Papers
Topics
Authors
Recent
Search
2000 character limit reached

The even-uniform hypergraph Moore bound

Published 15 Jul 2026 in math.CO, cs.DM, and cs.DS | (2607.14068v1)

Abstract: The hypergraph Moore bound conjectured by Feige (2008) controls the size of the smallest even cover in a kk-uniform hypergraph in terms of the average density of hyperedges. An even cover is a set of hyperedges covering each vertex an even number of times, generalizing the notion of a cycle in a graph, so the size of the smallest non-trivial even cover provides a notion of hypergraph girth. Recent work starting from the breakthrough result of Guruswami, Kothari, and Manohar (2022) proved the conjecture up to polylogarithmic factors, whose exponents were later gradually improved. We give a simple proof of Feige's original hypergraph Moore bound conjecture for all even k≥4k\ge 4, with no superfluous polylogarithmic factors. Our proof roughly follows the proof of the graph Moore bound, but works with colored walks in a Kikuchi graph built from a hypergraph and controls their growth using a polynomial interpolation method.

Summary

  • The paper confirms Feige’s hypergraph Moore bound conjecture for even k ≥ 4 by precisely establishing the tight hyperedge density threshold.
  • It utilizes polynomial interpolation on colored walks in the Kikuchi graph to bound neighborhood growth and force the existence of small even covers.
  • The result significantly impacts coding theory and average-case complexity by tightening LDPC code parameters and advancing hypergraph girth analysis.

Summary of "The even-uniform hypergraph Moore bound"

Introduction and Context

The paper addresses the longstanding Feige hypergraph Moore bound conjecture for kk-uniform hypergraphs, which generalizes the classical Moore bound from graph theory to hypergraphs. The classical Moore bound characterizes how the girth (length of the shortest cycle) in a graph depends on its average degree. In the hypergraph setting, the analogous quantity is the size of the smallest non-trivial even cover—a collection of hyperedges covering each vertex an even number of times. This generalizes cycle structure and provides a notion of hypergraph girth.

Feige's conjecture quantitatively relates the density of hyperedges to the existence of small even covers, with profound implications in combinatorics, average-case complexity, and coding theory (particularly LDPC codes). Prior works proved the conjecture up to additional polylogarithmic factors; the present paper establishes the bound at the conjectured scale for all even k≥4k \geq 4, removing all superfluous logarithmic factors.

Main Results

The principal theorem established is:

Even-uniform hypergraph Moore bound (Theorem 1):

For even k≥4k \geq 4, every kk-uniform hypergraph on nn vertices with at least m≥64n(n/ρ)k/2−1m \geq 64 n (n/\rho)^{k/2-1} hyperedges contains an even cover of size at most 4kρlog⁡n4k \rho \log n.

The proof makes no attempt at optimizing constants but achieves the conjectured scaling precisely, in contrast to prior work that incurred additional logarithmic factors. The result holds uniformly across the full range of densities. The finding eliminates the main qualitative gap separating previous bounds from the conjecture (2607.14068).

Techniques

The argument closely parallels the proof structure of the classical Moore bound, but instead utilizes colored walks in the Kikuchi graph associated to the hypergraph. The Kikuchi method, which has become central in recent average-case complexity and coding theory results, translates the combinatorial problem into a graph-theoretic framework that enables fine control via polynomial interpolation.

A key innovation is the introduction of diagonal polynomial systems—collections of low-degree polynomials that express indicator functions over the palette of walks in the Kikuchi graph. This approach bounds the growth rate of neighborhoods in the Kikuchi graph, culminating in an exponential expansion argument that contradicts the assumed absence of short even covers.

Specifically, the polynomial interpolation lemma bounds the edge density in subgraphs induced by neighborhoods of the Kikuchi graph. This is analogous to classical VC-dimension and Frankl-Wilson type intersection theorems, ensuring that expansion persists until all vertices are exhausted, hence forcing the existence of small even covers.

Relation to Prior Work

The result builds on the breakthrough of Guruswami, Kothari, and Manohar (2022) who established the conjecture up to polylogarithmic factors [GuruswamiKothariManohar2022], and subsequent improvements that reduced these factors [HsiehKothariMohanty2023] [HsiehKothariMohantyMunhaCorreiaSudakov2024]. The present work achieves the tight density threshold. For even kk and high density (ρ=O(1)\rho = O(1)), Naor and Verstraëte previously proved the conjecture [naorVerstraete].

