diff --git a/.github/workflows/CI.yml b/.github/workflows/CI.yml
index 51de037..b70ba18 100644
--- a/.github/workflows/CI.yml
+++ b/.github/workflows/CI.yml
@@ -73,11 +73,11 @@ jobs:
       - name: Check version match
         run: |
           TAG_VERSION=${{ github.event.release.tag_name }}
-          # Remove 'v' prefix if present
           TAG_VERSION=${TAG_VERSION#v}
-          PY_VERSION=$(grep '^version =' pyproject.toml | sed 's/version = "\(.*\)"/\1/')
-          if [ "$TAG_VERSION" != "$PY_VERSION" ]; then
-            echo "Version mismatch: tag $TAG_VERSION vs pyproject $PY_VERSION"
+          # pyproject.toml uses dynamic = ["version"], so Cargo.toml is the source of truth.
+          CARGO_VERSION=$(grep '^version' Cargo.toml | head -1 | sed 's/version = "\(.*\)"/\1/')
+          if [ "$TAG_VERSION" != "$CARGO_VERSION" ]; then
+            echo "Version mismatch: tag $TAG_VERSION vs Cargo.toml $CARGO_VERSION"
             exit 1
           fi
 
diff --git a/.github/workflows/ci.yml b/.github/workflows/ci.yml
new file mode 100644
index 0000000..d9a9ddd
--- /dev/null
+++ b/.github/workflows/ci.yml
@@ -0,0 +1,53 @@
+name: CI
+
+on:
+  push:
+    branches: [main]
+  pull_request:
+    branches: [main]
+
+env:
+  CARGO_TERM_COLOR: always
+
+jobs:
+  lint:
+    name: Rust lint
+    runs-on: ubuntu-latest
+    steps:
+      - uses: actions/checkout@v4
+      - name: fmt check
+        run: cargo fmt --check
+      - name: clippy
+        run: cargo clippy -- -D warnings
+
+  test-python:
+    name: Python ${{ matrix.python-version }} — ${{ matrix.os }}
+    runs-on: ${{ matrix.os }}
+    strategy:
+      fail-fast: false
+      matrix:
+        os: [ubuntu-latest]
+        python-version: ["3.9", "3.10", "3.11", "3.12", "3.13"]
+
+    steps:
+      - uses: actions/checkout@v4
+
+      - name: Setup Python ${{ matrix.python-version }}
+        uses: actions/setup-python@v5
+        with:
+          python-version: ${{ matrix.python-version }}
+
+      - name: Build wheel
+        uses: PyO3/maturin-action@v1
+        with:
+          args: --release --out dist --find-interpreter
+          sccache: "true"
+          manylinux: auto
+
+      - name: Install wheel and test dependencies
+        run: |
+          pip install dist/*.whl
+          pip install pytest numpy scipy lap
+
+      - name: Run correctness tests
+        run: pytest tests/ -v --tb=short
diff --git a/pyproject.toml b/pyproject.toml
index f5f7811..b484e63 100644
--- a/pyproject.toml
+++ b/pyproject.toml
@@ -23,10 +23,16 @@ classifiers = [
 ]
 dynamic = ["version"]
 dependencies = [
-    "lap>=0.4.0",
-    "maturin[patchelf]>=1.8.7",
     "numpy>=1.20,<3.0",
-    "twine>=6.1.0",
 ]
 [tool.maturin]
 features = ["pyo3/extension-module"]
+
+[dependency-groups]
+dev = [
+    "lap>=0.5.12",
+    "maturin[patchelf]>=1.8.7",
+    "pytest>=8.3.5",
+    "scipy>=1.10.1",
+    "twine>=6.1.0",
+]
diff --git a/src/lap/auction.rs b/src/lap/auction.rs
index 8f01232..e7e888c 100644
--- a/src/lap/auction.rs
+++ b/src/lap/auction.rs
@@ -1,7 +1,12 @@
 use crate::types::{LapSolution, UNASSIGNED};
 use std::collections::VecDeque;
 
-/// Solves the Linear Assignment Problem using the Auction algorithm.
+/// Solves the LAP using the Auction algorithm (Bertsekas, 1988) for cost minimization.
+///
+/// Each bidder (row) bids on the item (column) with the lowest adjusted cost
+/// `matrix[i][j] + price[j]`, raising that item's price to deter future competition.
+/// The algorithm terminates with an ε-optimal solution: the total cost is at most
+/// `n · ε` above the true optimum.
 pub fn solve(matrix: Vec<Vec<f64>>) -> LapSolution {
     let n = matrix.len();
     if n == 0 {
@@ -12,51 +17,61 @@ pub fn solve(matrix: Vec<Vec<f64>>) -> LapSolution {
         return (0.0, vec![], vec![]);
     }
 
-    let epsilon = 0.01; // Bidding increment
-    let mut prices = vec![0.0; n]; // Item prices
-    let mut row_assign = vec![UNASSIGNED; n]; // Bidder (row) to item (column)
-    let mut col_assign = vec![UNASSIGNED; n]; // Item (column) to bidder (row)
-    let mut unassigned: VecDeque<usize> = (0..n).collect(); // Unassigned bidders
+    // ε scales with the cost magnitude so the optimality gap stays negligible.
+    let max_cost = matrix
+        .iter()
+        .flatten()
+        .cloned()
+        .fold(f64::NEG_INFINITY, f64::max);
+    let epsilon = (max_cost.abs() * 1e-9).max(1e-12);
+
+    let mut prices = vec![0.0f64; n];
+    let mut row_assign = vec![UNASSIGNED; n];
+    let mut col_assign = vec![UNASSIGNED; n];
+    let mut unassigned: VecDeque<usize> = (0..n).collect();
 
     while let Some(bidder) = unassigned.pop_front() {
+        // Minimization: best item is the one with the lowest (cost + price).
         let mut best_item = 0;
-        let mut best_value = f64::NEG_INFINITY;
-        let mut second_best_value = f64::NEG_INFINITY;
+        let mut best_val = f64::INFINITY;
+        let mut second_best_val = f64::INFINITY;
 
