MOTIVATION: Multiplex PCR-based enrichment is widely used in viral genome sequencing and pathogen surveillance. However, designing large sets of primers that maximize genome coverage while minimizing primer-primer interactions remains a major computational challenge. Existing methods such as SADDLE and Olivar use heuristics to optimize a Badness score for primer dimers but lack theoretical guarantees on solution quality. RESULTS: We introduce PRISM, a new framework that formulates multiplex primer design as a constrained submodular maximization problem. Our method defines an objective that balances genome coverage and dimer risk, and applies a local search algorithm with a constant-factor approximation guarantee. Evaluations on viral genome datasets demonstrate that PRISM consistently achieves lower Badness scores compared to PrimalScheme, Olivar, and primerJinn. These results highlight the scalability and theoretical rigor of submodular optimization in primer design. AVAILABILITY: PRISM is open-source and available at https://github.com/yhhan19/PRISM-new. The experimental data, scripts, and results used in this paper are archived on Figshare at https://doi.org/10.6084/m9.figshare.32806499.