No Constant-Cost Protocol for Point--Line Incidence

Abstract: Alice and Bob are given $n$-bit integer pairs $(x,y)$ and $(a,b)$, respectively, and they must decide if $y=ax+b$. We prove that the randomised communication complexity of this Point--Line Incidence problem is $Θ(\log n)$. This confirms a conjecture of Cheung, Hatami, Hosseini, and Shirley (CCC 2023) that the complexity is super-constant, and gives the first example of a communication problem with constant support-rank but super-constant randomised complexity.

• The paper proves that the randomized communication complexity of the Point–Line Incidence problem is Θ(log n) by applying a sophisticated discrepancy method.
• It introduces a structure–randomness decomposition that separates bounded functions into structured and pseudorandom components to rigorously control line averages.
• The results resolve longstanding conjectures on the separation between constant support-rank and randomized protocols, with extensions to Integer Inner Product problems.

Communication Complexity Lower Bounds for Point–Line Incidence

Problem Statement and Main Results

The paper rigorously establishes the tight randomized communication complexity of the Point–Line Incidence (PL) problem, where Alice holds $n$-bit integers $(x, y)$ and Bob holds $n$-bit integers $(a, b)$, and their task is to decide whether $y = a x + b$. The authors prove that the public-coin randomized communication complexity $\Rcc(PL)$ is $\Theta(\log n)$, confirming the conjecture of Cheung, Hatami, Hosseini, and Shirley (CCC 2023). The separation is strong: although PL has constant support-rank, it attains super-constant $\Rcc$, marking the first known instance of such a separation.

In addition, they settle the randomized communication complexity of the Integer Inner Product ($IIP^k_n$) problem to be $\Theta(k \log n)$ for appropriate parameter regimes. The work introduces new lower-bound techniques via a decomposition lemma, with further consequences for constant-cost communication complexity and its associated algebraic rank measures.

Technical Contributions

Randomized Lower Bound via Discrepancy

The lower bound for $(x, y)$0 employs the (public-coin) discrepancy method. The core challenge arises because previous separations between randomized complexity and algebraic rank measures have only been established for sign-rank and not support-rank. For functions with low support-rank, it has remained open whether randomized protocols can achieve constant communication, and PL was the primary candidate problem to resolve this question.

The main theorem asserts that for the PL problem, any rectangle's discrepancy under a carefully designed pair of distributions is at most $(x, y)$1. As a consequence, the randomized communication complexity must be at least $(x, y)$2, via the standard discrepancy lower bound.

Structure–Randomness Decomposition

A crucial component of the argument is a decomposition lemma for bounded functions $(x, y)$3, separating $(x, y)$4 into (i) a structured component that is nearly invariant under shifts by a large period $(x, y)$5, and (ii) a pseudorandom component whose averages over random lines (modulo small primes) are negligibly small. This decomposition is achieved via a major/minor arc analysis of the Fourier spectrum, setting $(x, y)$6 as the product of small integers and partitioning Fourier coefficients accordingly. The technique is reminiscent of the circle method, but specialized to enable quantitative control of line averages and periodicity properties.

The main line lemma (Lemma 1) asserts that for appropriately chosen $(x, y)$7 and $(x, y)$8, and any bounded function $(x, y)$9, the average $n$0-norm difference between line-averages of $n$1 and their $n$2-shifted versions, when restricted to lines of random small prime slope, is $n$3.

Implications for Support-Rank and Communication Complexity

Prior to this work, separation results such as those for the Greater-Than function related to sign-rank but not support-rank. The PL function, with negated support-rank 3 but randomized communication $n$4, establishes the optimal (qualitative) separation known for support-rank. Specifically, the result demonstrates that constant support-rank does not guarantee constant-cost randomized communication protocols.

Quantitatively, any further separation would require functions of even lower support-rank and higher complexity, but PL appears tight in this respect given that support-rank 2 functions reduce to Equality—which is computable with $n$5 bits.

Integer Inner Product

The result extends to the $n$6-argument Integer Inner Product problem ($n$7), showing that public-coin randomized communication complexity is $n$8, matching upper and lower bounds for $n$9. The lower bound leverages the AND composition method and discrepancy amplification, connecting it back to the line incidence problem.

Algebraic Rank Measures

The separation between randomized communication complexity and support-rank provided by PL fills a prior gap. While functions with low sign-rank may have high communication complexity, this was not known for support-rank. The paper's result shows that randomized communication does not collapse to support-rank for total functions, resolving an open problem.

Broader Implications and Future Directions

Constant-Cost Communication Complexity Classes

The result impacts the understanding of “constant-cost” classes such as $(a, b)$0 (functions solvable with $(a, b)$1 randomized communication) and $(a, b)$2 (functions of constant support-rank), showing that $(a, b)$3. The separation is significant for the structural study of communication classes, impacting the characterization of the extent to which algebraic structure governs randomized communication.

The techniques developed are applicable in the analysis of other algebraic communication problems, particularly in understanding how randomness interacts with group/field invariance.

Open Problems

The possibility of improving the separation quantitatively—for instance, producing a total function with constant support-rank and polynomial randomized complexity—remains open. Moreover, the paper revisits the challenge of characterizing $(a, b)$4 in a syntactic way, as the present techniques demonstrate the intricacy of searching for such a characterization.

The outstanding open problems outlined include finding a function in $(a, b)$5 not in $(a, b)$6, and further understanding separations between classes such as $(a, b)$7, $(a, b)$8, and their combinations.

Conclusion

This work conclusively determines the randomized communication complexity of the Point–Line Incidence problem as $(a, b)$9, confirms a major conjecture about the separation between support-rank and randomized complexity, and introduces a decomposition lemma of independent interest. The results have substantial implications for the communication complexity landscape, particularly in constant-cost regimes and the algebraic characterization of communication lower bounds. The introduced analytic and combinatorial techniques will serve as a foundation for further investigation into the hierarchy of algebraic rank measures and the complexity of natural arithmetic communication problems.

Reference:

"No Constant-Cost Protocol for Point--Line Incidence" (2604.03805)