Multi-Party Set Disjointness and Intersection with Bounded Dependence

In the multi-party set disjointness problem, k players receive private inputs in the form of sets X¹, ..., X^k ⊆ [n], and their goal is to check whether their sets intersect. The set intersection problem is similar, except that the players are required to output the full intersection of their sets rather than just checking whether it is empty. We study the communication complexity of these two problems in the shared-blackboard model of communication complexity, where players communicate with one another by broadcast.Set disjointness and set intersection are two of the most well-studied problems in communication complexity. It has long been known that two-party set disjointness is significantly easier when the players' inputs are independent of one another, and similar results have recently been established for multi-party set disjointness and intersection; however, these results do not apply when the players' inputs have even a small amount of dependence. In this work we close this gap, and give nearly-tight upper and lower bounds for set disjointness and set intersection as a function of the amount of dependence between the players' inputs. Our work explores two existing notions of correlation between the inputs to a multi-party communication protocol, total correlation and dual total correlation, and shows how each is useful in deriving lower and upper bounds, respectively.