feat: add FeasibleBasisExtension model (#530) (#818)

* feat: add FeasibleBasisExtension model (closes #530)

Add the Feasible Basis Extension problem: given m x n integer matrix A,
vector a_bar, and required columns S, determine if a feasible basis
extending S exists. Uses exact rational arithmetic (Bareiss + rational
back-substitution) to avoid floating-point issues.

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>

* Fix paper citation, Bareiss overflow, and solve command

- Add missing Murty1972 BibTeX entry (fixes make paper failure)
- Correct paper: "3-SAT" → "Hamiltonian Circuit", @murty1980 → @Murty1972
- Widen Bareiss elimination to i128 to prevent overflow on large coefficients
- Fix pred solve command to use --solver brute-force (no ILP path exists)

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>

---------

Co-authored-by: Claude Opus 4.6 (1M context) <noreply@anthropic.com>

Author: isPANN

docs/paper/reductions.typ

@@ -154,6 +154,7 @@
   "ConsecutiveBlockMinimization": [Consecutive Block Minimization],
   "ConsecutiveOnesMatrixAugmentation": [Consecutive Ones Matrix Augmentation],
   "ConsecutiveOnesSubmatrix": [Consecutive Ones Submatrix],
+  "FeasibleBasisExtension": [Feasible Basis Extension],
   "SparseMatrixCompression": [Sparse Matrix Compression],
   "DirectedTwoCommodityIntegralFlow": [Directed Two-Commodity Integral Flow],
   "IntegralFlowHomologousArcs": [Integral Flow with Homologous Arcs],
@@ -6607,6 +6608,101 @@ A classical NP-complete problem from Garey and Johnson @garey1979[Ch.~3, p.~76],
   ]
 }
 
+#{
+  let x = load-model-example("FeasibleBasisExtension")
+  let A = x.instance.matrix
+  let m = A.len()
+  let n = A.at(0).len()
+  let rhs = x.instance.rhs
+  let S = x.instance.required_columns
+  let cfg = x.optimal_config
+  // Free column indices (those not in S)
+  let free-cols = range(n).filter(j => j not in S)
+  // Selected free columns from config
+  let selected = cfg.enumerate().filter(((i, v)) => v == 1).map(((i, v)) => free-cols.at(i))
+  // Full basis: required + selected
+  let basis = S + selected
+  [
+    #problem-def("FeasibleBasisExtension")[
+      Given an $m times n$ integer matrix $A$ with $m < n$, a column vector $overline(a) in bb(Z)^m$, and a subset $S$ of column indices with $|S| < m$, determine whether there exists a _feasible basis_ $B$ --- a set of $m$ column indices including $S$ --- such that the $m times m$ submatrix $A_B$ is nonsingular and $A_B^(-1) overline(a) >= 0$ (componentwise).
+    ][
+      The Feasible Basis Extension problem arises in linear programming theory and the study of simplex method pivoting rules. It was shown NP-complete by Murty @Murty1972 via a reduction from Hamiltonian Circuit, establishing that determining whether a partial basis can be extended to a feasible one is computationally intractable in general. The problem is closely related to the question of whether a given linear program has a feasible basic solution containing specified variables. The best known exact algorithm is brute-force enumeration of all $binom(n - |S|, m - |S|)$ candidate extensions, testing each for nonsingularity and non-negativity of the solution in $O(m^3)$ time.#footnote[No algorithm improving on brute-force enumeration is known for the general Feasible Basis Extension problem.]
+
+      *Example.* Consider the $#m times #n$ matrix $A = mat(#A.map(row => row.map(v => str(v)).join(", ")).join("; "))$ with $overline(a) = (#rhs.map(str).join(", "))^top$ and required columns $S = \{#S.map(str).join(", ")\}$. We need $#(m - S.len())$ additional column from the free set $\{#free-cols.map(str).join(", ")\}$. Selecting column #selected.at(0) gives basis $B = \{#basis.map(str).join(", ")\}$, which yields $A_B^(-1) overline(a) = (4, 5, 3)^top >= 0$. Column 4 makes $A_B$ singular, and column 5 produces a negative component.
+
+      #pred-commands(
+        "pred create --example " + problem-spec(x) + " -o feasible-basis-extension.json",
+        "pred solve feasible-basis-extension.json --solver brute-force",
+        "pred evaluate feasible-basis-extension.json --config " + x.optimal_config.map(str).join(","),
+      )
+
+      #figure(
+        canvas(length: 0.7cm, {
+          import draw: *
+          let cell-size = 0.9
+          let gap = 0.15
+          // Draw the matrix
+          for i in range(m) {
+            for j in range(n) {
+              let val = A.at(i).at(j)
+              let is-basis = j in basis
+              let is-required = j in S
+              let f = if is-required {
+                graph-colors.at(1).transparentize(50%)
+              } else if is-basis {
+                graph-colors.at(0).transparentize(30%)
+              } else {
+                white
+              }
+              rect(
+                (j * cell-size, -i * cell-size),
+                (j * cell-size + cell-size - gap, -i * cell-size - cell-size + gap),
+                fill: f,
+                stroke: 0.3pt + luma(180),
+              )
+              content(
+                (j * cell-size + (cell-size - gap) / 2, -i * cell-size - (cell-size - gap) / 2),
+                text(8pt, str(val)),
+              )
+            }
+          }
+          // Column labels
+          for j in range(n) {
+            let label-color = if j in S { graph-colors.at(1) } else if j in basis { graph-colors.at(0) } else { black }
+            content(
+              (j * cell-size + (cell-size - gap) / 2, 0.4),
+              text(7pt, fill: label-color)[$c_#j$],
+            )
+          }
+          // Row labels
+          for i in range(m) {
+            content(
+              (-0.5, -i * cell-size - (cell-size - gap) / 2),
+              text(7pt)[$r_#(i + 1)$],
+            )
+          }
+          // RHS vector
+          let rhs-x = n * cell-size + 0.6
+          content((rhs-x + (cell-size - gap) / 2, 0.4), text(7pt, weight: "bold")[$overline(a)$])
+          for i in range(m) {
+            rect(
+              (rhs-x, -i * cell-size),
+              (rhs-x + cell-size - gap, -i * cell-size - cell-size + gap),
+              fill: luma(240),
+              stroke: 0.3pt + luma(180),
+            )
+            content(
+              (rhs-x + (cell-size - gap) / 2, -i * cell-size - (cell-size - gap) / 2),
+              text(8pt, str(rhs.at(i))),
+            )
+          }
+        }),
+        caption: [Feasible Basis Extension instance ($#m times #n$). Orange columns are required ($S = \{#S.map(str).join(", ")\}$), blue column is the selected extension. Together they form a nonsingular basis with non-negative solution.],
+      ) <fig:feasible-basis-extension>
+    ]
+  ]
+}
+
 // Completeness check: warn about problem types in JSON but missing from paper
 #{
   let json-models = {

