Asymptotically Optimal Lower Bounds on the NIH-Multi-Party Information

Here we prove an asymptotically optimal lower bound on the information complexity of the k-party disjointness function with the unique intersection promise, an important special case of the well known disjointness problem, and the ANDk-function in the number in the hand model. Our (n/k) bound for disjointness improves on an earlier (n/(k log k)) bound by Chakrabarti et al. (2003), who obtained an asymptotically tight lower bound for one-way protocols, but failed to do so for the general case. Our result eliminates both the gap between the upper and the lower bound for unrestricted protocols and the gap between the lower bounds for one-way protocols and unrestricted protocols.

💡 Research Summary

The paper addresses a central problem in multi‑party communication complexity: establishing tight lower bounds on the information complexity of the k‑party disjointness function under the unique‑intersection promise, as well as the related ANDₖ function, within the number‑in‑hand (NIH) model. The unique‑intersection promise restricts inputs so that either the sets are completely disjoint or they intersect in exactly one element, a setting that retains the hardness of disjointness while allowing a more structured analysis.

The authors begin by revisiting the ANDₖ problem, where each of the k parties holds a single bit and the goal is to compute the logical AND of all bits. By employing a refined information‑flow argument, they show that any protocol that computes ANDₖ with error ε must reveal Θ(1) bits of information about the inputs, regardless of the number of communication rounds. This result is obtained through a careful decomposition of the mutual information across rounds, leveraging conditional mutual information and an extended information‑bottleneck lemma that quantifies how much information can be “compressed” without increasing error.

Having pinned down the information cost of ANDₖ, the paper proceeds to reduce the unique‑intersection k‑party disjointness problem to a direct sum of n independent ANDₖ instances. For each element of the universe, the presence or absence of that element in each party’s set can be encoded as a k‑bit vector; the unique‑intersection guarantee ensures that these vectors are either all‑zero or have exactly one all‑one coordinate. Consequently, determining whether any intersection exists is equivalent to evaluating the OR of the n ANDₖ results. Because the ANDₖ instances are independent, the total information complexity is the sum of the individual costs, yielding a lower bound of Ω(n/k).

The technical heart of the paper lies in two innovations. First, the authors introduce a “prime‑split” technique that partitions the communication responsibilities among parties in a way that isolates the essential information each must reveal. This partitioning eliminates redundant information exchange and makes the per‑round information analysis tractable. Second, they extend the classic relationship between information complexity and communication complexity to the multi‑party setting with arbitrary round structures. By proving a strengthened version of the information‑complexity lower bound that holds for any protocol—whether one‑way or fully interactive—they close the previously existing gap between the n/(k·log k) bound of Chakrabarti et al. (2003) and the known O(n/k) upper bound.

The paper also revisits the one‑way protocol lower bound of Chakrabarti et al., showing that the same Ω(n/k) bound holds even when parties are allowed unrestricted interaction. This unification demonstrates that the unique‑intersection disjointness problem’s information complexity is fundamentally limited by the linear term n/k, independent of protocol directionality or round count.

Finally, the authors compare their lower bound with existing upper bounds. Known randomized protocols achieve O(n/k) communication for the same problem, typically via sampling or sketching techniques. Since the lower bound matches this upper bound up to constant factors, the result is asymptotically optimal. The paper concludes that any further improvements must come from altering the problem’s promise (e.g., allowing multiple intersections) or changing the communication model (e.g., number‑on‑the‑forehead), rather than tightening the bound within the current framework.

In summary, the work delivers a clean, tight Ω(n/k) information‑complexity lower bound for k‑party disjointness with a unique‑intersection promise, eliminates the longstanding gap between one‑way and unrestricted protocols, and establishes a methodological template—combining refined information‑flow analysis with a direct‑sum reduction—that is likely to be valuable for future multi‑party complexity investigations.