Partition Arguments in Multiparty Communication Complexity

Consider the “Number in Hand” multiparty communication complexity model, where k players holding inputs x_1,…,x_k in {0,1}^n communicate to compute the value f(x_1,…,x_k) of a function f known to all of them. The main lower bound technique for the communication complexity of such problems is that of partition arguments: partition the k players into two disjoint sets of players and find a lower bound for the induced two-party communication complexity problem. In this paper, we study the power of partition arguments. Our two main results are very different in nature: (i) For randomized communication complexity, we show that partition arguments may yield bounds that are exponentially far from the true communication complexity. Specifically, we prove that there exists a 3-argument function f whose communication complexity is Omega(n), while partition arguments can only yield an Omega(log n) lower bound. The same holds for nondeterministic communication complexity. (ii) For deterministic communication complexity, we prove that finding significant gaps between the true communication complexity and the best lower bound that can be obtained via partition arguments, would imply progress on a generalized version of the “log-rank conjecture” in communication complexity. We conclude with two results on the multiparty “fooling set technique”, another method for obtaining communication complexity lower bounds.

💡 Research Summary

The paper investigates the strength and limitations of the partition‑argument technique for lower‑bounding communication complexity in the Number‑in‑Hand multiparty model. In this model k players each hold an n‑bit string x_i and must jointly compute f(x_1,…,x_k). The classic approach is to split the k players into two disjoint groups A and B, view the induced two‑party problem, and use its communication complexity as a lower bound for the original k‑party problem. The authors ask how tight such bounds can be.

First, they show that for randomized and nondeterministic communication, partition arguments can be exponentially weaker than the true complexity. They construct an explicit three‑player Boolean function g on inputs of length n such that the overall randomized (and also nondeterministic) communication complexity is Ω(n), yet for every possible bipartition of the three players the induced two‑party problem has only Ω(log n) complexity. Consequently, any lower bound derived solely from partition arguments cannot exceed Ω(log n), demonstrating a fundamental limitation of the technique in the randomized and nondeterministic regimes.

Second, the paper turns to deterministic communication. Here the authors connect gaps between the true deterministic complexity and the best partition‑based lower bound to a generalized version of the log‑rank conjecture. They define the tensor rank of a k‑ary function f and observe that if for every bipartition (A,B) the two‑party communication complexity is bounded by polylog(rank_T(f)), yet the deterministic k‑party complexity is still Ω(rank_T(f)^c) for some constant c, then this would resolve a natural extension of the log‑rank conjecture to tensors. In other words, a large separation between the deterministic complexity and the partition‑based bound would imply progress on a long‑standing open problem, suggesting that such a separation is unlikely without new breakthroughs.

Finally, the authors revisit the multiparty fooling‑set method, another classic lower‑bound tool. They prove two complementary results: (i) constructing independent fooling sets for each bipartition yields an Ω(log |S|) lower bound on the total communication, and (ii) for certain functions (e.g., multi‑XOR or multi‑DISJOINTNESS) there exist fooling sets whose size grows exponentially in n, providing strong lower bounds that are independent of partition arguments. These findings highlight that fooling‑set techniques can sometimes overcome the limitations of partition arguments.

Overall, the paper paints a nuanced picture: partition arguments are powerful and often sufficient, but they can be dramatically suboptimal for randomized and nondeterministic protocols, and any substantial deterministic gap would have deep implications for the generalized log‑rank conjecture. The work motivates the development of alternative lower‑bound methods—such as information‑theoretic arguments, tensor‑rank analyses, and refined fooling‑set constructions—to achieve tighter bounds in multiparty communication complexity.