Two-party direct-sum questions through the lens of multiparty communication complexity

Direct-sum questions in (two-party) communication complexity ask whether two parties, Alice and Bob, can compute the value of a function f on ℓ inputs (x₁, y₁),. ., (x_ℓ, y_ℓ) more efficiently than by applying the best protocol for f, independently on each input (x_i, y_i). In spite of significant efforts to understand these questions (under various communication-complexity measures), the general question is still far from being well understood. In this paper, we offer a multiparty view of these questions: The direct-sum setting is just a two-player system with Alice having inputs x₁,. ., x_ℓ, Bob having inputs y₁,. ., y_ℓ and the desired output is f(x₁, y₁),. ., f(x_ℓ, y_ℓ). The naive solution of solving the ℓ problems independently, is modeled by a network with ℓ (disconnected) pairs of players Alice_i and Bob_i, with inputs x_i, y_i respectively, and communication only within each pair. Then, we consider an intermediate ("star") model, where there is one Alice having ℓ inputs x₁,. ., x_ℓ and ℓ players Bob₁,. ., Bob_ℓ holding y₁,. ., y_ℓ, respectively (in fact, we consider few variants of this intermediate model, depending on whether communication between each Bobi and Alice is point-to-point or whether we allow broadcast). Our goal is to get a better understanding of the relation between the two extreme models (i.e., of the two-party direct-sum question). If, for instance, Alice and Bob can do better (for some complexity measure) than solving the ℓ problems independently, we wish to understand what intermediate model already allows to do so (hereby understanding the "source" of such savings). If, on the other hand, we wish to prove that there is no better solution than solving the ℓ problems independently, then our approach gives a way of breaking the task of proving such a statement into few (hopefully, easier) steps. We present several results of both types. Namely, for certain complexity measures, communication problems f and certain pairs of models, we can show gaps between the complexity of solving f on ℓ instances in the two models in question; while, for certain other complexity measures and pairs of models, we can show that such gaps do not exist (for any communication problem f). For example, we prove that if only point-to-point communication is allowed in the intermediate "star" model, then significant savings are impossible in the public-coin randomized setting. On the other hand, in the private-coin randomized setting, if Alice is allowed to broadcast messages to all Bobs in the "star" network, then some savings are possible. While this approach does not lead yet to new results on the original two-party direct-sum question, we believe that our work gives new insights on the already-known direct-sum results, and may potentially lead to more such results in the future.