docs/paper/references.bib

@@ -1454,3 +1454,13 @@ @techreport{plaisted1976
   number      = {STAN-CS-76-583},
   year        = {1976}
 }
+
+@article{Murty1972,
+  author  = {Murty, Katta G.},
+  title   = {A fundamental problem in linear inequalities with applications to the travelling salesman problem},
+  journal = {Mathematical Programming},
+  year    = {1972},
+  volume  = {3},
+  pages   = {326--370},
+  doi     = {10.1007/BF01584550}
+}

problemreductions-cli/src/cli.rs

@@ -272,6 +272,7 @@ Flags by problem type:
   ConsecutiveOnesMatrixAugmentation --matrix (0/1), --bound
   ConsecutiveOnesSubmatrix        --matrix (0/1), --k
   SparseMatrixCompression         --matrix (0/1), --bound
+  FeasibleBasisExtension          --matrix (JSON 2D i64), --rhs, --required-columns
   SteinerTree                     --graph, --edge-weights, --terminals
   MultipleCopyFileAllocation      --graph, --usage, --storage
   AcyclicPartition                --arcs [--weights] [--arc-costs] --weight-bound --cost-bound [--num-vertices]
@@ -734,6 +735,12 @@ pub struct CreateArgs {
     /// Query attribute index for PrimeAttributeName
     #[arg(long)]
     pub query: Option<usize>,
+    /// Right-hand side vector for FeasibleBasisExtension (comma-separated, e.g., "7,5,3")
+    #[arg(long)]
+    pub rhs: Option<String>,
+    /// Required column indices for FeasibleBasisExtension (comma-separated, e.g., "0,1")
+    #[arg(long)]
+    pub required_columns: Option<String>,
     /// Number of groups for SumOfSquaresPartition
     #[arg(long)]
     pub num_groups: Option<usize>,

problemreductions-cli/src/commands/create.rs

@@ -9,7 +9,7 @@ use anyhow::{bail, Context, Result};
 use problemreductions::export::{ModelExample, ProblemRef, ProblemSide, RuleExample};
 use problemreductions::models::algebraic::{
     ClosestVectorProblem, ConsecutiveBlockMinimization, ConsecutiveOnesMatrixAugmentation,
-    ConsecutiveOnesSubmatrix, SparseMatrixCompression, BMF,
+    ConsecutiveOnesSubmatrix, FeasibleBasisExtension, SparseMatrixCompression, BMF,
 };
 use problemreductions::models::formula::Quantifier;
 use problemreductions::models::graph::{
@@ -193,6 +193,8 @@ fn all_data_flags_empty(args: &CreateArgs) -> bool {
         && args.conjuncts_spec.is_none()
         && args.deps.is_none()
         && args.query.is_none()
+        && args.rhs.is_none()
+        && args.required_columns.is_none()
 }
 
