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feat: add QuadraticDiophantineEquations model (#543) (#812)
* feat: add QuadraticDiophantineEquations model (#543) Add decision problem: given positive integers a, b, c, determine whether there exist positive integers x, y such that ax^2 + by = c. Single variable x with y derived; complexity O(sqrt(c)) by trial. Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com> * Fix agentic review issues: rename c_val→c getter, fix paper solve command - Rename `c_val()` getter to `c()` so user-facing catalog output shows the mathematical parameter name instead of an implementation detail - Add `--solver brute-force` to paper's `pred solve` command since QuadraticDiophantineEquations has no ILP reduction path Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com> * fix: rename duplicate rhs CLI field to qde_rhs for QuadraticDiophantineEquations * fix: remove OLA→STMWCT rule file accidentally included from merge The file was brought in by a contaminated merge commit and does not belong to the QuadraticDiophantineEquations model PR. Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com> * fix: rename --qde-rhs to --coeff-c for consistency All three QDE parameters now use the coeff- prefix: --coeff-a, --coeff-b, --coeff-c. Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com> --------- Co-authored-by: Claude Opus 4.6 (1M context) <noreply@anthropic.com> Co-authored-by: zazabap <sweynan@icloud.com>
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‎docs/paper/reductions.typ‎

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@@ -178,6 +178,7 @@
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"PrecedenceConstrainedScheduling": [Precedence Constrained Scheduling],
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"PrimeAttributeName": [Prime Attribute Name],
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"QuadraticAssignment": [Quadratic Assignment],
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"QuadraticDiophantineEquations": [Quadratic Diophantine Equations],
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"QuantifiedBooleanFormulas": [Quantified Boolean Formulas (QBF)],
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"RectilinearPictureCompression": [Rectilinear Picture Compression],
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"ResourceConstrainedScheduling": [Resource Constrained Scheduling],
@@ -3257,6 +3258,53 @@ A classical NP-complete problem from Garey and Johnson @garey1979[Ch.~3, p.~76],
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]
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}
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#{
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let x = load-model-example("QuadraticDiophantineEquations")
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let a = x.instance.a
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let b = x.instance.b
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let c = x.instance.c
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let config = x.optimal_config
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let xval = config.at(0) + 1
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let yval = int((c - a * xval * xval) / b)
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// Enumerate all valid x values for the table
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let max-x = calc.floor(calc.sqrt(c / a))
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let rows = range(1, max-x + 1).map(xi => {
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let rem = c - a * xi * xi
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let feasible = rem > 0 and calc.rem(rem, b) == 0
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let yi = if feasible { int(rem / b) } else { none }
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(xi, rem, feasible, yi)
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})
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[
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#problem-def("QuadraticDiophantineEquations")[
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Given positive integers $a$, $b$, $c$, determine whether there exist positive integers $x$, $y$ such that $a x^2 + b y = c$.
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][
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Quadratic Diophantine equations of the form $a x^2 + b y = c$ form one of the simplest families of mixed-degree Diophantine problems. The variable $y$ is entirely determined by $x$ via $y = (c - a x^2) slash b$, so the decision problem reduces to checking whether any $x in {1, dots, floor(sqrt(c slash a))}$ yields a positive integer $y$. This can be done in $O(sqrt(c))$ time by trial#footnote[No algorithm improving on brute-force trial of all candidate $x$ values is known; the registered complexity `sqrt(c)` reflects this direct enumeration bound.].
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*Example.* Let $a = #a$, $b = #b$, $c = #c$. Then $x$ ranges over $1, dots, #max-x$:
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#pred-commands(
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"pred create --example QuadraticDiophantineEquations -o qde.json",
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"pred solve qde.json --solver brute-force",
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"pred evaluate qde.json --config " + config.map(str).join(","),
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)
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#align(center, table(
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columns: 4,
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align: center,
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table.header([$x$], [$c - a x^2$], [Divisible by $b$?], [$y$]),
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..rows.map(((xi, rem, ok, yi)) => (
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[$#xi$],
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[$#rem$],
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[#if ok [Yes] else [No]],
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[#if yi != none [$#yi$] else [$dash$]],
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)).flatten(),
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))
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The instance is satisfiable: $x = #xval, y = #yval$ gives $#a dot #xval^2 + #b dot #yval = #c$.
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]
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]
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}
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#{
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let x = load-model-example("ClosestVectorProblem")
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let basis = x.instance.basis

‎problemreductions-cli/src/cli.rs‎

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@@ -249,6 +249,7 @@ Flags by problem type:
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ProductionPlanning --num-periods, --demands, --capacities, --setup-costs, --production-costs, --inventory-costs, --cost-bound
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SubsetSum --sizes, --target
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ThreePartition --sizes, --bound
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QuadraticDiophantineEquations --coeff-a, --coeff-b, --coeff-c
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SumOfSquaresPartition --sizes, --num-groups
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ExpectedRetrievalCost --probabilities, --num-sectors
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PaintShop --sequence
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/// Target string for StringToStringCorrection (comma-separated symbol indices, e.g., "0,1,3,2")
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#[arg(long)]
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pub target_string: Option<String>,
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/// Coefficient a for QuadraticDiophantineEquations (coefficient of x²)
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#[arg(long)]
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pub coeff_a: Option<u64>,
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/// Coefficient b for QuadraticDiophantineEquations (coefficient of y)
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#[arg(long)]
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pub coeff_b: Option<u64>,
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/// Constant c for QuadraticDiophantineEquations (right-hand side of ax² + by = c)
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#[arg(long)]
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pub coeff_c: Option<u64>,
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}
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#[derive(clap::Args)]

