Graphics Processing Units (GPUs) excel at regular data-parallel workloads where massive hardware parallelism can be readily exploited. In contrast, many important irregular applications are naturally expressed as task parallelism with a fork-join control structure. While CPU runtimes for fork-join task parallelism are mature, it remains challenging to efficiently support it on GPUs. We propose GTaP, a GPU-resident runtime that supports fork-join task parallelism. GTaP is based on the persistent kernel model, and supports two worker granularities: thread blocks and individual threads. To realize fork-join on GPUs, GTaP represents joins as continuations and executes each task as a state machine that can be split into multiple execution segments. We also extend Clang's frontend with a pragma-based programming model that enables programmers to express fork-join without exposing low-level mechanisms. GTaP employs work stealing for load balancing, providing better scalability than a global-queue approach. For thread-level workers, we further introduce Execution-Path-Aware Queueing (EPAQ), which allows programmers to partition task queues using user-defined criteria, reducing warp divergence caused by mixing heterogeneous control flows within a warp. Across representative irregular applications, GTaP outperforms OpenMP task-parallel execution on a 72-core CPU in many cases, especially for large problem sizes with compute-intensive tasks. We also show that GTaP's design choices outperform naive GPU alternatives. The benefit of EPAQ is workload-dependent: it can improve performance for some benchmarks while having little effect on others; on Fibonacci, EPAQ achieves up to a 1.8$\times$ speedup.

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