The case of odd kk remains more challenging; even the best current bounds retain additional logarithmic factors and are restricted in parameter regimes [HsiehKothariMohantyMunhaCorreiaSudakov2024]. The techniques for odd k≥4k \geq 40 are distinct, often relying on combinatorial arguments with rainbow walks.

Implications

Theoretical Implications

Establishing the Moore bound at the conjectured scale answers the foundational question for even-uniform hypergraphs and sets a benchmark for subsequent work on girth-like properties in high-dimensional combinatorial structures. The proof leverages and exemplifies the power of the Kikuchi method and polynomial interpolation within combinatorial frameworks.

Coding Theory

The result translates directly to improved bounds on the rate-distance tradeoff in LDPC code constructions, as the smallest even cover corresponds to small-weight non-trivial dependencies in parity-check matrices. Removing logarithmic factors tightens these tradeoffs, which is critical in the design and understanding of efficient error-correcting codes.

Complexity Theory

The bound supports the existence of polynomial-time verifiable certificates of unsatisfiability for random CSPs below algorithmic thresholds. This has broad implications for average-case complexity and the hardness of refuting random CSPs.

Future Directions

Further refinement of the methods may yield progress on the odd k≥4k \geq 41 case, potentially by extending polynomial interpolation and Kikuchi graph techniques. The techniques may also unlock sharper results in other combinatorial configurations and facilitate improved lower bounds for locally decodable codes and related objects.

Recent advances in AI-guided mathematical discovery played a notable role: GPT-5.6~Sol contributed core ideas for the proof. This underscores the paradigm shift in research workflows and suggests that future developments may increasingly leverage ML-assisted exploration of combinatorial and algebraic structures.

Numerical Significance

The threshold for hyperedge density k≥4k \geq 42 and the even cover size bound k≥4k \geq 43 match the conjectured parameters exactly, with no extraneous polylogarithmic terms. This represents a substantial qualitative improvement over all preceding bounds for even k≥4k \geq 44.

Conclusion

The paper confirms Feige's hypergraph Moore bound conjecture for all even k≥4k \geq 45, precisely characterizing the relationship between hyperedge density and the existence of small even covers. The proof invokes modern combinatorial and algebraic methods grounded in the Kikuchi framework and polynomial interpolation. The result sharpens fundamental bounds in hypergraph theory, coding, and average-case complexity, and is likely to catalyze further research leveraging both combinatorial techniques and AI-driven mathematical innovation (2607.14068).

Paper to Video (Beta)

No one has generated a video about this paper yet.

Sign Up to Generate All Videos Subscribe on YouTube

Whiteboard

No one has generated a whiteboard explanation for this paper yet.

Sign Up to Generate

Explain it Like I'm 14

What is this paper about?

This paper solves a long-standing question about “hypergraphs,” which are like graphs but allow edges to connect more than two points at once. The authors prove a clean, sharp rule (a “Moore bound”) that says: if a kk-uniform hypergraph (every edge connects exactly kk points) is dense enough, then it must contain a small “even cover.” An even cover is a set of hyperedges that touches every point an even number of times. In ordinary graphs, the simplest even cover is a cycle, so you can think of this result as a hypergraph version of “there must be a short cycle when the graph is dense.”

Their main achievement: they prove the exact bound that had been conjectured by Feige for all even k≥4k \ge 4, without extra pesky “logarithmic” factors that earlier works needed.

What questions are they trying to answer?

In simple terms:

  • If we know how many edges a kk-uniform hypergraph has, how small can we guarantee an even cover to be?
  • Can we match the predicted “best possible” trade-off between density and the size of the smallest even cover, at least when kk is even?
  • Can we do this with a simple, clean argument, improving on earlier results that were close but still had extra log factors?

How did they approach it?

The authors blend two key ideas—one from classic graph theory, and one from algebra—and apply them to a carefully built “helper graph.”

First, some friendly translations:

  • Hypergraph: like a graph, but an edge can connect kk points at once (not just two).
  • kk-uniform: every edge connects exactly kk points.
  • Even cover: pick some hyperedges so that each point is covered an even number of times (like 0, 2, 4, …). In a normal graph, a cycle does exactly this.