 fn emit_problem_output(output: &ProblemJsonOutput, out: &OutputConfig) -> Result<()> {
@@ -766,6 +768,9 @@ fn example_for(canonical: &str, graph_type: Option<&str>) -> &'static str {
         "StringToStringCorrection" => {
             "--source-string \"0,1,2,3,1,0\" --target-string \"0,1,3,2,1\" --bound 2"
         }
+        "FeasibleBasisExtension" => {
+            "--matrix '[[1,0,1,2,-1,0],[0,1,0,1,1,2],[0,0,1,1,0,1]]' --rhs '7,5,3' --required-columns '0,1'"
+        }
         _ => "",
     }
 }
@@ -893,6 +898,9 @@ fn help_flag_hint(
         }
         ("ConsecutiveOnesSubmatrix", "matrix") => "semicolon-separated 0/1 rows: \"1,0;0,1\"",
         ("SparseMatrixCompression", "matrix") => "semicolon-separated 0/1 rows: \"1,0;0,1\"",
+        ("FeasibleBasisExtension", "matrix") => "JSON 2D integer array: '[[1,0,1],[0,1,0]]'",
+        ("FeasibleBasisExtension", "rhs") => "comma-separated integers: \"7,5,3\"",
+        ("FeasibleBasisExtension", "required_columns") => "comma-separated column indices: \"0,1\"",
         ("TimetableDesign", "craftsman_avail") | ("TimetableDesign", "task_avail") => {
             "semicolon-separated 0/1 rows: \"1,1,0;0,1,1\""
         }
@@ -2831,6 +2839,53 @@ pub fn create(args: &CreateArgs, out: &OutputConfig) -> Result<()> {
             )
         }
 
+        // FeasibleBasisExtension
+        "FeasibleBasisExtension" => {
+            let usage = "Usage: pred create FeasibleBasisExtension --matrix '[[1,0,1],[0,1,0]]' --rhs '7,5' --required-columns '0'";
+            let matrix_str = args.matrix.as_deref().ok_or_else(|| {
+                anyhow::anyhow!(
+                    "FeasibleBasisExtension requires --matrix (JSON 2D i64 array), --rhs, and --required-columns\n\n{usage}"
+                )
+            })?;
+            let matrix: Vec<Vec<i64>> = serde_json::from_str(matrix_str).map_err(|err| {
+                anyhow::anyhow!(
+                    "FeasibleBasisExtension requires --matrix as a JSON 2D integer array (e.g., '[[1,0,1],[0,1,0]]')\n\n{usage}\n\nFailed to parse --matrix: {err}"
+                )
+            })?;
+            let rhs_str = args.rhs.as_deref().ok_or_else(|| {
+                anyhow::anyhow!(
+                    "FeasibleBasisExtension requires --rhs (comma-separated integers)\n\n{usage}"
+                )
+            })?;
+            let rhs: Vec<i64> = rhs_str
+                .split(',')
+                .map(|s| s.trim().parse::<i64>())
+                .collect::<Result<Vec<_>, _>>()
+                .map_err(|err| {
+                    anyhow::anyhow!(
+                        "Failed to parse --rhs as comma-separated integers: {err}\n\n{usage}"
+                    )
+                })?;
+            let required_str = args.required_columns.as_deref().unwrap_or("");
+            let required_columns: Vec<usize> = if required_str.is_empty() {
+                vec![]
+            } else {
+                required_str
+                    .split(',')
+                    .map(|s| s.trim().parse::<usize>())
+                    .collect::<Result<Vec<_>, _>>()
+                    .map_err(|err| {
+                        anyhow::anyhow!(
+                            "Failed to parse --required-columns as comma-separated indices: {err}\n\n{usage}"
+                        )
+                    })?
+            };
+            (
+                ser(FeasibleBasisExtension::new(matrix, rhs, required_columns))?,
+                resolved_variant.clone(),
+            )
+        }
+
         // LongestCommonSubsequence
         "LongestCommonSubsequence" => {
             let usage =
@@ -7612,6 +7667,8 @@ mod tests {
             storage: None,
             quantifiers: None,
             homologous_pairs: None,
+            rhs: None,
+            required_columns: None,
         }
     }