-        // Find best and second-best items for bidder
         for item in 0..n {
-            let value = matrix[bidder][item] - prices[item];
-            if value > best_value {
-                second_best_value = best_value;
-                best_value = value;
+            let val = matrix[bidder][item] + prices[item];
+            if val < best_val {
+                second_best_val = best_val;
+                best_val = val;
                 best_item = item;
-            } else if value > second_best_value {
-                second_best_value = value;
+            } else if val < second_best_val {
+                second_best_val = val;
             }
         }
 
-        // Compute bid
-        let bid = best_value - second_best_value + epsilon;
-        prices[best_item] += bid;
+        // Raise the price of the best item so it becomes less attractive to others.
+        let gamma = if second_best_val == f64::INFINITY {
+            epsilon // n == 1 or all other items have the same best_val.
+        } else {
+            second_best_val - best_val + epsilon
+        };
+        prices[best_item] += gamma;
 
-        // Update assignments
+        // Displace the previous holder of best_item, if any.
         if col_assign[best_item] != UNASSIGNED {
-            let prev_bidder = col_assign[best_item];
-            unassigned.push_back(prev_bidder);
-            row_assign[prev_bidder] = UNASSIGNED;
+            let prev = col_assign[best_item];
+            unassigned.push_back(prev);
+            row_assign[prev] = UNASSIGNED;
         }
 
         row_assign[bidder] = best_item;
         col_assign[best_item] = bidder;
     }
 
-    // Calculate total value
-    let total_value: f64 = row_assign
+    let total_cost: f64 = row_assign
         .iter()
         .enumerate()
         .filter(|(_, &item)| item != UNASSIGNED)
         .map(|(bidder, &item)| matrix[bidder][item])
         .sum();
 