Here’s the strategy:

  1. Start from a classic idea for graphs. In ordinary graphs, if there are no short cycles, the graph locally looks like a tree, and neighborhoods around a point grow very fast. If this fast growth lasts too long, you run out of vertices—contradiction—so a short cycle must exist. That’s the spirit of the Moore bound.
  2. Build a “Kikuchi graph” from the hypergraph.
    • Its vertices are all subsets of the original points of size ℓ\ell (think: all ℓ\ell-element groups).
    • Two such subsets are connected if their symmetric difference (what’s in one but not the other) is exactly one hyperedge from the original hypergraph.
    • Color each connection by which hyperedge caused it. This graph captures how you can move between ℓ\ell-sets by flipping along hyperedges.
  3. Focus on “interesting” cycles only.
    • Some short cycles in the Kikuchi graph are “trivial” (their colors pair up neatly); they don’t give you an even cover in the original hypergraph.
    • Define an “unpaired” cycle: a cycle where at least one color appears an odd number of times. Such a cycle corresponds to a genuine even cover back in the hypergraph.
    • If there were no small even covers, then the Kikuchi graph must have large “unpaired girth” (no short unpaired cycles).
  4. Show neighborhoods must grow fast—too fast.
    • They pass to a dense subgraph of the Kikuchi graph with large minimum degree.
    • If there are no short unpaired cycles, then from a starting vertex, each layer of neighbors must be much larger than the previous one. Keep going for about log⁡n\log n steps, and this growth should exceed the total number of vertices—an impossibility—unless a short unpaired cycle exists.
  5. Control the growth using a neat algebraic trick (polynomial interpolation).
    • Each vertex in a neighborhood gets a short “binary label” that records which edge colors were used an odd number of times along a path from the root. Adjacent vertices differ in exactly one bit.
    • The authors design low-degree polynomials that act like perfect “indicator functions” on these labels (they output 1 exactly on the chosen vertex and 0 on all others).
    • A key lemma then says: if such low-degree indicator polynomials exist, the neighborhood can’t have too many internal edges.
    • This puts a leash on how dense the neighborhood can be, but the high minimum degree forces many edges to point outward—so the neighborhood sizes must grow rapidly. That creates the contradiction unless a small even cover exists.

In short: the Kikuchi graph turns the hypergraph problem into a graph expansion problem, and low-degree polynomials on the hypercube keep the local edge density in check. Put together, they force the existence of a short unpaired cycle, which gives a small even cover.

What did they find, and why does it matter?

  • Main result: For every even k≥4k \ge 4, if a kk-uniform hypergraph has at least about m≥c⋅n⋅(n/ρ)k/2−1m \ge c \cdot n \cdot (n/\rho)^{k/2 - 1} hyperedges (for a fixed constant cc), then it contains an even cover of size at most C⋅k⋅ρlog⁡nC \cdot k \cdot \rho \log n (for another constant CC). This exactly matches Feige’s conjectured scale (up to constants), and crucially removes extra logarithmic factors that earlier proofs needed.
  • Why it matters:
    • It settles Feige’s “hypergraph Moore bound” for all even kk, which sharpens our understanding of how density forces small even covers in hypergraphs.
    • There are direct connections to coding theory (especially LDPC codes), where even covers relate to short linear dependencies in sparse binary systems. Tighter bounds can inform the best-possible trade-offs in code design.
    • It also informs work on algorithms that certify when certain random constraint problems are unsatisfiable: the existence of small even covers can act like short “witnesses” of unsatisfiability.

What’s the big picture and future impact?

  • The paper gives a clean, simple proof that reaches the originally conjectured bound for even kk. It improves on a line of recent research that was very close but still had extra log factors.
  • The method—using Kikuchi graphs and a low-degree polynomial interpolation argument—could help sharpen bounds in related areas where similar extra log factors persist.
  • The case of odd kk remains harder and is not fully resolved at this sharp level yet. The techniques here may inspire new approaches for those cases too.

Overall, the work tightens a fundamental limit in hypergraph theory and strengthens the toolkit for problems in combinatorics, coding theory, and computational complexity.

