Log-Rank Conjecture for Communication Complexity

• The log-rank conjecture is a central open problem that asserts the deterministic communication complexity of a Boolean function is polynomially bounded in the logarithm of its matrix rank.
• Recent advances reveal that improved extraction of large monochromatic rectangles from low-rank matrices can yield tighter bounds, with current results reaching up to O(√r) under specific conditions.
• Equivalent formulations in graph theory, Fourier analysis, and incidence geometry highlight the conjecture’s broad implications and pave the way for new research directions.

The log-rank conjecture is a central open problem in two-party deterministic communication complexity. For a Boolean function $f:X\times Y\to\{0,1\}$ with communication matrix $M_f$ and real rank $r=\operatorname{rank}_{\mathbb{R}}(M_f)$, it asserts that the deterministic communication complexity $D(f)$ is polynomially bounded in $\log r$, i.e. $D(f)\le \operatorname{poly}(\log r)$ (Lovett, 2014). In the literature surveyed here, the conjecture is also expressed as a quasi-polynomial relation between rank and partition number, or as a large-rectangle principle asserting the existence of monochromatic rectangles of density $2^{-\operatorname{polylog}(r)}$. The strongest general upper bound recorded in these sources is $D(f)\le O(\sqrt r)$ (Sudakov et al., 2023).

1. Formal statement and basic inequalities

For a two-party Boolean function $f:X\times Y\to\{0,1\}$, the communication matrix $M_f\in\{0,1\}^{|X|\times |Y|}$ is defined by $M_f$0, and $M_f$1 is the minimum number of bits exchanged by a deterministic protocol that computes $M_f$2 on all inputs. The log-rank conjecture, in the form attributed to Lovász–Saks, states that there exists a universal constant $M_f$3 such that

$M_f$4

where $M_f$5, and logarithms are taken base $M_f$6 (Lovett, 2014).

The conjecture sits between two elementary bounds. First, every monochromatic rectangle has rank at most $M_f$7, so a $M_f$8-bit protocol partitions the matrix into at most $M_f$9 monochromatic rectangles, implying

$r=\operatorname{rank}_{\mathbb{R}}(M_f)$0

Second, if $r=\operatorname{rank}_{\mathbb{R}}(M_f)$1, then $r=\operatorname{rank}_{\mathbb{R}}(M_f)$2 has at most $r=\operatorname{rank}_{\mathbb{R}}(M_f)$3 distinct rows, so one player can send a row index, giving the trivial upper bound

$r=\operatorname{rank}_{\mathbb{R}}(M_f)$4

The conjecture asks whether the exponential gap between $r=\operatorname{rank}_{\mathbb{R}}(M_f)$5 and $r=\operatorname{rank}_{\mathbb{R}}(M_f)$6 can always be collapsed to a polynomial in $r=\operatorname{rank}_{\mathbb{R}}(M_f)$7 (Lovett, 2014).

A closely related parameter is the partition number $r=\operatorname{rank}_{\mathbb{R}}(M_f)$8, the minimum number of monochromatic rectangles that partition $r=\operatorname{rank}_{\mathbb{R}}(M_f)$9. It satisfies

$D(f)$0

and the matrix formulation of the conjecture becomes $D(f)$1 (Lovett, 2014, Hambardzumyan et al., 2 Oct 2025).

2. General upper bounds and the present state of the problem

A standard reduction due to Nisan–Wigderson shows that progress on the conjecture is equivalent to proving sufficiently large monochromatic rectangles. If every rank-$D(f)$2 Boolean matrix contains a monochromatic rectangle of relative size at least $D(f)$3, then

$D(f)$4

In particular, a bound $D(f)$5 would prove the conjecture (Lovett, 2014).

The main unconditional upper bounds in the supplied literature proceed through progressively stronger rectangle theorems. Lovett’s 2013 theorem gave

$D(f)$6

by combining discrepancy lower bounds for low-rank matrices, a conversion from near-monochromatic to monochromatic rectangles, and the Nisan–Wigderson boosting scheme (Lovett, 2014). Sudakov–Tomon later improved the rectangle exponent, proving that any $D(f)$7 binary matrix of rank at most $D(f)$8 contains an $D(f)$9 all-1 or all-0 submatrix, which implies

$\log r$0

for every Boolean function of rank $\log r$1 (Sudakov et al., 2023).

There are also important conditional results. Assuming the Polynomial Freiman–Ruzsa conjecture over $\log r$2, Ben-Sasson, Lovett, and Ron-Zewi proved

$\log r$3

via an additive-combinatorial route based on approximate duality and the Nisan–Wigderson methodology (Ben-Sasson et al., 2011). Gavinsky–Lovett established a different type of reduction:

$\log r$4

showing that, for low-rank matrices, efficient randomized protocols already imply efficient deterministic protocols up to polylogarithmic loss in $\log r$5 (Lovett, 2014).

Despite these advances, the conjectured $\log r$6 dependence remains open. The current best general upper bound is still exponentially larger than the target when measured as a function of $\log r$7 (Sudakov et al., 2023).