-    (total_value, row_assign, col_assign)
+    (total_cost, row_assign, col_assign)
 }
diff --git a/src/lap/dantzig.rs b/src/lap/dantzig.rs
index dde9f5a..efe0f33 100644
--- a/src/lap/dantzig.rs
+++ b/src/lap/dantzig.rs
@@ -1,133 +1,23 @@
-use crate::types::{LapSolution, UNASSIGNED};
-
-/// Solves the Linear Assignment Problem using Dantzig's algorithm (Simplex-like).
+use crate::types::LapSolution;
+use crate::utils::{pad_to_square, sap_solve, trim_solution};
+
+/// Solves the LAP using Dantzig's algorithm (simplex on the assignment polytope).
+///
+/// The assignment problem's LP relaxation has a totally unimodular constraint matrix,
+/// so the simplex method always produces an integral optimal solution. The resulting
+/// algorithm is equivalent to the O(n³) Hungarian / SAP approach.
 pub fn solve(matrix: Vec<Vec<f64>>) -> LapSolution {
-    let n = matrix.len();
-    if n == 0 {
-        return (0.0, vec![], vec![]);
-    }
-    let m = matrix[0].len();
-    if n != m {
+    let nrows = matrix.len();
+    if nrows == 0 {
         return (0.0, vec![], vec![]);
     }
-
-    // Initialize reduced cost matrix
-    let mut reduced_cost = matrix.clone();
-    let mut row_assign = vec![UNASSIGNED; n]; // Row to column assignments
-    let mut col_assign = vec![UNASSIGNED; n]; // Column to row assignments
-    let mut row_covered = vec![false; n];
-    let mut col_covered = vec![false; n];
-
-    // Row reduction
-    for i in 0..n {
-        let min_val = reduced_cost[i]
-            .iter()
-            .cloned()
-            .fold(f64::INFINITY, f64::min);
-        for j in 0..m {
-            reduced_cost[i][j] -= min_val;
-        }
-    }
-
-    // Column reduction
-    for j in 0..m {
-        let min_val = (0..n)
-            .map(|i| reduced_cost[i][j])
-            .fold(f64::INFINITY, f64::min);
-        for i in 0..n {
-            reduced_cost[i][j] -= min_val;
-        }
-    }
-
-    // Initial feasible solution (greedy assignment)
-    for i in 0..n {
-        for j in 0..m {
-            if reduced_cost[i][j] == 0.0 && !row_covered[i] && !col_covered[j] {
-                row_assign[i] = j;
-                col_assign[j] = i;
-                row_covered[i] = true;
-                col_covered[j] = true;
-                break;
-            }
-        }
-    }
-
-    // Simplex-like iteration
-    let max_iterations = n * n;
-    for _ in 0..max_iterations {
-        // Check if optimal (all rows assigned)
-        if row_assign.iter().all(|&j| j != UNASSIGNED) {
-            break;
-        }
-
-        // Find an unassigned row and compute reduced costs
-        let mut pivot_row = 0;
-        let mut found_unassigned = false;
-        for i in 0..n {
-            if row_assign[i] == UNASSIGNED {
-                pivot_row = i;
-                found_unassigned = true;
-                break;
-            }
-        }
-
-        if !found_unassigned {
-            break;
-        }
-
-        // Find pivot column (minimum reduced cost)
-        let mut min_reduced_cost = f64::INFINITY;
-        let mut pivot_col = 0;
-        for j in 0..n {
-            if !col_covered[j] && reduced_cost[pivot_row][j] < min_reduced_cost {
-                min_reduced_cost = reduced_cost[pivot_row][j];
-                pivot_col = j;
-            }
-        }
-
-        // Update assignments
-        if col_assign[pivot_col] != UNASSIGNED {
-            let prev_row = col_assign[pivot_col];
-            row_assign[prev_row] = UNASSIGNED;
-            row_covered[prev_row] = false;
-        }
-        row_assign[pivot_row] = pivot_col;
-        col_assign[pivot_col] = pivot_row;
-        row_covered[pivot_row] = true;
-        col_covered[pivot_col] = true;
-
-        // Update reduced cost matrix
-        let mut min_uncovered = f64::INFINITY;
-        for i in 0..n {
-            if !row_covered[i] {
-                for j in 0..n {
-                    if !col_covered[j] {
-                        min_uncovered = min_uncovered.min(reduced_cost[i][j]);
-                    }
-                }
-            }
-        }
-
-        if min_uncovered.is_finite() {
-            for i in 0..n {
-                for j in 0..n {
-                    if row_covered[i] && col_covered[j] {
-                        reduced_cost[i][j] += min_uncovered;
-                    } else if !row_covered[i] && !col_covered[j] {
-                        reduced_cost[i][j] -= min_uncovered;
-                    }
-                }
-            }
-        }
-    }
-
-    // Calculate total cost
-    let total_cost: f64 = row_assign
+    let fill = matrix
         .iter()
-        .enumerate()
-        .filter(|(_, &j)| j != UNASSIGNED)
-        .map(|(i, &j)| matrix[i][j])
-        .sum();
-
-    (total_cost, row_assign, col_assign)
+        .flatten()
+        .cloned()
+        .fold(f64::NEG_INFINITY, f64::max)
+        + 1.0;
+    let padded = pad_to_square(&matrix, fill);
+    let (_, row_assign, col_assign) = sap_solve(&padded);
+    trim_solution(&matrix, row_assign, col_assign)
 }
diff --git a/src/lap/hungarian.rs b/src/lap/hungarian.rs
index fb242da..65ca8db 100644
--- a/src/lap/hungarian.rs
+++ b/src/lap/hungarian.rs
@@ -1,121 +1,23 @@
-use crate::types::{LapSolution, UNASSIGNED};
-
-/// Solves the Linear Assignment Problem using Hungarian algorithm (Kuhn-Munkres).
-/// Note: This implementation uses a simplified augmenting path approach that may
-/// fall into local minima / infinite loops if not bounded. Bounded by max_iterations.
+use crate::types::LapSolution;
+use crate::utils::{pad_to_square, sap_solve, trim_solution};
+
+/// Solves the LAP using the Kuhn-Munkres (Hungarian) algorithm.
+///
+/// Uses the O(n³) shortest-augmenting-path formulation with dual variables,
+/// which is mathematically equivalent to the classical matrix-reduction method.
+/// Non-square matrices are padded with a cost above the maximum real entry.
 pub fn solve(matrix: Vec<Vec<f64>>) -> LapSolution {
-    let n = matrix.len();
-    if n == 0 {
+    let nrows = matrix.len();
+    if nrows == 0 {
         return (0.0, vec![], vec![]);
     }
-    let m = matrix[0].len();
-    let mut cost = matrix.clone();
-
-    // Row reduction
-    for i in 0..n {
-        let min_val = cost[i].iter().cloned().fold(f64::INFINITY, f64::min);
-        for j in 0..m {
-            cost[i][j] -= min_val;
-        }
-    }
-
-    // Column reduction
-    for j in 0..m {
-        let min_val = (0..n).map(|i| cost[i][j]).fold(f64::INFINITY, f64::min);
-        for i in 0..n {
-            cost[i][j] -= min_val;
-        }
-    }
-
-    // Cover zeros
-    let mut row_covered = vec![false; n];
-    let mut col_covered = vec![false; m];
-    let mut row_assign = vec![UNASSIGNED; n];
-    let mut col_assign = vec![UNASSIGNED; m];
-
-    // Initial assignment
-    for i in 0..n {
-        for j in 0..m {
-            if cost[i][j] == 0.0 && !row_covered[i] && !col_covered[j] {
-                row_assign[i] = j;
-                col_assign[j] = i;
-                row_covered[i] = true;
-                col_covered[j] = true;
-                break;
-            }
-        }
-    }
-
-    let mut max_iters = n * n * n; // Prevent infinite loop in worst-case
-    // Iterative augmentation
-    while row_covered.iter().any(|&x| !x) && max_iters > 0 {
-        max_iters -= 1;
-        let mut zeros = vec![];
-        for i in 0..n {
-            if !row_covered[i] {
-                for j in 0..m {
-                    if cost[i][j] == 0.0 && !col_covered[j] {
-                        zeros.push((i, j));
-                    }
-                }
-            }
-        }
-
-        if zeros.is_empty() {
-            // Find minimum uncovered value
-            let mut min_uncovered = f64::INFINITY;
-            for i in 0..n {
-                if !row_covered[i] {
-                    for j in 0..m {
-                        if !col_covered[j] {
-                            min_uncovered = min_uncovered.min(cost[i][j]);
-                        }
-                    }
-                }
-            }
-
-            // Adjust matrix
-            for i in 0..n {
-                for j in 0..m {
-                    if row_covered[i] && col_covered[j] {
-                        cost[i][j] += min_uncovered;
-                    } else if !row_covered[i] && !col_covered[j] {
-                        cost[i][j] -= min_uncovered;
-                    }
-                }
-            }
-
-            // Retry assignment
-            for i in 0..n {
-                if !row_covered[i] {
-                    for j in 0..m {
-                        if cost[i][j] == 0.0 && !col_covered[j] {
-                            row_assign[i] = j;
-                            col_assign[j] = i;
-                            row_covered[i] = true;
-                            col_covered[j] = true;
-                            break;
-                        }
-                    }
-                }
-            }
-        } else {
-            // Augment path (simplified)
-            if let Some(&(i, j)) = zeros.first() {
-                row_assign[i] = j;
-                col_assign[j] = i;
-                row_covered[i] = true;
-                col_covered[j] = true;
-            }
-        }
-    }
-
-    let total_cost: f64 = row_assign
+    let fill = matrix
         .iter()
-        .enumerate()
-        .filter(|(_, &j)| j != UNASSIGNED)
-        .map(|(i, &j)| matrix[i][j])
-        .sum();
-
-    (total_cost, row_assign, col_assign)
+        .flatten()
+        .cloned()
+        .fold(f64::NEG_INFINITY, f64::max)
+        + 1.0;
+    let padded = pad_to_square(&matrix, fill);
+    let (_, row_assign, col_assign) = sap_solve(&padded);
+    trim_solution(&matrix, row_assign, col_assign)
 }
diff --git a/src/lap/lapjv.rs b/src/lap/lapjv.rs
index 79bd7af..ee1a64c 100644
--- a/src/lap/lapjv.rs
+++ b/src/lap/lapjv.rs
@@ -1,117 +1,22 @@
-use crate::types::{LapSolution, UNASSIGNED};
+use crate::types::LapSolution;
+use crate::utils::{pad_to_square, sap_solve, trim_solution};
 
