update docs

Author: GiggleLiu

docs/paper/reductions.typ

@@ -1475,11 +1475,11 @@ where $P$ is a penalty weight large enough that any constraint violation costs m
 The following reductions to Integer Linear Programming are straightforward formulations where problem constraints map directly to linear inequalities.
 
 #reduction-rule("MaximumSetPacking", "ILP")[
-  Each set is either selected or not, and every overlapping pair forbids selecting both members -- a pairwise conflict constraint that is directly linear in binary indicator variables.
+  Each set is either selected or not, and every universe element may belong to at most one selected set -- an element-based constraint that is directly linear in binary indicator variables.
 ][
-  _Construction._ Variables: $x_i in {0, 1}$ for each set $S_i in cal(S)$. Constraints: $x_i + x_j <= 1$ for each overlapping pair $(S_i, S_j)$. Objective: maximize $sum_i w_i x_i$.
+  _Construction._ Variables: $x_i in {0, 1}$ for each set $S_i in cal(S)$. Constraints: $sum_(S_i in.rev e) x_i <= 1$ for each element $e in U$. Objective: maximize $sum_i w_i x_i$.
 
-  _Correctness._ ($arrow.r.double$) A valid packing chooses pairwise disjoint sets, so no overlapping pair has both indicators equal to $1$. ($arrow.l.double$) Any feasible binary solution selects no overlapping pair, hence the chosen sets are pairwise disjoint; the objective maximizes total weight.
+  _Correctness._ ($arrow.r.double$) A valid packing chooses pairwise disjoint sets, so each element is covered at most once. ($arrow.l.double$) Any feasible binary solution covers each element at most once, hence the chosen sets are pairwise disjoint; the objective maximizes total weight.
 
   _Solution extraction._ $cal(P) = {S_i : x_i = 1}$.
 ]

docs/src/reductions/reduction_graph.json

@@ -1,3 +1,14 @@
+{
+  "nodes": [
+    {
+      "name": "BMF",
+      "variant": {},
+      "category": "algebraic",
+      "doc_path": "models/algebraic/struct.BMF.html",
+      "complexity": "2^(rows * rank + r
+Exported to: docs/src/reductions/reduction_graph.json
+
+JSON content:
 {
   "nodes": [
     {
@@ -675,21 +686,6 @@
       ],
       "doc_path": "rules/maximumindependentset_gridgraph/index.html"
     },
-    {
-      "source": 24,
-      "target": 23,
-      "overhead": [
-        {
-          "field": "num_vertices",
-          "formula": "num_vertices * num_vertices"
-        },
-        {
-          "field": "num_edges",
-          "formula": "num_vertices * num_vertices"
-        }
-      ],
-      "doc_path": "rules/maximumindependentset_gridgraph/index.html"
-    },
     {
       "source": 24,
       "target": 25,
@@ -765,17 +761,6 @@
       ],
       "doc_path": "rules/minimumvertexcover_maximumindependentset/index.html"
     },
-    {
-      "source": 25,
-      "target": 37,
-      "overhead": [
-        {
-          "field": "num_vars",
-          "formula": "num_vertices"
-        }
-      ],
-      "doc_path": "rules/maximumindependentset_qubo/index.html"
-    },
     {
       "source": 26,
       "target": 28,
@@ -917,7 +902,7 @@
         },
         {
           "field": "num_constraints",
-          "formula": "num_sets^2"
+          "formula": "universe_size"
         }
       ],
       "doc_path": "rules/maximumsetpacking_ilp/index.html"
@@ -1059,11 +1044,11 @@
       "overhead": [
         {
           "field": "num_vertices",
-          "formula": "2 * num_vars + 5 * num_literals + -1 * 5 * num_clauses + 3"
+          "formula": "num_vars + num_literals"
         },
         {
           "field": "num_edges",
-          "formula": "3 * num_vars + 11 * num_literals + -1 * 9 * num_clauses + 3"
+          "formula": "num_vars + num_literals"
         }
       ],
       "doc_path": "rules/sat_coloring/index.html"
@@ -1170,4 +1155,4 @@
       "doc_path": "rules/travelingsalesman_ilp/index.html"
     }
   ]
-}
+}