more it allows the elimination from the search tree that are not promising. Therefore, the efficiency of a B\&B algorithm depends strongly on the quality of its lower bound function. In this chapter, we use the lower bound proposed by Lenstra et al.~\cite{ch8:Lenstra_1978} for FSP, based on the Johnson's algorithm~\cite{ch8:Johnson_1954}.\\
\subsection{The thread divergence issue}
-During the execution of an application on GPU, one or more thread block(s) are assigned to each GPU multiprocessor to execute. Those threads are partitioned into warps that get scheduled for execution. For each instruction of the flow, the multiprocessor selects a warp that is ready to be run. A warp executes one common instruction at a time, so full efficiency is realized when all threads of a warp agree on their execution path. In this chapter, the G80 model, in which a warp is a pool of 32 threads, is used. If threads of a warp diverge via a data-dependent conditional branch, the warp serially executes each branch path taken. Threads that are not on the taken path are disabled, and when all paths are complete, the threads converge back to the same execution path. This phenomenon is called thread/branch divergence\index{Thread divergence} and often causes serious performance degradations. Branch divergence occurs only within a warp; different warps execute independently regardless of whether they are executing common or disjointed code paths.\\
+During the execution of an application on GPU, one or more thread block(s) are assigned to each GPU multiprocessor to execute. Those threads are partitioned into warps that get scheduled for execution. For each instruction of the flow, the multiprocessor selects a warp that is ready to be run. A warp executes one common instruction at a time, so full efficiency is realized when all threads of a warp agree on their execution path. In this chapter, the G80 model, in which a warp is a pool of 32 threads, is used. If threads of a warp diverge via a data-dependent conditional branch, the warp serially executes each branch path taken. Threads that are not on the taken path are disabled, and when all paths are complete, the threads converge back to the same execution path. This phenomenon is called thread/branch divergence\index{GPU!thread divergence} and often causes serious performance degradations. Branch divergence occurs only within a warp; different warps execute independently regardless of whether they are executing common or disjointed code paths.\\
This section discusses thread divergence issues encountered when computing the bounds by GPU. The thread divergence occurs for two main reasons, namely, the locations of nodes in the search tree and the control flow instructions within the bounding operator. \\
\section{Memory access optimization}
\label{ch8:DataAccessOpt}
-Memory access optimizations \index{Memory access optimizations} are by far the most studied area for improving GPU-based application performances. Indeed, adjusting the pattern of accesses to the GPU device memory allows programmers to further improve the throughput of many high-performance CUDA applications. The goal of memory access optimizations is generally to use as much fast-access memory and as little slow-access memory as possible. This section discusses how best to set up data LB items on the various kinds of memory on the device. \\
+Memory access optimizations \index{memory access optimizations} are by far the most studied area for improving GPU-based application performances. Indeed, adjusting the pattern of accesses to the GPU device memory allows programmers to further improve the throughput of many high-performance CUDA applications. The goal of memory access optimizations is generally to use as much fast-access memory and as little slow-access memory as possible. This section discusses how best to set up data LB items on the various kinds of memory on the device. \\
CUDA-enabled devices use several memory spaces, which have different characteristics in term of sizes and access latencies. These memory spaces include global memory, local memory , shared memory, texture memory , and registers. Devices of compute capability 2.0 also have an L1/L2 cache hierarchy that is used to cache local and global memory accesses.
