Parallel Vertex Cover Algorithms on GPUs

Abstract

Finding small vertex covers in a graph has applications in numerous domains. Two common formulations of the problem include: Minimum Vertex Cover, which finds the smallest vertex cover in a graph, and Parameterized Vertex Cover, which finds a vertex cover whose size is less than or equal to some parameter $k$. Algorithms for both formulations traverse a search tree, which grows exponentially with the size of the graph or the value of $k$. Parallelizing the traversal of the vertex cover search tree on GPUs is challenging for multiple reasons. First, the search tree is a narrow binary tree which makes it difficult to extract enough sub-trees to process in parallel to fully utilize the GPU's resources. Second, the search tree is highly imbalanced which makes load balancing across a massive number of parallel GPU workers challenging. Third, keeping around all the intermediate state needed to traverse many sub-trees in parallel puts high pressure on the GPU's memory resources and may act as a limiting factor to parallelism. To address these challenges, we propose an approach to traverse the vertex cover search tree in parallel using GPUs while handling dynamic load balancing. Each thread block traverses a different sub-tree using a local stack, however, we also use a global worklist to balance load. Blocks contribute branches of their sub-trees to the global worklist on an as-needed basis, while blocks that finish their sub-trees get new ones from the global worklist. We use degree arrays to represent intermediate graphs so that the representation is compact in memory to avoid limiting parallelism, but self-contained which is necessary for load balancing. Our evaluation shows that compared to prior work, our hybrid approach of using local stacks and a global worklist substantially improves performance and reduces load imbalance, especially on difficult instances of the problem.