Learn about the computational complexity and algorithmic challenges in resource scheduling through this 35-minute conference talk that explores both approximation hardness results and online algorithm design. Examine the theoretical foundations of scheduling problems where resources must be allocated efficiently under various constraints, with particular focus on understanding when problems become computationally intractable and how to design algorithms that perform well without complete future information. Discover the trade-offs between solution quality and computational efficiency in scheduling scenarios, including analysis of competitive ratios for online algorithms and inapproximability bounds for offline problems. Gain insights into advanced techniques for proving hardness results and developing practical algorithmic solutions for resource management in parallel computing environments.