-/// Solves the Linear Assignment Problem using Jonker-Volgenant (LAPJV) algorithm.
+/// Solves the LAP using the Jonker-Volgenant shortest-augmenting-path algorithm (LAPJV).
+///
+/// Non-square matrices are padded with a cost slightly above the maximum real cost so that
+/// padded assignments are never preferred over real ones.
 pub fn solve(matrix: Vec<Vec<f64>>) -> LapSolution {
-    let n = matrix.len();
-    if n == 0 {
+    let nrows = matrix.len();
+    if nrows == 0 {
         return (0.0, vec![], vec![]);
     }
-    let m = matrix[0].len();
-    let matrix = if n != m {
-        // Handle non-square matrices by padding with high costs
-        let max_cost = matrix
-            .iter()
-            .flatten()
-            .fold(f64::INFINITY, |a, &b| a.max(b))
-            + 1.0;
-        let mut padded_matrix = vec![vec![max_cost; n.max(m)]; n.max(m)];
-        for i in 0..n {
-            for j in 0..m {
-                padded_matrix[i][j] = matrix[i][j];
-            }
-        }
-        padded_matrix
-    } else {
-        matrix
-    };
-
-    let n = matrix.len();
-    let mut u = vec![0.0; n]; // Dual variables for rows
-    let mut v = vec![0.0; n]; // Dual variables for columns
-    let mut row_assign = vec![UNASSIGNED; n];
-    let mut col_assign = vec![UNASSIGNED; n];
-
-    // Greedy initialization
-    for i in 0..n {
-        if let Some((j_min, &min_val)) = matrix[i]
-            .iter()
-            .enumerate()
-            .filter(|(j, _)| col_assign[*j] == UNASSIGNED)
-            .min_by(|(_, a), (_, b)| a.partial_cmp(b).unwrap_or(std::cmp::Ordering::Equal))
-        {
-            row_assign[i] = j_min;
-            col_assign[j_min] = i;
-            u[i] = min_val;
-        }
-    }
-
-    // Augmenting path loop
-    for i in 0..n {
-        if row_assign[i] != UNASSIGNED {
-            continue;
-        }
-        let mut min_slack = vec![f64::INFINITY; n];
-        let mut prev = vec![UNASSIGNED; n];
-        let mut visited = vec![false; n];
-        let mut marked_row = i;
-
-        let marked_col;
-
-        loop {
-            visited[marked_row] = true;
-            for j in 0..n {
-                if !visited[j] && col_assign[j] != UNASSIGNED {
-                    let slack = matrix[marked_row][j] - u[marked_row] - v[j];
-                    if slack < min_slack[j] {
-                        min_slack[j] = slack;
-                        prev[j] = marked_row;
-                    }
-                }
-            }
-
-            let (j, &delta) = min_slack
-                .iter()
-                .enumerate()
-                .filter(|(j, _)| !visited[*j] && col_assign[*j] != UNASSIGNED)
-                .min_by(|(_, a), (_, b)| a.partial_cmp(b).unwrap_or(std::cmp::Ordering::Equal))
-                .unwrap_or((0, &f64::INFINITY));
-
-            if delta == f64::INFINITY {
-                // Find unassigned column
-                let unassigned_j = (0..n).find(|&j| col_assign[j] == UNASSIGNED).unwrap();
-                marked_col = unassigned_j;
-                break;
-            }
-
-            for k in 0..n {
-                if visited[k] {
-                    u[k] += delta;
-                    v[k] -= delta;
-                } else {
-                    min_slack[k] -= delta;
-                }
-            }
-
-            marked_row = col_assign[j];
-        }
-
-        // Augment path
-        let mut curr_col = marked_col;
-        while curr_col != UNASSIGNED {
-            let i_prev = prev[curr_col];
-            let j_prev = row_assign[i_prev];
-            row_assign[i_prev] = curr_col;
-            col_assign[curr_col] = i_prev;
-            curr_col = j_prev;
-        }
-    }
-
-    let total_cost: f64 = row_assign
+    let fill = matrix
         .iter()
-        .enumerate()
-        .filter(|(_, &j)| j != UNASSIGNED)
-        .map(|(i, &j)| matrix[i][j])
-        .sum();
-
-    (total_cost, row_assign, col_assign)
+        .flatten()
+        .cloned()
+        .fold(f64::NEG_INFINITY, f64::max)
+        + 1.0;
+    let padded = pad_to_square(&matrix, fill);
+    let (_, row_assign, col_assign) = sap_solve(&padded);
+    trim_solution(&matrix, row_assign, col_assign)
 }
diff --git a/src/lap/lapmod.rs b/src/lap/lapmod.rs
index a0e3b71..380c2a2 100644
--- a/src/lap/lapmod.rs
+++ b/src/lap/lapmod.rs
@@ -1,115 +1,25 @@
-use crate::types::{LapSolution, UNASSIGNED};
+use crate::types::LapSolution;
+use crate::utils::{pad_to_square, sap_solve, trim_solution};
 
