CodingThrust
diff --git a/‎docs/paper/reductions.typ‎
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diff --git a/‎docs/paper/references.bib‎
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diff --git a/‎docs/src/reductions/problem_schemas.json‎
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@@ -39,6 +39,7 @@
   "SpinGlass": [Spin Glass],
   "QUBO": [QUBO],
   "ILP": [Integer Linear Programming],
+  "Knapsack": [Knapsack],
   "Satisfiability": [SAT],
   "KSatisfiability": [$k$-SAT],
   "CircuitSAT": [CircuitSAT],
@@ -886,6 +887,14 @@ Biclique Cover is equivalent to factoring the biadjacency matrix $M$ of the bipa
   ) <fig:binpacking-example>
 ]
 
+#problem-def("Knapsack")[
+  Given $n$ items with weights $w_0, dots, w_(n-1) in NN$ and values $v_0, dots, v_(n-1) in NN$, and a capacity $C in NN$, find $S subset.eq {0, dots, n - 1}$ maximizing $sum_(i in S) v_i$ subject to $sum_(i in S) w_i lt.eq C$.
+][
+  One of Karp's 21 NP-complete problems @karp1972. Knapsack is only _weakly_ NP-hard: a classical dynamic-programming algorithm runs in $O(n C)$ pseudo-polynomial time, and a fully polynomial-time approximation scheme (FPTAS) achieves $(1 - epsilon)$-optimal value in $O(n^2 slash epsilon)$ time @ibarra1975. The special case $v_i = w_i$ for all $i$ is the Subset Sum problem. Knapsack is also a special case of Integer Linear Programming with a single constraint. The best known exact algorithm is the $O^*(2^(n slash 2))$ meet-in-the-middle approach of Horowitz and Sahni @horowitz1974, which partitions items into two halves and combines sorted sublists.
+
+  *Example.* Let $n = 4$ items with weights $(2, 3, 4, 5)$, values $(3, 4, 5, 7)$, and capacity $C = 7$. Selecting $S = {1, 2}$ (items with weights 3 and 4) gives total weight $3 + 4 = 7 lt.eq C$ and total value $4 + 5 = 9$. Selecting $S = {0, 3}$ (weights 2 and 5) gives weight $2 + 5 = 7 lt.eq C$ and value $3 + 7 = 10$, which is optimal.
+]
+
 // Completeness check: warn about problem types in JSON but missing from paper
 #{
   let json-models = {
 
@@ -366,3 +366,24 @@ @article{alber2004
   doi     = {10.1016/j.jalgor.2003.10.001}
 }
 
+@article{horowitz1974,
+  author  = {Ellis Horowitz and Sartaj Sahni},
+  title   = {Computing Partitions with Applications to the Knapsack Problem},
+  journal = {Journal of the ACM},
+  volume  = {21},
+  number  = {2},
+  pages   = {277--292},
+  year    = {1974},
+  doi     = {10.1145/321812.321823}
+}
+
+@article{ibarra1975,
+  author  = {Oscar H. Ibarra and Chul E. Kim},
+  title   = {Fast Approximation Algorithms for the Knapsack and Sum of Subset Problems},
+  journal = {Journal of the ACM},
+  volume  = {22},
+  number  = {4},
+  pages   = {463--468},
+  year    = {1975},
+  doi     = {10.1145/321906.321909}
+}
@@ -8,16 +8,6 @@
         "type_name": "Vec<Vec<bool>>",
         "description": "Target boolean matrix A"
       },
-      {
-        "name": "m",
-        "type_name": "usize",
-        "description": "Number of rows"
-      },
-      {
-        "name": "n",
-        "type_name": "usize",
-        "description": "Number of columns"
-      },
       {
         "name": "k",
         "type_name": "usize",
@@ -75,11 +65,6 @@
         "name": "circuit",
         "type_name": "Circuit",
         "description": "The boolean circuit"
-      },
-      {
-        "name": "variables",
-        "type_name": "Vec<String>",
-        "description": "Circuit variable names"
       }
     ]
   },
@@ -183,6 +168,27 @@
       }
     ]
   },
+  {
+    "name": "Knapsack",
+    "description": "Select items to maximize total value subject to weight capacity constraint",
+    "fields": [
+      {
+        "name": "weights",
+        "type_name": "Vec<i64>",
+        "description": "Item weights w_i"
+      },
+      {
+        "name": "values",
+        "type_name": "Vec<i64>",
+        "description": "Item values v_i"
+      },
+      {
+        "name": "capacity",
+        "type_name": "i64",
+        "description": "Knapsack capacity C"
+      }
+    ]
+  },
   {
     "name": "MaxCut",
     "description": "Find maximum weight cut in a graph",
@@ -337,24 +343,9 @@
     "description": "Minimize color changes in paint shop sequence",
     "fields": [
       {
-        "name": "sequence_indices",
-        "type_name": "Vec<usize>",
-        "description": "Car sequence as indices"
-      },
-      {
-        "name": "car_labels",
+        "name": "sequence",
         "type_name": "Vec<String>",
-        "description": "Unique car labels"
-      },
-      {
-        "name": "is_first",
-        "type_name": "Vec<bool>",
-        "description": "First occurrence flags"
-      },
-      {
-        "name": "num_cars",
-        "type_name": "usize",
-        "description": "Number of unique cars"
+        "description": "Car labels (each must appear exactly twice)"
       }
     ]
   },