Multitask Efficiencies in the Decision Tree Model

TL;DR

This paper investigates how the joint complexity of multiple functions evaluated on a common input behaves in the decision tree model, revealing that the obvious constraints are essentially the only possibilities, and introduces a new cryptographic data structure called 'mystery bin' to support this analysis.

Contribution

It demonstrates that in the decision tree model, the joint complexity constraints are nearly complete and introduces the 'mystery bin' data structure using cryptographic techniques.

Findings

Joint complexity constraints are essentially the only constraints in the decision tree model.

Introduces the 'mystery bin' cryptographic data structure for complexity analysis.

Connects the results to the Direct Sum Conjecture and potential extensions to communication models.

Abstract

In Direct Sum problems [KRW], one tries to show that for a given computational model, the complexity of computing a collection 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 functions 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 [KRW] which, if proved for a single family of functions, could yield an analogous result for models such as the communication model.