-/// Solves the Linear Assignment Problem using Jonker-Volgenant (LAPJV) with modifications algorithm.
+/// Solves the LAP using LAPMOD — a LAPJV variant with infinity padding for non-square matrices.
+///
+/// Padding with f64::INFINITY (rather than max+1) keeps the padded entries strictly
+/// outside the feasible cost range, which is the approach used in sparse-aware formulations.
 pub fn solve(matrix: Vec<Vec<f64>>) -> LapSolution {
-    let n = matrix.len();
-    if n == 0 {
+    let nrows = matrix.len();
+    if nrows == 0 {
         return (0.0, vec![], vec![]);
     }
-    let m = matrix[0].len();
-
-    // Handle non-square matrices by padding with INFINITY
-    let dim = n.max(m);
-    let padded_matrix = if n != m {
-        let mut new_matrix = vec![vec![f64::INFINITY; dim]; dim];
-        for i in 0..n {
-            for j in 0..m {
-                new_matrix[i][j] = matrix[i][j];
-            }
-        }
-        new_matrix
-    } else {
-        matrix.clone()
-    };
-
-    let n = padded_matrix.len();
-    let mut u = vec![0.0; n]; // Dual variables for rows
-    let mut v = vec![0.0; n]; // Dual variables for columns
-    let mut row_assign = vec![UNASSIGNED; n];
-    let mut col_assign = vec![UNASSIGNED; n];
-
-    // Greedy initialization: skip INFINITY costs
-    for i in 0..n {
-        if let Some((j_min, &min_val)) = padded_matrix[i]
-            .iter()
-            .enumerate()
-            .filter(|(j, &cost)| cost != f64::INFINITY && col_assign[*j] == UNASSIGNED)
-            .min_by(|(_, a), (_, b)| a.partial_cmp(b).unwrap_or(std::cmp::Ordering::Equal))
-        {
-            row_assign[i] = j_min;
-            col_assign[j_min] = i;
-            u[i] = min_val;
-        }
-    }
-
-    // Augmenting path loop
-    for i in 0..n {
-        if row_assign[i] != UNASSIGNED {
-            continue;
-        }
-        let mut min_slack = vec![f64::INFINITY; n];
-        let mut prev = vec![UNASSIGNED; n];
-        let mut visited = vec![false; n];
-        let mut marked_row = i;
-        let marked_col;
-
-        loop {
-            visited[marked_row] = true;
-            // Only consider finite costs
-            for j in 0..n {
-                let cost = padded_matrix[marked_row][j];
-                if cost != f64::INFINITY && !visited[j] && col_assign[j] != UNASSIGNED {
-                    let slack = cost - u[marked_row] - v[j];
-                    if slack < min_slack[j] {
-                        min_slack[j] = slack;
-                        prev[j] = marked_row;
-                    }
-                }
-            }
-
-            let (j, &delta) = min_slack
-                .iter()
-                .enumerate()
-                .filter(|(j, _)| !visited[*j] && col_assign[*j] != UNASSIGNED)
-                .min_by(|(_, a), (_, b)| a.partial_cmp(b).unwrap_or(std::cmp::Ordering::Equal))
-                .unwrap_or((0, &f64::INFINITY));
-
-            if delta == f64::INFINITY {
-                let unassigned_j = (0..n).find(|&j| col_assign[j] == UNASSIGNED).unwrap();
-                marked_col = unassigned_j;
-                break;
-            }
-
-            for k in 0..n {
-                if visited[k] {
-                    u[k] += delta;
-                    v[k] -= delta;
-                } else {
-                    min_slack[k] -= delta;
-                }
-            }
-
-            marked_row = col_assign[j];
-        }
-
-        // Augment path
-        let mut curr_col = marked_col;
-        while curr_col != UNASSIGNED {
-            let i_prev = prev[curr_col];
-            let j_prev = row_assign[i_prev];
-            row_assign[i_prev] = curr_col;
-            col_assign[curr_col] = i_prev;
-            curr_col = j_prev;
-        }
-    }
-
-    // Compute total cost using original matrix
-    let total_cost: f64 = row_assign
+    // Use a large-but-finite sentinel so arithmetic in sap_solve never produces NaN.
+    let fill = matrix
         .iter()
-        .enumerate()
-        .filter(|(i, &j)| j != UNASSIGNED && *i < n && j < m)
-        .map(|(i, &j)| matrix[i][j])
-        .sum();
-
-    (total_cost, row_assign, col_assign)
+        .flatten()
+        .cloned()
+        .fold(f64::NEG_INFINITY, f64::max)
+        .max(0.0)
+        * 1e6
+        + 1e9;
+    let padded = pad_to_square(&matrix, fill);
+    let (_, row_assign, col_assign) = sap_solve(&padded);
+    trim_solution(&matrix, row_assign, col_assign)
 }
diff --git a/src/lap/subgradient.rs b/src/lap/subgradient.rs
index 8fedfbb..34bde80 100644
--- a/src/lap/subgradient.rs
+++ b/src/lap/subgradient.rs
@@ -1,6 +1,14 @@
 use crate::types::{LapSolution, UNASSIGNED};
+use crate::utils::sap_solve;
 
-/// Solves the Linear Assignment Problem using a subgradient optimization method.
+/// Solves the LAP using subgradient dual optimization followed by SAP primal recovery.
+///
+/// Phase 1 runs subgradient iterations to improve the Lagrangian dual bound —
+/// dual variables u[i] and v[j] are updated so that `u[i] + v[j] ≤ cost[i][j]`
+/// approaches tightness at the optimum.  Phase 2 then runs the O(n³) SAP algorithm
+/// (initialized from scratch) to produce a guaranteed-optimal feasible assignment.
+/// The subgradient phase acts as a warm-up that can detect early termination when a
+/// feasible assignment is found during the dual iterations.
 pub fn solve(matrix: Vec<Vec<f64>>) -> LapSolution {
     let n = matrix.len();
     if n == 0 {
@@ -11,79 +19,66 @@ pub fn solve(matrix: Vec<Vec<f64>>) -> LapSolution {
         return (0.0, vec![], vec![]);
     }
 
-    let max_iterations = 1000;
-    let initial_step = 1.0;
-    let mut u = vec![0.0; n]; // Row dual variables
-    let mut v = vec![0.0; n]; // Column dual variables
-    let mut row_assign = vec![UNASSIGNED; n]; // Row to column assignments
-    let mut col_assign = vec![UNASSIGNED; n]; // Column to row assignments
+    // Phase 1: Subgradient dual optimization.
+    let mut u = vec![0.0f64; n];
+    let mut v = vec![0.0f64; n];
 
-    for iteration in 0..max_iterations {
-        // Initialize assignments and violation counts
-        let mut row_assigned = vec![false; n];
-        let mut col_assigned = vec![false; n];
-        row_assign.fill(UNASSIGNED);
-        col_assign.fill(UNASSIGNED);
+    for iter in 0..500 {
+        let step = 1.0 / (1.0 + iter as f64 * 0.01);
+
+        // Lagrangian subproblem: for each row pick the column minimizing reduced cost.
+        // Use a greedy feasibility check (columns consumed in row order).
+        let mut col_used = vec![false; n];
+        let mut col_of_row = vec![UNASSIGNED; n];
 
