One-Round Multi-Party Communication Complexity of Distinguishing Sums

We consider an instance of the following problem: Parties P_1,..., P_k each receive an input x_i, and a coordinator (distinct from each of these parties) wishes to compute f(x_1,..., x_k) for some predicate f. We are interested in one-round protocols where each party sends a single message to the coordinator; there is no communication between the parties themselves. What is the minimum communication complexity needed to compute f, possibly with bounded error? We prove tight bounds on the one-round communication complexity when f corresponds to the promise problem of distinguishing sums (namely, determining which of two possible values the {x_i} sum to) or the problem of determining whether the {x_i} sum to a particular value. Similar problems were studied previously by Nisan and in concurrent work by Viola. Our proofs rely on basic theorems from additive combinatorics, but are otherwise elementary.