Auto-tuning is a popular approach to program optimization: it automatically finds good configurations of a program’s so-called tuning parameters whose values are crucial for achieving high performance for a particular parallel architecture and characteristics of input/output data. We present three new contributions of the Auto-Tuning Framework (ATF), which enable a key advantage in general-purpose auto-tuning : efficiently optimizing programs whose tuning parameters have interdependencies among them. We make the following contributions to the three main phases of general-purpose auto-tuning: (1) ATF generates the search space of interdependent tuning parameters with high performance by efficiently exploiting parameter constraints; (2) ATF stores such search spaces efficiently in memory, based on a novel chain-of-trees search space structure; (3) ATF explores these search spaces faster, by employing a multi-dimensional search strategy on its chain-of-trees search space representation. Our experiments demonstrate that, compared to the state-of-the-art, general-purpose auto-tuning frameworks, ATF substantially improves generating, storing, and exploring the search space of interdependent tuning parameters, thereby enabling an efficient overall auto-tuning process for important applications from popular domains, including stencil computations, linear algebra routines, quantum chemistry computations, and data mining algorithms.