Complexity-Theoretic Regularity Lemma

• Complexity-Theoretic Regularity Lemma is a framework that approximates complex structures by decomposing them into structured components and negligible pseudorandom remainders.
• It extends classical results like Szemerédi’s lemma to arithmetic, algebraic, analytic, and model-theoretic settings, enabling efficient algorithmic property testing.
• It provides explicit complexity parameters that quantify the trade-offs between structure and randomness, impacting graph theory, Boolean function analysis, and tensor decompositions.

The complexity-theoretic regularity lemma captures a spectrum of quantitative and structural decompositions in combinatorics, analysis, and theoretical computer science. Initially motivated by Szemerédi’s regularity lemma for graphs, modern developments extend to arithmetic, analytic, algebraic, model-theoretic, and computational settings, focusing on structural decompositions with explicitly controlled complexity parameters and algorithmic amenability. The central theme is that “complex” combinatorial or functional objects can be efficiently approximated by structured (low-complexity) components plus pseudorandom or negligible remainders, with these decompositions governed by explicit complexity-theoretic invariants.

1. Classical and Generalized Regularity Lemmas

Szemerédi’s Regularity Lemma

Szemerédi’s regularity lemma states that every large dense graph can be partitioned into a bounded number of parts such that most bipartite subgraphs between parts are “ε-regular,” i.e., within each pair, all large enough vertex subsets have nearly constant edge density. The lemma ensures there exists, for every $\epsilon > 0$, a partition of the vertex set into $K = \operatorname{tower}(O(1/\epsilon))$ parts such that the number of irregular pairs is small. However, the tower-type quantitative dependence between $\epsilon$ and $K$ (see Gowers’ lower bound (Moshkovitz et al., 2013)) is inherent.

Extensions and Algorithmic Aspects

The regularity paradigm is extended through:

• Weak regularity lemmas (Frieze–Kannan): Graphs can be approximated in cut-norm by a sum of $O(1/\epsilon^2)$ cut matrices, with corresponding polynomial-time algorithms (Fox et al., 2016).
• Banach/Hilbert space regularity: Any bounded subset of an appropriate Banach or Hilbert space (e.g., graphons under the cut norm) admits weakly regular approximations (step functions or low-rank tensors), which is operationalized via compactness of orbit spaces under group actions (Regts, 2015).
• Polynomial and distal regularity: In “tame” classes (semi-algebraic, distal, or bounded VC-dimension graphs), one can obtain polynomial or doubly exponential bounds, and, crucially, partitions with very strong atomicity or indivisibility properties (cf. (Fox et al., 2015, Chernikov et al., 2015, Towsner, 2013, Malliaris et al., 2011)).

2. Complexity-Informed Structure Theorems

Arithmetic Regularity Lemma

Green–Tao’s arithmetic regularity lemma (Green et al., 2010) demonstrates that a bounded function $f:[N] \to [0,1]$ can be decomposed as

$$f = f_{\text{nil}} + f_{\text{sml}} + f_{\text{unf}}$$

where

• $f_{\text{nil}}$ is a polynomial nilsequence, encoding highly structured, equidistributed dynamics (quantified via the irrationality parameter $(F(M), N)$ and complexity $M$);
• $\|f_{\text{sml}}\|_{L^2[N]} < \epsilon$ is a small $L^2$-error;
• $\|f_{\text{unf}}\|_{U^{s+1}[N]} < 1/F(M)$ is negligible in Gowers uniformity.

This decomposition enables not only structure/randomness analysis but also precise counting lemmas for arithmetic patterns—provided the underlying system of linear forms satisfies the “flag property,” a nontrivial symmetry/equidistribution condition. The regularity and counting lemmas are parametrized by explicit complexity quantities: the degree and dimension of the nilmanifold, filtration degree $s$, and Gowers–Wolf versus Cauchy–Schwarz complexities of the pattern system.

Algebraic/Definable Regularity Lemmas

In definable group settings over finite fields, one can obtain partitions into bounded-index normal subgroups such that coset-induced graphs are highly pseudorandom, with “power-saving” error rates, e.g., $O_M(|\mathbf{F}|^{-1/2})$-quasirandomness (Pillay et al., 15 Dec 2024). These results rely on model theory (due to uniformity across definable classes) and yield spectral control (small nontrivial Fourier coefficients).

In stable graphs, the complexity-theoretic regularity lemma (Malliaris et al., 2011) yields partitions into “excellent” or “indivisible” sets, with the number of parts depending linearly (or even polynomially) on $\epsilon^{-1}$, and the complete elimination of irregular pairs—possible only because unstable half-graph configurations are absent.

