A direct product theorem for bounded-round public-coin randomized communication complexity

In this paper, we show a direct product theorm in the model of two-party bounded-round public-coin randomized communication complexity. For a relation f subset of X times Y times Z (X,Y,Z are finite sets), let R^{(t), pub}e (f) denote the two-party t-message public-coin communication complexity of f with worst case error e. We show that for any relation f and positive integer k: R^{(t), pub}{1 - 2^{-Omega(k/t^2)}}(f^k) = Omega(k/t (R^{(t), pub}_{1/3}(f) - O(t^2))) . In particular, it implies a strong direct product theorem for the two-party constant-message public-coin randomized communication complexity of all relations f. Our result for example implies a strong direct product theorem for the pointer chasing problem. This problem has been well studied for understanding round v/s communication trade-offs in both classical and quantum communication protocols. We show our result using information theoretic arguments. Our arguments and techniques build on the ones used in [Jain 2011], where a strong direct product theorem for the two-party one-way public-coin communication complexity of all relations is shown (that is the special case of our result when t=1). One key tool used in our work and also in [Jain 2011] is a message compression technique due to [Braverman and Rao 2011], who used it to show a direct sum theorem for the two-party bounded-round public-coin randomized communication complexity of all relations. Another important tool that we use is a correlated sampling protocol, which for example, has been used in [Holenstein 2007] for proving a parallel repetition theorem for two-prover games.

💡 Research Summary

The paper establishes a strong direct product theorem for two‑party bounded‑round public‑coin randomized communication complexity. For a relation f ⊆ X × Y × Z, let R^{(t),pub}ε(f) denote the minimum number of bits exchanged in a protocol that uses at most t messages, public randomness, and has worst‑case error at most ε. The main result states that for any integer k ≥ 1,
R^{(t),pub}{1‑2^{-Ω(k/t^2)}}(f^k) = Ω!\Big(\frac{k}{t}\big(R^{(t),pub}_{1/3}(f)‑O(t^2)\big)\Big).
In words, if one attempts to solve k independent copies of f simultaneously with a bounded‑round protocol, then either the total communication must be roughly k/t times the cost of solving a single instance (up to an additive O(t^2) term), or the overall success probability drops exponentially in k/t^2. When t is a constant, this yields a “strong direct product theorem”: achieving non‑trivial success on all k instances requires essentially k times the communication needed for one instance.

The proof builds on three central ingredients:

1. Message Compression (Braverman–Rao 2011).
   The authors apply a compression scheme that replaces each transmitted message by a short description whose length is proportional to the conditional mutual information between the message and the sender’s input given the receiver’s view. This transformation incurs only an additive O(t^2) overhead in total communication while preserving the error probability up to a small additive loss. Consequently, the total communication of any t‑message protocol can be lower‑bounded by its information cost.

2. Information‑Cost Analysis.
   By tracking the conditional mutual information I(M;X^k|Y^k) and I(M;Y^k|X^k) where M is the concatenation of all compressed messages, the authors show that the information revealed by the protocol must be at least Ω(k·R^{(t),pub}_{1/3}(f) − O(t^2 k)). This step uses the data‑processing inequality and a careful Markov‑chain decomposition across the t rounds. The bound essentially says that the protocol cannot “share” information between the k copies without paying a linear cost in k.

3. Correlated Sampling (Holenstein 2007).
   To convert the information‑theoretic lower bound into an actual communication lower bound for the original (uncompressed) protocol, a correlated‑sampling subroutine is employed. This protocol enables the two parties, each holding their own compressed description, to jointly reconstruct the original messages with high probability, without additional public randomness. The error introduced by this step is bounded by 2^{-Ω(k/t^2)}, which explains the specific error term in the theorem.

The combination of these tools yields the desired direct product bound. The theorem immediately implies a strong direct product result for any constant‑round public‑coin protocol, regardless of the underlying relation f. As a concrete application, the authors consider the pointer‑chasing problem, a canonical benchmark for round‑communication trade‑offs. Using the theorem, they show that solving k independent pointer‑chasing instances with t rounds requires Ω(k·t·log n) bits of communication, matching the known lower bound for a single instance up to constant factors and demonstrating that no “amortization” across copies is possible.

The work extends Jain’s 2011 strong direct product theorem for one‑way public‑coin protocols (the case t = 1) to the full bounded‑round setting. It also refines the direct‑sum theorem of Braverman and Rao by strengthening the error dependence and providing a multiplicative k/t factor rather than a mere additive term. The paper discusses limitations: the error guarantee is of the form 1 − 2^{-Ω(k/t^2)}; achieving a constant‑error strong direct product (e.g., error ≤ 1/3) remains open. Moreover, extending the result to quantum or private‑coin models would likely require new techniques.

In summary, the paper delivers a robust, information‑theoretic framework for proving strong direct product theorems in bounded‑round public‑coin communication complexity. By integrating message compression, fine‑grained information‑cost analysis, and correlated sampling, it shows that any protocol attempting to solve many independent instances simultaneously must pay a near‑linear communication penalty, thereby deepening our understanding of the fundamental limits of interactive computation.