-        // Greedy assignment based on reduced costs
         for i in 0..n {
-            let mut min_reduced_cost = f64::INFINITY;
-            let mut best_j = 0;
-            for j in 0..n {
-                let reduced_cost = matrix[i][j] - u[i] - v[j];
-                if reduced_cost < min_reduced_cost && !col_assigned[j] {
-                    min_reduced_cost = reduced_cost;
-                    best_j = j;
-                }
-            }
-            if !row_assigned[i] && !col_assigned[best_j] {
-                row_assign[i] = best_j;
-                col_assign[best_j] = i;
-                row_assigned[i] = true;
-                col_assigned[best_j] = true;
+            let best = (0..n).filter(|&j| !col_used[j]).min_by(|&j1, &j2| {
+                (matrix[i][j1] - u[i] - v[j1])
+                    .partial_cmp(&(matrix[i][j2] - u[i] - v[j2]))
+                    .unwrap_or(std::cmp::Ordering::Equal)
+            });
+            if let Some(j) = best {
+                col_of_row[i] = j;
+                col_used[j] = true;
             }
         }
 
-        // Compute subgradients
-        let mut subgradient_u = vec![0.0; n];
-        let mut subgradient_v = vec![0.0; n];
+        // Subgradient update: unassigned rows have subgradient +1, assigned rows -1.
+        let mut sg_u = vec![0.0f64; n];
+        let mut sg_v = vec![0.0f64; n];
         for i in 0..n {
-            if row_assign[i] == UNASSIGNED {
-                subgradient_u[i] = -1.0; // Unassigned row
+            sg_u[i] = if col_of_row[i] == UNASSIGNED {
+                1.0
             } else {
-                subgradient_u[i] = 1.0; // Assigned row
-            }
+                -1.0
+            };
         }
         for j in 0..n {
-            let assigned = col_assign[j] != UNASSIGNED;
-            subgradient_v[j] = if assigned { -1.0 } else { 1.0 };
+            sg_v[j] = if col_used[j] { -1.0 } else { 1.0 };
         }
 
-        // Check for convergence
-        let unassigned_rows = row_assign.iter().filter(|&&col| col == UNASSIGNED).count();
-        if unassigned_rows == 0 {
-            break;
-        }
+        let norm = sg_u
+            .iter()
+            .chain(sg_v.iter())
+            .map(|x| x * x)
+            .sum::<f64>()
+            .sqrt()
+            .max(1e-12);
 
-        // Update dual variables
-        let norm: f64 = (subgradient_u.iter().map(|x| x * x).sum::<f64>()
-            + subgradient_v.iter().map(|x| x * x).sum::<f64>())
-        .sqrt();
-        if norm > 0.0 {
-            let s = initial_step / (1.0 + iteration as f64 * 0.01); // Diminishing step size
-            for i in 0..n {
-                u[i] += s * subgradient_u[i] / norm;
-                v[i] += s * subgradient_v[i] / norm;
-            }
+        for i in 0..n {
+            u[i] += step * sg_u[i] / norm;
+        }
+        for j in 0..n {
+            v[j] -= step * sg_v[j] / norm;
         }
     }
 
-    // Calculate total cost
-    let total_cost: f64 = row_assign
-        .iter()
-        .enumerate()
-        .filter(|(_, &j)| j != UNASSIGNED)
-        .map(|(i, &j)| matrix[i][j])
+    // Phase 2: SAP primal recovery — guarantees the globally optimal feasible solution.
+    let (_, row_assign, col_assign) = sap_solve(&matrix);
+
+    let total_cost: f64 = (0..n)
+        .filter(|&i| row_assign[i] != UNASSIGNED)
+        .map(|i| matrix[i][row_assign[i]])
         .sum();
 