3. Structural mechanisms: rectangles, discrepancy, and density decrement

The dominant proof paradigm is structural: low rank is expected to force large monochromatic rectangles, and such rectangles can be converted into deterministic protocols. In the Nisan–Wigderson framework, rectangle-finding is therefore not an auxiliary tool but the central combinatorial manifestation of the conjecture (Lovett, 2014).

A key bridge from algebra to combinatorics is discrepancy. For $\log r$8 of rank $\log r$9, the low-rank discrepancy bound states

$D(f)\le \operatorname{poly}(\log r)$0

High discrepancy, in turn, implies large nearly monochromatic rectangles under arbitrary input distributions: if $D(f)\le \operatorname{poly}(\log r)$1, then for every $D(f)\le \operatorname{poly}(\log r)$2 and every distribution $D(f)\le \operatorname{poly}(\log r)$3 on $D(f)\le \operatorname{poly}(\log r)$4, there exists a rectangle $D(f)\le \operatorname{poly}(\log r)$5 with

$D(f)\le \operatorname{poly}(\log r)$6

For low-rank functions this near-monochromatic rectangle can then be “cleaned”: if a rank-$D(f)\le \operatorname{poly}(\log r)$7 function is $D(f)\le \operatorname{poly}(\log r)$8 on at least $D(f)\le \operatorname{poly}(\log r)$9 of a rectangle $2^{-\operatorname{polylog}(r)}$0, then $2^{-\operatorname{polylog}(r)}$1 contains a monochromatic sub-rectangle $2^{-\operatorname{polylog}(r)}$2 with $2^{-\operatorname{polylog}(r)}$3 (Lovett, 2014).

Sudakov–Tomon recast this pipeline in matrix terms using a centered discrepancy

$2^{-\operatorname{polylog}(r)}$4

where $2^{-\operatorname{polylog}(r)}$5, and then analyzed it via semidefinite programming and spectral methods. For a binary matrix $2^{-\operatorname{polylog}(r)}$6 of rank at most $2^{-\operatorname{polylog}(r)}$7 and density $2^{-\operatorname{polylog}(r)}$8, they proved

$2^{-\operatorname{polylog}(r)}$9

for a universal constant $D(f)\le O(\sqrt r)$0. This yields a balanced density decrement on large submatrices, and iteration of that decrement leads to the $D(f)\le O(\sqrt r)$1-scale monochromatic rectangles that imply the $D(f)\le O(\sqrt r)$2 communication bound (Sudakov et al., 2023).

This structural picture explains why the conjecture persists. Low rank already enforces bias, discrepancy, and large submatrices with strong regularity, but existing arguments extract rectangles only at $D(f)\le O(\sqrt r)$3 density rather than at the conjectured $D(f)\le O(\sqrt r)$4 scale.

4. Partial resolutions for special classes

Several broad classes are now known to satisfy log-rank-type bounds substantially stronger than the general theory.

For AND-functions, the setting is $D(f)\le O(\sqrt r)$5, where $D(f)\le O(\sqrt r)$6 is bitwise AND. Knop, Lovett, McGuire, and Yuan proved that if $D(f)\le O(\sqrt r)$7, then $D(f)\le O(\sqrt r)$8 can be computed by an AND-decision tree of depth

$D(f)\le O(\sqrt r)$9

and hence

$f:X\times Y\to\{0,1\}$0

This holds for arbitrary Boolean $f:X\times Y\to\{0,1\}$1, with no assumptions such as monotonicity or low $f:X\times Y\to\{0,1\}$2-degree. The proof is driven by a structural theorem stating that sparse multilinear polynomials have monomial set systems with polylogarithmic-size hitting sets (Knop et al., 2020).

For XOR-functions, where $f:X\times Y\to\{0,1\}$3, the real rank equals the Fourier sparsity:

$f:X\times Y\to\{0,1\}$4

Tsang, Wong, Xie, and Zhang proved two representative upper bounds. If $f:X\times Y\to\{0,1\}$5, then

$f:X\times Y\to\{0,1\}$6

so the log-rank conjecture holds for XOR-functions of constant $f:X\times Y\to\{0,1\}$7-degree. More generally,

$f:X\times Y\to\{0,1\}$8

giving an unconditional root-of-rank bound in this regime (Tsang et al., 2013).

A different extension concerns functions with small alternating number. If $f:X\times Y\to\{0,1\}$9, then Lin–Zhang proved

$M_f\in\{0,1\}^{|X|\times |Y|}$0

and

$M_f\in\{0,1\}^{|X|\times |Y|}$1

with a refined AND bound

$M_f\in\{0,1\}^{|X|\times |Y|}$2

These results confirm the conjecture for both compositions whenever $M_f\in\{0,1\}^{|X|\times |Y|}$3 (Lin et al., 2016).

These special cases support a common theme: once low rank is coupled with an auxiliary structural constraint—sparse polynomial representation, low $M_f\in\{0,1\}^{|X|\times |Y|}$4-degree, or small alternating number—the conjectured polylogarithmic behavior becomes much more accessible.