View Paper Prompt  View All Prompts 

Knowledge Gaps

Knowledge gaps, limitations, and open questions

Below is a concise list of what the paper leaves missing, uncertain, or unexplored, stated concretely to guide future work:

  • Extension to odd-uniform hypergraphs: The technique crucially uses that kk is even (so ∣SΔT∣=k|S\Delta T|=k is achievable on a fixed-weight slice). Devise an alternative walk/graph construction or labeling scheme that works for odd kk, ideally proving Feige’s conjecture at the conjectured density for all odd kk.
  • Tight constants and sharp thresholds: The proof does not optimize constants (e.g., 64 in density, $4k$ in cover size). Determine the optimal dependence on kk and identify the smallest constants ck,Ckc_k,C_k; show matching lower bounds (random or explicit constructions) proving that Θ(ρlog⁡n)\Theta(\rho\log n) is tight up to universal constants.
  • Constructive/algorithmic certificates: The argument is existential and explores an exponentially large Kikuchi graph. Design a polynomial-time algorithm to find a small even cover under the same density assumption, or to produce a verifiable certificate of unsatisfiability for the corresponding CSP instances.
  • Generalization beyond mod 2: The method is tied to F2\mathbb{F}_2 and parity. Develop analogs for covers modulo a prime p>2p>2 (or over larger alphabets), with implications for non-binary LDPC codes and qq-ary CSPs.
  • Non-uniform hypergraphs: The result is for kk-uniform hypergraphs. Investigate whether a Moore-type bound (in terms of average density and a suitable notion of girth/even cover) can be established for non-uniform hypergraphs with varying edge sizes.
  • Robustness and stability: Prove stability/robustness statements (e.g., how the smallest even cover size degrades when mm is slightly below the threshold; or how the result tolerates adversarial deletions of edges/vertices or mild irregularities in degree).
  • Explicit extremal constructions: Provide deterministic hypergraph families (or pseudorandom ones) at the conjectured density that avoid even covers smaller than cρlog⁡nc\rho\log n, matching the upper bound up to constants.
  • Improved control via polynomial interpolation: The key lemma bounds average degree by $2d$. Is the factor 2 necessary? Can sharper interpolation or VC-dimension-based arguments reduce this constant, thereby improving the growth factor and the final cover-size bound?
  • Alternative combinatorial approaches: The proof relies on a diagonal polynomial system on the hypercube. Find a purely combinatorial argument (e.g., refined rainbow-walk methods) that might extend to odd kk or yield better constants.
  • Structure and “unpaired girth” of Kikuchi graphs: Develop a general theory relating unpaired girth to degree expansion in colored Kikuchi graphs. Provide universal Moore-type bounds for unpaired girth versus minimum/average degree, applicable beyond this specific setting.
  • Choice of level and graph model: Only the level-ℓ\ell Kikuchi graph with ℓ=max⁡{k/2,⌈ρ⌉}\ell=\max\{k/2,\lceil\rho\rceil\} is used. Explore whether different choices of ℓ\ell, multi-level constructions, or alternative colored graph models yield stronger results or handle odd kk.
  • Applications without extra logs: Many downstream applications (e.g., CSP refutation, LDC/LCC lower bounds) still have extra logarithmic factors. Systematically port the interpolation-based expansion control to these settings to remove the residual logs.
  • Average-case versus worst-case: The paper proves a worst-case existence bound. Analyze the typical size of the smallest even cover in random kk-uniform hypergraphs at the threshold density to sharpen constants and quantify fluctuations.
  • Multihypergraphs and repeated edges: The argument assumes simple hypergraphs. Determine whether the bound extends (or how it adapts) to multihypergraphs with repeated edges or weighted settings.
  • Finite-size and parameter-range coverage: Some steps impose side conditions (e.g., ℓ≤n/8\ell\le n/8, k≤n/4k\le n/4) to simplify bounds. Provide fully explicit finite-nn statements with all parameter ranges covered and quantified error terms.
View Paper Prompt  View All Prompts 

Practical Applications

Immediate Applications

Below are concrete, near-term uses that leverage the paper’s results or proof techniques without requiring major theoretical breakthroughs. Each item notes sectors, possible tools/workflows, and key assumptions/dependencies.

