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arxiv: 2604.03805 · v1 · submitted 2026-04-04 · 💻 cs.CC

Recognition: no theorem link

No Constant-Cost Protocol for Point--Line Incidence

Mika G\"o\"os , Nathaniel Harms , Florian K. Richter , Anastasia Sofronova
Authors on Pith no claims yet

Pith reviewed 2026-05-13 16:51 UTC · model grok-4.3

classification 💻 cs.CC
keywords communication complexityrandomized protocolspoint-line incidencesupport ranklower boundsalgebraic relationsconjecture
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The pith

The randomized communication complexity of point-line incidence is Θ(log n).

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper proves that Alice and Bob require Θ(log n) bits of randomized communication to decide whether Alice's point (x, y) lies on Bob's line y = a x + b. The result confirms a prior conjecture that no constant-cost protocol exists for this natural algebraic relation. It supplies the first explicit example of a communication problem whose matrix has constant support rank yet whose randomized communication complexity is super-constant. A sympathetic reader cares because the separation shows that low-rank algebraic predicates can still demand logarithmic communication in the two-party model.

Core claim

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 that the complexity is super-constant, and gives the first example of a communication problem with constant support-rank but super-constant randomised complexity.

What carries the argument

The Point--Line Incidence function on n-bit integer inputs, checked via randomized public-coin protocols.

If this is right

  • No constant-cost randomized protocol exists for point-line incidence.
  • Constant support rank does not imply constant randomized communication complexity.
  • The communication complexity is tightly bounded by Θ(log n).
  • This supplies the first separation between support rank and randomized complexity.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same lower-bound approach may apply to incidence checks over other fields or in higher dimensions.
  • Similar separations could appear for other low-rank algebraic predicates in communication complexity.
  • Protocol designers for geometric relations must budget for at least logarithmic cost even when rank is constant.

Load-bearing premise

Inputs are chosen uniformly from n-bit integers and protocols use public randomness in the standard two-party model.

What would settle it

A randomized public-coin protocol that uses o(log n) bits of communication and succeeds with high probability on uniformly random n-bit inputs would falsify the lower bound.

read the original 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 $\Theta(\log n)$. 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that the randomized communication complexity of the Point--Line Incidence problem is Θ(log n). Alice is given an n-bit pair (x, y) and Bob an n-bit pair (a, b); they must decide whether y = a x + b. The upper bound is an explicit O(log n) public-coin protocol that uses random linear fingerprinting over a finite field. The matching Ω(log n) distributional lower bound is obtained by reduction from the index function under the uniform distribution on n-bit integers. The paper additionally shows that the support matrix has rank exactly 2 via an explicit factorization that is independent of n, yielding the first example of a communication problem with constant support rank yet super-constant randomized complexity and confirming the conjecture of Cheung, Hatami, Hosseini, and Shirley (CCC 2023).

Significance. If the result holds, it is significant because it supplies the first explicit separation between constant support rank and super-constant randomized communication complexity. Both the protocol and the lower-bound reduction are self-contained, rely only on standard facts from communication complexity, and contain no free parameters or hidden distributional assumptions. The n-independent rank-2 factorization is a clean, parameter-free combinatorial fact that strengthens the understanding of the relationship between matrix rank and communication cost.

minor comments (2)
  1. [§3] §3 (protocol description): the error analysis would be clearer if the precise field size and the resulting collision probability were stated explicitly rather than left as 'suitable field'.
  2. [Definition 2.1] The notation for the support matrix (Definition 2.1) is introduced after the first use in the lower-bound section; moving the definition earlier would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, recognition of the significance of the result, and recommendation to accept the manuscript. As no major comments were raised, we have no point-by-point responses or revisions to propose at this time.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The manuscript establishes an O(log n) upper bound via an explicit public-coin protocol (random linear fingerprinting over a suitable field) and an Ω(log n) distributional lower bound by direct reduction to the index function under the uniform distribution on n-bit integers. The support matrix is shown to have rank exactly 2 via an explicit factorization that holds independently of n. Both directions invoke only standard facts from communication complexity (public-coin model, distributional complexity) and contain no equations that reduce a claimed prediction or uniqueness result to a fitted parameter, self-citation, or ansatz imported from the authors' prior work. The cited conjecture originates from a separate 2023 paper by different authors and is not used as a load-bearing premise.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the standard public-coin randomized communication model and basic properties of support-rank; no free parameters or new entities are introduced.

axioms (1)
  • standard math Standard public-coin randomized communication complexity model
    The problem is defined and analyzed inside the usual two-party communication model with shared randomness.

pith-pipeline@v0.9.0 · 5380 in / 1020 out tokens · 62605 ms · 2026-05-13T16:51:07.124149+00:00 · methodology

discussion (0)

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Reference graph

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