‎problemreductions-cli/src/commands/create.rs‎

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@@ -9,7 +9,8 @@ use anyhow::{bail, Context, Result};
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use problemreductions::export::{ModelExample, ProblemRef, ProblemSide, RuleExample};
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use problemreductions::models::algebraic::{
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ClosestVectorProblem, ConsecutiveBlockMinimization, ConsecutiveOnesMatrixAugmentation,
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ConsecutiveOnesSubmatrix, FeasibleBasisExtension, SparseMatrixCompression, BMF,
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ConsecutiveOnesSubmatrix, FeasibleBasisExtension, QuadraticDiophantineEquations,
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SparseMatrixCompression, BMF,
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};
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use problemreductions::models::formula::Quantifier;
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use problemreductions::models::graph::{
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&& args.conjuncts_spec.is_none()
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&& args.deps.is_none()
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&& args.query.is_none()
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&& args.coeff_a.is_none()
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&& args.coeff_b.is_none()
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&& args.rhs.is_none()
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&& args.coeff_c.is_none()
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&& args.required_columns.is_none()
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}
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"IntegerKnapsack" => "--sizes 3,4,5,2,7 --values 4,5,7,3,9 --capacity 15",
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"SubsetSum" => "--sizes 3,7,1,8,2,4 --target 11",
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"ThreePartition" => "--sizes 4,5,6,4,6,5 --bound 15",
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"QuadraticDiophantineEquations" => "--coeff-a 3 --coeff-b 5 --coeff-c 53",
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"BoyceCoddNormalFormViolation" => {
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"--n 6 --sets \"0,1:2;2:3;3,4:5\" --target 0,1,2,3,4,5"
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}
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)
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}
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// QuadraticDiophantineEquations
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"QuadraticDiophantineEquations" => {
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let a = args.coeff_a.ok_or_else(|| {
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anyhow::anyhow!(
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"QuadraticDiophantineEquations requires --coeff-a, --coeff-b, and --coeff-c\n\n\
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Usage: pred create QuadraticDiophantineEquations --coeff-a 3 --coeff-b 5 --coeff-c 53"
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)
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})?;
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let b = args.coeff_b.ok_or_else(|| {
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anyhow::anyhow!(
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"QuadraticDiophantineEquations requires --coeff-b\n\n\
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Usage: pred create QuadraticDiophantineEquations --coeff-a 3 --coeff-b 5 --coeff-c 53"
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)
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})?;
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let c = args.coeff_c.ok_or_else(|| {
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anyhow::anyhow!(
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"QuadraticDiophantineEquations requires --coeff-c\n\n\
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Usage: pred create QuadraticDiophantineEquations --coeff-a 3 --coeff-b 5 --coeff-c 53"
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)
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})?;
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(
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ser(QuadraticDiophantineEquations::try_new(a, b, c).map_err(anyhow::Error::msg)?)?,
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resolved_variant.clone(),
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)
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}
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// SumOfSquaresPartition
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"SumOfSquaresPartition" => {
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let sizes_str = args.sizes.as_deref().ok_or_else(|| {
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storage: None,
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quantifiers: None,
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homologous_pairs: None,
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coeff_a: None,
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coeff_b: None,
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rhs: None,
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coeff_c: None,
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required_columns: None,
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}
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}

‎src/models/algebraic/mod.rs‎

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//! - [`ConsecutiveBlockMinimization`]: Consecutive Block Minimization
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//! - [`ConsecutiveOnesSubmatrix`]: Consecutive Ones Submatrix (column selection with C1P)
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//! - [`QuadraticAssignment`]: Quadratic Assignment Problem
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//! - [`QuadraticDiophantineEquations`]: Decide ax² + by = c in positive integers
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//! - [`SparseMatrixCompression`]: Sparse Matrix Compression by row overlay
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pub(crate) mod bmf;
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pub(crate) mod feasible_basis_extension;
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pub(crate) mod ilp;
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pub(crate) mod quadratic_assignment;
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pub(crate) mod quadratic_diophantine_equations;
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pub(crate) mod qubo;
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pub(crate) mod sparse_matrix_compression;
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pub use feasible_basis_extension::FeasibleBasisExtension;
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pub use ilp::{Comparison, LinearConstraint, ObjectiveSense, VariableDomain, ILP};
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pub use quadratic_assignment::QuadraticAssignment;
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pub use quadratic_diophantine_equations::QuadraticDiophantineEquations;
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pub use qubo::QUBO;
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pub use sparse_matrix_compression::SparseMatrixCompression;
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specs.extend(consecutive_ones_submatrix::canonical_model_example_specs());
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specs.extend(feasible_basis_extension::canonical_model_example_specs());
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specs.extend(quadratic_assignment::canonical_model_example_specs());
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specs.extend(quadratic_diophantine_equations::canonical_model_example_specs());
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specs.extend(sparse_matrix_compression::canonical_model_example_specs());
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specs
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}

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