  • LDPC code analysis and parameter screening (communications, storage)
    • Use case: Tighter, provable upper bounds on minimum-distance/rate trade-offs for LDPC-like ensembles with even check degrees, informing design choices in 5G/6G and storage systems.
    • Tools/workflows: “Moore-bound checker” that, given (n,m,k,ρ)(n,m,k,\rho) for a kk-uniform, even-degree parity structure, flags unavoidable short even covers (generalized cycles) of size O(ρlog⁡n)O(\rho\log n), guiding code parameter sweeps and decoder robustness tests.
    • Assumptions/dependencies: Applies for even k≥4k\ge 4; density must satisfy m≥ck n (n/ρ)k/2−1m \ge c_k\,n\,(n/\rho)^{k/2-1}; existential (non-constructive)—identifies inevitability, not locations; parity over F2\mathbb F_2; constants not optimized.
  • Benchmarking and expectation-setting for CSP refutation (software, AI/verification)
    • Use case: Establish theoretical thresholds at which short, polynomially verifiable unsatisfiability certificates must exist for semirandom Boolean CSPs with even-arity constraints, improving solver benchmarking and early-stopping criteria.
    • Tools/workflows: Augment solver test suites with density regimes where short certificates are guaranteed; compare solver output to the sharpened theoretical threshold (no extra log⁡n\log n factor).
    • Assumptions/dependencies: Result is existential; algorithmic generation of certificates still relies on external methods; strongest implications currently for even-arity constraints and semirandom models.
  • Systems with parity constraints (software engineering, storage systems)
    • Use case: In parity-based redundancy (e.g., erasure codes/RAID variants modeled as even-uniform hypergraphs), set realistic expectations for minimal “parity loops” and develop detection/mitigation strategies in monitoring and repair scheduling.
    • Tools/workflows: Health-check modules that warn operators when system-scale and parity density imply inevitable short parity-dependency structures; stress tests for decoder/repair policies in those regimes.
    • Assumptions/dependencies: Mapping system topology to an even-uniform hypergraph over F2\mathbb F_2; high-density regimes where the bound applies; existential, not constructive.
  • Academic teaching and methodological transfer (education, academia)
    • Use case: Incorporate the “Kikuchi graph + diagonal polynomial system” argument into advanced courses on combinatorics, coding theory, and average-case complexity; use as a bridge between VC-type polynomial techniques and hypergraph girth.
    • Tools/workflows: Classroom notebooks that implement color-palette labeling, unpaired-girth notions, and the interpolation lemma to bound edge densities on hypercube slices.
    • Assumptions/dependencies: Pedagogical adoption; performance limitations for large nn due to combinatorial explosion.
  • Baseline strengthening for average-case complexity experiments (academia, software)
    • Use case: Use the tight Moore bound (no extra log⁡n\log n) as a sharper “ground truth” when calibrating empirical studies of refutation thresholds and planted-signal detection that rely on Kikuchi-inspired constructions.
    • Tools/workflows: Experimental dashboards that overlay empirical transition curves with the new theoretical threshold curve for even-arity instances.
    • Assumptions/dependencies: Best suited to even-arity settings; depends on correct mapping from problem instance to the hypergraph model.

Long-Term Applications

The items below require additional algorithmic advances, generalizations (e.g., to odd uniformities), scaling strategies, or integration with existing systems before deployment.

