Multitask Efficiencies in the Decision Tree Model

In Direct Sum problems |8|, one tries to show that for a given computational model, the complexity of computing a collection F = {f₁(x₁),_hellip f₁(x₁)} of finite functions on independent inputs is approximately the sum of their individual complexities. In this paper, by contrast, we study the diversity of ways in which the joint computational complexity can behave when all the f_i are evaluated on a common input. We focus on the deterministic decision tree model, with depth as the complexity measure; in this model we prove a result to the effect that the ´obvious´ constraints on joint computational complexity are essentially the only ones. The proof uses an intriguing new type of cryptographic data structure called a `mystery bin´ which we construct using a small polynomial separation between deterministic and unambiguous query complexity shown by Savicky. We also pose a variant of the Direct Sum Conjecture of |8| which, if proved for a single family of functions, could yield an analogous result for models such as the communication model.