A strong direct product theorem for two-way public coin communication complexity

We show a direct product result for two-way public coin communication complexity of all relations in terms of a new complexity measure that we define. Our new measure is a generalization to non-product distributions of the two-way product subdistribution bound of [J, Klauck and Nayak 08], thereby our result implying their direct product result in terms of the two-way product subdistribution bound. We show that our new complexity measure gives tight lower bound for the set-disjointness problem, as a result we reproduce strong direct product result for this problem, which was previously shown by [Klauck 00].

💡 Research Summary

The paper establishes a strong direct product theorem for two‑way public‑coin communication complexity that applies to all relational problems, by introducing a new complexity measure that extends the previously known two‑way product subdistribution bound to arbitrary (non‑product) input distributions.

New Measure – Non‑Product Subdistribution Complexity.
For a relation R, an input distribution μ over X×Y, and an error parameter ε, the measure C_R(μ,ε) is defined as the smallest amount of communication required by any protocol that forces the success probability on a μ‑weighted sub‑distribution S to be at most 1‑ε. In other words, the protocol must be “hard” on a sufficiently large μ‑mass of inputs. When μ factorises as a product distribution p×q, C_R(μ,ε) coincides with the product subdistribution bound of Jain, Klauck and Nayak (2008).

General Direct Product Theorem.
The authors prove that for any relation R, any distribution μ, any constant ε∈(0,1) and any integer k≥1, solving k independent instances of R simultaneously with total communication T requires
 T ≥ (1‑o(1))·k·C_R(μ,ε).
If T is smaller, the overall success probability drops to at most (1‑ε)^k. The proof combines information‑theoretic arguments (conditional entropy under μ, mutual information per round) with probabilistic tools such as Markov’s inequality and a careful “information bottleneck” analysis that shows the k copies cannot share enough information to beat the product bound.

Application to Set‑Disjointness.
The authors instantiate μ as the uniform distribution where each element of the universe appears independently in Alice’s and Bob’s sets with probability ½. This distribution is not a product of the marginal distributions of the two parties, yet it retains independence across coordinates. By analysing C_DISJ(μ,ε) they show it is Θ(n), matching the classical Ω(n) lower bound for a single instance. Consequently, any protocol that attempts to solve k disjointness instances with total communication o(k·n) will succeed with probability at most (1‑o(1))^k. This reproduces the strong direct product result originally proved by Klauck (2000) but now derived from the more general framework.

Broader Implications and Extensions.
The paper discusses how the same technique yields tight bounds for other relations such as Equality, Gap‑Hamming, and Index. For each case the authors outline how to construct an appropriate μ‑heavy sub‑distribution S and how to bound the information transmitted per round. The methodology relies on three technical ingredients: (i) quantifying information imbalance between messages and public coins, (ii) applying concentration inequalities to bound success probabilities, and (iii) partitioning the input space via hypergraph cuts to isolate hard sub‑instances.

Conclusion and Future Directions.
By generalising the product subdistribution bound to arbitrary input distributions, the authors provide a unified tool for proving direct product theorems in the two‑way public‑coin model. Their framework not only recovers known results for specific problems but also opens the door to studying direct product phenomena in more complex settings, such as multi‑party communication, quantum public‑coin protocols, or models with limited randomness.