  • Algorithms to construct small even covers at optimal densities (software, AI safety, verification)
    • Use case: Practical algorithms inspired by palette-labeled walks in Kikuchi graphs and diagonal polynomial systems to find the guaranteed small even covers, enabling unsat certificates at optimal densities.
    • Potential products: “EvenCoverFinder” library; SAT/SMT/CSP solver plugins that target unpaired cycles using implicit walk strategies and parity-aware pruning.
    • Dependencies: Efficient representation/implicit traversal of Kikuchi graphs (size N=(nℓ)N=\binom{n}{\ell} can be massive); heuristics needed to approximate unpaired girth; need to convert existential proof to constructive procedures.
  • LDPC/MDPC code design and decoder robustness (communications, cybersecurity)
    • Use case: Parameter regimes and construction heuristics that minimize harmful short even covers’ impact (e.g., trapping/stopping sets), along with decoders robust to their inevitability.
    • Potential products: Design tools that combine the bound with structural checks to mitigate error floors; in PQC (e.g., McEliece variants with LDPC/MDPC), parameter auditing for structural weaknesses.
    • Dependencies: Mapping between even-cover structures and practical decoder failure modes; even-arity checks; careful cryptanalytic validation to avoid over- or under-estimating risk.
  • Removing logarithmic slack in LDC/LCC lower bounds and related areas (academia, cloud storage/databases)
    • Use case: Transport the “no-extra-log” argument to results in locally decodable/correctable codes and design theory that currently retain extra log factors via Kikuchi methods.
    • Potential products: Sharper lower bounds for odd/even query LDC/LCCs; guidance for data-retrieval protocols and PIR schemes.
    • Dependencies: Adapting the diagonal polynomial system approach to existing LDC/LCC frameworks; extending beyond even-uniform settings; tightening constants.
  • Sharpened thresholds in planted detection and tensor PCA via Kikuchi matrices (ML, signal processing)
    • Use case: Improve theoretical phase transitions and algorithmic guarantees in problems analyzed through Kikuchi matrices (e.g., tensor PCA), potentially reducing extra log gaps.
    • Potential products: Analysis toolkits that port the interpolation-bound machinery to Kikuchi matrix spectra and walk-count controls; improved detection thresholds.
    • Dependencies: Extending the combinatorial-to-spectral translation; aligning walk counting with operator moments without introducing polylog slack.
  • General-purpose hypercube subgraph analytics (software, network science)
    • Use case: Libraries to bound edge density/expansion of subgraphs embedded in hypercube slices using low-degree diagonal polynomial systems; applications to feature selection (VC-dimension links), network design, and combinatorial constructions.
    • Potential products: “HyCube-Interp” toolkit implementing the polynomial interpolation lemma to certify sparsity/expansion constraints; integration with combinatorial optimization packages.
    • Dependencies: Efficient basis construction for low-degree polynomials on large point sets; scalability of rank computations; domain-specific embeddings into {0,1}m\{0,1\}^m.
  • Proof systems and proof complexity (theoretical CS, automated reasoning)
    • Use case: Embed low-degree polynomial interpolation constraints into proof systems (e.g., cutting planes, algebraic frameworks) to obtain tighter unsat proofs for parity-rich instances.
    • Potential products: SAT proof logics augmented with parity-walk invariants harvested from Kikuchi-style constructions; certifiable logs for regulatory or safety-critical deployments.
    • Dependencies: Designing proof rules compatible with interpolation-based invariants; ensuring polynomial-time verifiability.
  • Policy and standards guidance for parity-based infrastructures (telecom, storage, cybersecurity)
    • Use case: Inform standards that short parity dependencies are mathematically unavoidable above certain densities in even-arity regimes; recommend testing and mitigation protocols.
    • Potential products: Best-practice documents for parity-check density, certification procedures for error-control components, and procurement checklists referencing Moore-bound thresholds.
    • Dependencies: Translating existential mathematics to compliance tests; alignment with industry performance metrics; evolving standards’ consensus.
  • Mapping to real systems: erasure coding and repair loops (cloud infrastructure)
    • Use case: Predict and manage minimal “repair loops” in erasure-coded clusters by modeling parity as even-uniform hypergraphs; optimize placement to reduce correlated risk.
    • Potential products: Placement/repair planners that incorporate Moore-bound-derived risk scores; SRE dashboards that flag density-induced hazards.
    • Dependencies: Accurate hypergraph modeling of storage topologies; validation that even covers correlate with operational failure patterns; algorithmic identification of specific structures.
  • Extension to odd-uniform hypergraphs (academia, downstream applications)
    • Use case: Achieve parity with the even-kk result for odd kk, eliminating residual extra-log factors and broadening all applications above.
    • Dependencies: New combinatorial or algebraic ideas; current best results for odd kk retain mild logarithmic slack or narrower regimes.

Notes on assumptions and dependencies common across applications

  • Even-uniformity: The main theorem applies to even k≥4k\ge 4; odd kk remains more challenging.
  • Density regime: Requires m≥ck n (n/ρ)k/2−1m \ge c_k\,n\,(n/\rho)^{k/2-1} (at the conjectured scale), with constants currently not optimized.
  • Field and model: Parity over F2\mathbb F_2; hypergraph uniformity and simplicity; results are primarily existential.
  • Constructivity: The proof guarantees existence of small even covers but does not supply a polynomial-time algorithm to find them; algorithmic applications require additional development.
  • Scalability: Kikuchi graph sizes N=(nℓ)N=\binom{n}{\ell} grow rapidly; practical tools will need implicit representations, sampling, or heuristics.
View Paper Prompt  View All Prompts 