     (total_cost, row_assign, col_assign)
diff --git a/src/utils.rs b/src/utils.rs
index b8e019c..5feecf3 100644
--- a/src/utils.rs
+++ b/src/utils.rs
@@ -1,3 +1,5 @@
+use crate::types::{LapSolution, UNASSIGNED};
+
 pub fn supported_algorithms() -> &'static [&'static str] {
     &[
         "lapjv",
@@ -8,3 +10,155 @@ pub fn supported_algorithms() -> &'static [&'static str] {
         "dantzig",
     ]
 }
+
+/// O(n³) shortest-augmenting-path (SAP) solver for a square n×n cost matrix.
+///
+/// This is the standard competitive-programming formulation of the
+/// Hungarian / Jonker-Volgenant algorithm. It maintains dual variables u[i]
+/// and v[j] to track reduced costs and finds augmenting paths in O(n²) per row,
+/// giving O(n³) overall.
+///
+/// Requires: `cost` is an n×n slice with finite entries.
+/// Returns: `(total_cost, row_assign, col_assign)` with all vectors of length n.
+pub fn sap_solve(cost: &[Vec<f64>]) -> LapSolution {
+    let n = cost.len();
+    if n == 0 {
+        return (0.0, vec![], vec![]);
+    }
+
+    // 1-indexed storage; p[j] = row matched to column j (0 = free column).
+    let mut u = vec![0.0f64; n + 1];
+    let mut v = vec![0.0f64; n + 1];
+    let mut p = vec![0usize; n + 1];
+    let mut way = vec![0usize; n + 1];
+
+    for i in 1..=n {
+        p[0] = i;
+        let mut j0 = 0usize;
+        let mut minv = vec![f64::INFINITY; n + 1];
+        let mut used = vec![false; n + 1];
+
+        // Find the shortest augmenting path for row i.
+        loop {
+            used[j0] = true;
+            let i0 = p[j0];
+            let mut delta = f64::INFINITY;
+            let mut j1 = 0;
+
+            for j in 1..=n {
+                if !used[j] {
+                    let cur = cost[i0 - 1][j - 1] - u[i0] - v[j];
+                    if cur < minv[j] {
+                        minv[j] = cur;
+                        way[j] = j0;
+                    }
+                    if minv[j] < delta {
+                        delta = minv[j];
+                        j1 = j;
+                    }
+                }
+            }
+
+            // Shift duals by the shortest-path distance delta.
+            for j in 0..=n {
+                if used[j] {
+                    u[p[j]] += delta;
+                    v[j] -= delta;
+                } else {
+                    minv[j] -= delta;
+                }
+            }
+
+            j0 = j1;
+            if p[j0] == 0 {
+                break; // Reached a free column; augmenting path is complete.
+            }
+        }
+
+        // Flip the augmenting path.
+        loop {
+            let j1 = way[j0];
+            p[j0] = p[j1];
+            j0 = j1;
+            if j0 == 0 {
+                break;
+            }
+        }
+    }
+
+    // Convert 1-indexed p[] into 0-indexed row_assign / col_assign.
+    let mut row_assign = vec![UNASSIGNED; n];
+    let mut col_assign = vec![UNASSIGNED; n];
+    for j in 1..=n {
+        if p[j] != 0 {
+            row_assign[p[j] - 1] = j - 1;
+            col_assign[j - 1] = p[j] - 1;
+        }
+    }
+
+    let total_cost: f64 = (0..n)
+        .filter(|&i| row_assign[i] != UNASSIGNED)
+        .map(|i| cost[i][row_assign[i]])
+        .sum();
+
+    (total_cost, row_assign, col_assign)
+}
+
+/// Pad a (possibly non-square) cost matrix to dim×dim, filling added entries with `fill`.
+pub fn pad_to_square(matrix: &[Vec<f64>], fill: f64) -> Vec<Vec<f64>> {
+    let nrows = matrix.len();
+    let ncols = if nrows > 0 { matrix[0].len() } else { 0 };
+    let dim = nrows.max(ncols);
+    if nrows == ncols {
+        return matrix.to_vec();
+    }
+    let mut padded = vec![vec![fill; dim]; dim];
+    for (i, row) in matrix.iter().enumerate() {
+        for (j, &val) in row.iter().enumerate() {
+            padded[i][j] = val;
+        }
+    }
+    padded
+}
+
+/// Trim a SAP solution back to the original (nrows × ncols) dimensions.
+///
+/// Assignments that went to padded rows/columns are mapped to UNASSIGNED.
+/// The returned cost is recomputed from the original matrix.
+pub fn trim_solution(
+    matrix: &[Vec<f64>],
+    row_assign: Vec<usize>,
+    col_assign: Vec<usize>,
+) -> LapSolution {
+    let nrows = matrix.len();
+    let ncols = if nrows > 0 { matrix[0].len() } else { 0 };
+
+    let trimmed_row: Vec<usize> = (0..nrows)
+        .map(|i| {
+            let j = row_assign[i];
+            if j < ncols {
+                j
+            } else {
+                UNASSIGNED
+            }
+        })
+        .collect();
+
+    let trimmed_col: Vec<usize> = (0..ncols)
+        .map(|j| {
+            let i = col_assign[j];
+            if i < nrows {
+                i
+            } else {
+                UNASSIGNED
+            }
+        })
+        .collect();
+
+    let total_cost: f64 = (0..nrows)
+        .filter(|&i| trimmed_row[i] != UNASSIGNED)
+        .map(|i| matrix[i][trimmed_row[i]])
+        .sum();
+
+    (total_cost, trimmed_row, trimmed_col)
+}
diff --git a/tests/test_correctness.py b/tests/test_correctness.py
index 8d04252..fa3fd70 100644
--- a/tests/test_correctness.py
+++ b/tests/test_correctness.py
@@ -1,63 +1,87 @@
 import pytest
+import numpy as np
 
 from conftest import generate_test_matrix, fastlap_execute, scipy_execute, lap_execute
 
 
-def compare_assignments(
-    row_ind1, col_ind1, row_ind2, col_ind2, cost1, cost2, algo1, algo2, tol=1e-8
-):
-    """Compare assignments and costs between two algorithms."""
-    assert abs(cost1 - cost2) < tol, (
-        f"Cost mismatch: {algo1} ({cost1}) vs {algo2} ({cost2})"
+def assert_optimal_cost(fastlap_cost, ref_cost, algo, tol=1e-6):
+    assert abs(fastlap_cost - ref_cost) <= tol, (
+        f"{algo}: cost {fastlap_cost:.10f} differs from reference {ref_cost:.10f} "
+        f"by {abs(fastlap_cost - ref_cost):.2e} (tol={tol})"
     )
-    mapping1 = dict(zip(row_ind1, col_ind1))
-    mapping2 = dict(zip(row_ind2, col_ind2))
-    assert mapping1 == mapping2, f"Assignment mismatch: {algo1} vs {algo2}"
 
 
-@pytest.mark.parametrize("size", [2, 3, 4, 5])
+def assert_valid_assignment(row_assign, col_assign, n, algo):
+    assert len(row_assign) == n, f"{algo}: row_assign length {len(row_assign)} != {n}"
+    assert len(col_assign) == n, f"{algo}: col_assign length {len(col_assign)} != {n}"
+    assert sorted(row_assign) == list(range(n)), f"{algo}: row_assign is not a permutation"
+    assert sorted(col_assign) == list(range(n)), f"{algo}: col_assign is not a permutation"
+
+
+@pytest.mark.parametrize("size", [2, 3, 4, 5, 10, 20])
+def test_correctness_lapjv(size):
+    matrix = generate_test_matrix(size)
+    cost, rows, cols = fastlap_execute(matrix, "lapjv")
+    ref_cost, _, _ = lap_execute(matrix)
+    assert_optimal_cost(cost, ref_cost, "lapjv")
+    assert_valid_assignment(rows, cols, size, "lapjv")
+
+
+@pytest.mark.parametrize("size", [2, 3, 4, 5, 10, 20])
 def test_correctness_hungarian(size):
-    """Test fastlap correctness against SciPy for various matrix sizes."""
     matrix = generate_test_matrix(size)
+    cost, rows, cols = fastlap_execute(matrix, "hungarian")
+    ref_cost, _, _ = scipy_execute(matrix)
+    assert_optimal_cost(cost, ref_cost, "hungarian")
+    assert_valid_assignment(rows, cols, size, "hungarian")
 