3. Quantitative Complexity Parameters

Lower Bounds and Optimality

• Gowers (Moshkovitz et al., 2013), Conlon–Fox–Sudakov (Conlon et al., 2011) establish sharp tower-type (and even wowzer-type) lower bounds for part counts in both regular and strong/induced variants.
• Polynomial/doubly exponential bounds are achievable for graphs with bounded VC-dimension (Towsner, 2013), semi-algebraic or distal structures (Fox et al., 2015, Chernikov et al., 2015), and “gentle” graph classes (Jiang et al., 2020).
• For Banach/Hilbert space and weak regularity, one confronts only exponential or polynomial part counts, and explicit low-rank decompositions/approximations (Regts, 2015).

Complexity Measures

• Combinatorial complexity: Number of partition classes (“parts”), explicit as a function of $\epsilon$ and domain parameters, degree, dimension, or covering property.
• Algebraic/nilmanifold complexity: Degree and dimension of nilmanifolds in arithmetic settings, complexity of Mal’cev basis, filtration index.
• Spectral/analytic complexity: Threshold rank, sum-squares of eigenvalues above threshold, rank, or step complexity in decompositions.
• Model-theoretic complexity: Stability, NIP, distality, order- and VC-dimension controlling the possible strength of the regularity statement.

4. Algorithmic and Property Testing Implications

• Polynomial or quasi-polynomial time algorithms for finding regular partitions are possible in the weak/frieze–kannan regime and in bounded-complexity semi-algebraic/distal graph families (Fox et al., 2015, Fox et al., 2016), and for low threshold rank graphs (Gharan et al., 2012, Bodwin et al., 2019).
• In many cases, deterministic algorithms for property testing, subgraph counting, and approximation (e.g., MAX-CUT, property-freeness) are feasible due to the low complexity of required decompositions (see (Bodwin et al., 2019, Fox et al., 2016, Fox et al., 2015)).
• For hereditary hypergraph properties in semi-algebraic settings, query complexity is polynomial in $1/\epsilon$, contrasting with the much higher bounds for arbitrary hypergraphs (Fox et al., 2015).
• The regularity method underpins much of the algorithmic machinery for extremal and probabilistic combinatorics, especially in contexts where combinatorial or model-theoretic “tameness” enables efficient regularity or partitioning.

5. Model-Theoretic and Geometric Approaches

• Stability theory, distality, and other model-theoretic ideas are leveraged to obtain “atomic” or indivisible regular partitions. The elimination of “half-graph” obstructions is essential for perfect regularity (Malliaris et al., 2011, Chernikov et al., 2015).
• Cell decomposition and o-minimality yield geometric partitionings with polynomial complexity for semi-algebraic or definable graphs and hypergraphs (Fox et al., 2015, Chernikov et al., 2015, Pillay et al., 15 Dec 2024).
• The translation of model-theoretic hierarchy (stable $\Rightarrow$ NIP $\Rightarrow$ “distal”) into algorithmic/complexity-theoretic regularity phenomena is fully realized in recent work (Chernikov et al., 2015, Jiang et al., 2020).

6. Limits, Banach Space and Analytic Generalizations

• In the analyst’s perspective, regularity lemmas become compactness statements in Banach spaces: any (weakly) compact $G$-invariant set can be approximated (in the $|{\cdot}|_R$ norm quotient by group action) by a finite sum of simple structured elements (Regts, 2015).
• These ideas explain and unify regularity and graph limit theories (graphons, sparse graph convergence) within the broader complexity-theoretic framework (Regts, 2015, Bodwin et al., 2019).

7. Future Directions and Open Problems

• Field-size sensitivity and algebraic structure: Failures of regularity phenomena (e.g., in tensor blow-up rank) over small fields call for precise classification and new invariants (Derksen et al., 2018).
• Multicalibration and function approximation: Complexity-theoretic regularity lemmas underpin new results in property testing, pseudo-randomness, and fairness in algorithms (cf. multicalibration, multiaccuracy, and their role in regular approximations of boolean functions (Casacuberta et al., 2023)).
• Refined sparsity, covering, and “gentle” class decompositions: New work explores how “2-covers” and reductions from gentle or sparse base classes yield efficient regularity for more complex structures (Jiang et al., 2020).
• Higher-dimensional and tensor extensions: Multidimensional regularity and counting require stronger (often recursively defined) regularity conditions to generalize hypergraph results and property testing to tensorial data (Taranenko, 2019).

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In summary, complexity-theoretic regularity lemmas form a flexible and quantitatively precise unifying principle. They enable fine-grained decompositions in diverse mathematical and CS settings, parameterized and governed by model-theoretic, analytic, algebraic, or combinatorial complexity, with far-reaching implications for both the structure vs. randomness dichotomy and the design of efficient algorithms in property testing, approximation, and learning. Sophisticated variants now eliminate tower-type blow-up for a range of “tame” structures, while deep lower bounds and counterexamples delineate the true limits of regularity-based methods.