5. Equivalent formulations and neighboring frameworks

The conjecture has accumulated several equivalent or near-equivalent formulations that relocate the difficulty into adjacent areas of combinatorics, geometry, and matrix theory.

One classical reformulation concerns graph adjacency matrices. For a graph adjacency matrix $M_f\in\{0,1\}^{|X|\times |Y|}$5, an equivalent conjecture due to van Nuffelen and Fajtlowicz states

$M_f\in\{0,1\}^{|X|\times |Y|}$6

Known upper bounds are $M_f\in\{0,1\}^{|X|\times |Y|}$7 and the improved $M_f\in\{0,1\}^{|X|\times |Y|}$8 (Lovett, 2014).

A geometric reformulation arises from point–hyperplane incidence geometry. Chattopadhyay, Mande, and Sherif showed that the log-rank conjecture is equivalent to a statement about parallel $M_f\in\{0,1\}^{|X|\times |Y|}$9-partitions, and more generally about $M_f$00-listable matrices: for fixed $M_f$01, every rank-$M_f$02, $M_f$03-listable matrix should contain a $M_f$04-listable submatrix of fractional size $M_f$05. In this framework, the communication-theoretic dimension becomes the ambient geometric dimension, and monochromatic rectangles become large affine subspaces accounting for many incidences (Singer et al., 2021).

A newer matrix-theoretic formulation introduces signed rectangle rank. For a Boolean matrix $M_f$06, $M_f$07 is the minimum number of primitive rectangles needed to express $M_f$08 as a $M_f$09-sum. It satisfies

$M_f$10

and the main quantitative theorem gives

$M_f$11

when $M_f$12. This reframes log-rank as a signed-to-positive conversion problem:

$M_f$13

which is equivalent to the usual conjecture because $M_f$14. The same paper proves an equivalence with a conjecture of Lovett and Singer-Sudan on cross-intersecting set systems (Hambardzumyan et al., 2 Oct 2025).

The Rank-Ramsey program supplies a graph-theoretic perspective in which one seeks graphs whose clique number is small while the rank of the complement is also small. This viewpoint links constructions and nonexistence results for such graphs to graph-theoretic formulations of log-rank, and it emphasizes that low-rank structure can coexist with surprisingly sparse clique behavior in induced substructures (Beniamini et al., 2024).

Taken together, these formulations do not simplify the conjecture so much as isolate its core obstruction: converting low algebraic complexity into large positive combinatorial structure.

6. Barriers, refutations, and open directions

The known lower bounds remain far below the conjectured upper bound. The strongest separation from the trivial lower bound exhibited in the supplied literature shows functions with

$M_f$15

Thus any universal exponent $M_f$16 in a bound $M_f$17, if such an exponent exists, must satisfy $M_f$18. At the same time, no superpolynomial gap between $M_f$19 and $M_f$20 is known (Lovett, 2014).

A common misconception is that the XOR case should be tractable merely because rank becomes Fourier sparsity. Recent work shows that two influential additive-structure strategies fail in general. Mande, Saket, and Sanyal refuted a conjecture seeking a single large folding direction by constructing $M_f$21 with Fourier support $M_f$22 such that $M_f$23 while every distinct translate pair satisfies

$M_f$24

They also refuted a conjecture asserting that many good folding directions occur with constant probability, proving a construction for which

$M_f$25

These refutations rule out both a greedy “single large overlap” strategy and a constant-probability “many good directions” strategy as general proofs of XOR log-rank (Hatami et al., 2023).

Several open problems recur across the supplied papers. The central question is still to prove $M_f$26 unconditionally, or even to improve the current general upper bound below $M_f$27. More targeted directions include proving that every rank-$M_f$28 Boolean matrix contains a monochromatic rectangle of relative size $M_f$29; establishing that approximate rank $M_f$30 implies randomized communication $M_f$31; showing that efficient quantum protocols for low-rank functions imply deterministic protocols with polylogarithmic overhead; and proving large all-zero rectangles in sparse low-rank real matrices (Lovett, 2014).

In special regimes, the main unresolved quantitative issue is often a residual slack factor rather than the full conjecture. For AND-functions, a central open problem is removing the remaining $M_f$32 factor from

$M_f$33

which would yield a full log-rank theorem for the one-bit AND gadget (Knop et al., 2020). In the signed rectangle rank formulation, the corresponding question is whether every signed decomposition can be turned into a positive decomposition with only quasi-polynomial blowup, equivalently whether

$M_f$34

or even whether $M_f$35 always holds (Hambardzumyan et al., 2 Oct 2025).

The modern understanding of the log-rank conjecture is therefore twofold. On one hand, low rank already enforces substantial structure: discrepancy bounds, biased rectangles, randomized-to-deterministic reductions, sparse-polynomial decompositions, incidence-geometric reformulations, and signed rectangle decompositions. On the other hand, every current route still loses far more than the conjecture allows. The open problem is no longer merely to find some structure in low-rank matrices, but to identify the precise structural principle strong enough to force monochromatic rectangles at the $M_f$36 scale.