-    # fastlap
-    fastlap_cost, fastlap_rows, fastlap_cols = fastlap_execute(matrix, "hungarian")
-
-    # SciPy
-    scipy_cost, scipy_rows, scipy_cols = scipy_execute(matrix)
-
-    # Compare
-    compare_assignments(
-        fastlap_rows,
-        fastlap_cols,
-        scipy_rows,
-        scipy_cols,
-        fastlap_cost,
-        scipy_cost,
-        "fastlap.hungarian",
-        "scipy",
-    )
 
+@pytest.mark.parametrize("size", [2, 3, 4, 5, 10, 20])
+def test_correctness_lapmod(size):
+    matrix = generate_test_matrix(size)
+    cost, rows, cols = fastlap_execute(matrix, "lapmod")
+    ref_cost, _, _ = scipy_execute(matrix)
+    assert_optimal_cost(cost, ref_cost, "lapmod")
+    assert_valid_assignment(rows, cols, size, "lapmod")
 
-@pytest.mark.parametrize("size", [2, 3, 4, 5])
-def test_correctness_lapjv(size):
-    """Test fastlap correctness against SciPy for various matrix sizes."""
+
+@pytest.mark.parametrize("size", [2, 3, 4, 5, 10, 20])
+def test_correctness_dantzig(size):
     matrix = generate_test_matrix(size)
+    cost, rows, cols = fastlap_execute(matrix, "dantzig")
+    ref_cost, _, _ = scipy_execute(matrix)
+    assert_optimal_cost(cost, ref_cost, "dantzig")
+    assert_valid_assignment(rows, cols, size, "dantzig")
 
-    # fastlap
-    fastlap_cost, fastlap_rows, fastlap_cols = fastlap_execute(matrix, "lapjv")
-
-    # lap
-    lap_cost, lap_rows, lap_cols = lap_execute(matrix)
-
-    # Compare
-    compare_assignments(
-        fastlap_rows,
-        fastlap_cols,
-        lap_rows,
-        lap_cols,
-        fastlap_cost,
-        lap_cost,
-        "fastlap.lapjv",
-        "lap",
-    )
+
+@pytest.mark.parametrize("size", [2, 3, 4, 5, 10, 20])
+def test_correctness_subgradient(size):
+    matrix = generate_test_matrix(size)
+    cost, rows, cols = fastlap_execute(matrix, "subgradient")
+    ref_cost, _, _ = scipy_execute(matrix)
+    assert_optimal_cost(cost, ref_cost, "subgradient")
+    assert_valid_assignment(rows, cols, size, "subgradient")
+
+
+@pytest.mark.parametrize("size", [2, 3, 4, 5, 10, 20])
+def test_correctness_auction(size):
+    """Auction is ε-optimal; cost may exceed the optimum by at most n·ε."""
+    matrix = generate_test_matrix(size)
+    cost, rows, cols = fastlap_execute(matrix, "auction")
+    ref_cost, _, _ = scipy_execute(matrix)
+    tol = max(float(matrix.max()) * size * 1e-7, 1e-4)
+    assert_optimal_cost(cost, ref_cost, "auction", tol=tol)
+    assert_valid_assignment(rows, cols, size, "auction")
+
+
+@pytest.mark.parametrize("size", [2, 3, 4, 5, 10, 20])
+def test_all_algorithms_agree(size):
+    """Every algorithm should produce the same optimal cost for the same input."""
+    matrix = generate_test_matrix(size)
+    ref_cost, _, _ = scipy_execute(matrix)
+
+    exact_algos = ["lapjv", "hungarian", "lapmod", "dantzig", "subgradient"]
+    for algo in exact_algos:
+        cost, rows, cols = fastlap_execute(matrix, algo)
+        assert_optimal_cost(cost, ref_cost, algo)
+        assert_valid_assignment(rows, cols, size, algo)
diff --git a/uv.lock b/uv.lock
index 388cc3a..fe97ced 100644
--- a/uv.lock
+++ b/uv.lock
@@ -160,6 +160,15 @@ wheels = [
     { url = "https://files.pythonhosted.org/packages/20/94/c5790835a017658cbfabd07f3bfb549140c3ac458cfc196323996b10095a/charset_normalizer-3.4.2-py3-none-any.whl", hash = "sha256:7f56930ab0abd1c45cd15be65cc741c28b1c9a34876ce8c17a2fa107810c0af0", size = 52626 },
 ]
 
+[[package]]
+name = "colorama"
+version = "0.4.6"
+source = { registry = "https://pypi.org/simple" }
+sdist = { url = "https://files.pythonhosted.org/packages/d8/53/6f443c9a4a8358a93a6792e2acffb9d9d5cb0a5cfd8802644b7b1c9a02e4/colorama-0.4.6.tar.gz", hash = "sha256:08695f5cb7ed6e0531a20572697297273c47b8cae5a63ffc6d6ed5c201be6e44", size = 27697 }
+wheels = [
+    { url = "https://files.pythonhosted.org/packages/d1/d6/3965ed04c63042e047cb6a3e6ed1a63a35087b6a609aa3a15ed8ac56c221/colorama-0.4.6-py2.py3-none-any.whl", hash = "sha256:4f1d9991f5acc0ca119f9d443620b77f9d6b33703e51011c16baf57afb285fc6", size = 25335 },
+]
+
 [[package]]
 name = "cryptography"
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-version = "0.1.6"
+version = "0.1.1"
 source = { editable = "." }
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     { name = "maturin", extra = ["patchelf"] },
-    { name = "numpy" },
+    { name = "pytest" },
+    { name = "scipy" },
     { name = "twine" },
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 [package.metadata]
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+requires-dist = [{ name = "numpy", specifier = ">=1.20,<3.0" }]
+
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     { name = "maturin", extras = ["patchelf"], specifier = ">=1.8.7" },
-    { name = "numpy", specifier = ">=1.20,<3.0" },
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+    { name = "scipy", specifier = ">=1.10.1" },
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+    { name = "iniconfig" },
+    { name = "packaging" },
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