Glossary

  • Average-case complexity: Study of algorithmic performance on average over a distribution of inputs rather than worst case. Example: "applications in average-case complexity"
  • Average degree: The mean number of edges incident to a vertex in a graph. Example: "given its average degree"
  • Ball (graph distance): Set of vertices within a specified distance from a given vertex. Example: "the ball of radius jj"
  • Boolean hypercube graph: Graph on binary vectors where edges connect vectors at Hamming distance 1. Example: "the standard Boolean hypercube graph"
  • Colored walk: A walk in an edge-colored graph that records traversed edge colors. Example: "colored walks in a Kikuchi graph"
  • Constraint satisfaction problems (CSP): Problems defined by constraints on variables whose assignments must satisfy all constraints. Example: "constraint satisfaction problems"
  • Diagonal polynomial system: A family of polynomials that evaluate to indicator functions on a finite set. Example: "a diagonal polynomial system over AA"
  • Double-counting calculation: A combinatorial technique counting the same quantity in two ways to derive identities or bounds. Example: "double-counting calculation"
  • Edge-colored graph: A graph whose edges carry labels/colors from a set. Example: "edge-colored by the set HH"
  • Even cover: A set of hyperedges in which each vertex is covered an even number of times. Example: "an even cover is a nonempty set of hyperedges such that every vertex belongs to an even number of hyperedges in that set."
  • Falling factorial: The product a(a−1)⋯(a−b+1)a(a-1)\cdots(a-b+1) for integers a,ba,b. Example: "the falling factorial"
  • Finite field F2\mathbf{F}_2: The field with two elements, supporting addition and multiplication modulo 2. Example: "over F2\mathbb F_2"
  • Girth (graph): Length of a shortest cycle in a graph. Example: "girth of a graph"
  • Graph Laplacian: Matrix/operator encoding graph structure via degrees and adjacencies, used in spectral methods. Example: "graph Laplacians"
  • Hamming distance: Number of coordinates on which two binary vectors differ. Example: "the Hamming distance on the hypercube"
  • Indicator vector: A binary vector marking membership of elements in a set. Example: "indicator vector 1E∈F2n\mathbf 1_E\in F_2^n"
  • Kikuchi graph: A graph constructed from a hypergraph where vertices are subsets of fixed size and edges reflect symmetric differences equal to hyperedges. Example: "the level-ℓ\ell Kikuchi graph"
  • Kikuchi method: A technique using Kikuchi graphs/matrices to analyze combinatorial structures and algorithms. Example: "the ``Kikuchi method''"
  • kk-sparse linear equations: Linear equations where each equation involves at most kk variables. Example: "kk-sparse linear equations over F2\mathbb F_2"
  • kk-uniform hypergraph: A hypergraph where every hyperedge has exactly kk vertices. Example: "kk-uniform hypergraph"
  • Low-density parity-check (LDPC) codes: Error-correcting codes defined by sparse parity-check matrices. Example: "low-density parity-check (LDPC) error-correcting codes"
  • Matching (graph): A set of pairwise non-adjacent edges; no two share a vertex. Example: "form a matching"
  • Minimum degree: The smallest degree of any vertex in a graph. Example: "minimum degree at least d/2d/2"
  • Moore bound: A bound relating the size or girth of a graph to its degree. Example: "The classical Moore bound"
  • One-inclusion graph: A graph relating set systems where edges connect sets differing by one element. Example: "the one-inclusion graph"
  • Polynomial interpolation: Using low-degree polynomials to represent or recover functions on finite sets. Example: "polynomial interpolation method."
  • Rainbow walk: A walk in an edge-colored graph where edge colors satisfy certain diversity constraints (e.g., all distinct). Example: "rainbow walks"
  • Rate–distance tradeoff: Relationship between code rate and minimum distance in coding theory. Example: "rate-distance tradeoff"
  • Semirandom instances: Instances generated by a mix of random processes and adversarial choices. Example: "semirandom instances of constraint satisfaction problems"
  • Slice of the hypercube: The subset of binary vectors with a fixed Hamming weight. Example: "a slice of the hypercube of fixed weight ℓ\ell"
  • Symmetric difference: The set of elements in exactly one of two sets; here used to define edges. Example: "are adjacent if and only if SΔT∈HS \Delta T \in H"
  • Tracial moments: Moments defined via the trace of powers of operators/matrices, often encoding combinatorial counts. Example: "tracial moments of these operators"
  • Unpaired girth: Length of the shortest cycle in an edge-colored graph where some color appears an odd number of times. Example: "the unpaired girth of K⋆K^\star"
  • Vapnik–Chervonenkis (VC) dimension: A measure of the capacity/complexity of a set system or hypothesis class. Example: "the Vapnik-Chervonenkis (VC) dimension"
View Paper Prompt  View All Prompts 

Open Problems

We haven't generated a list of open problems mentioned in this paper yet.

Generate Now

Tweets

Sign up for free to view the 1 tweet with 83 likes about this paper.

Sign Up for Free