Documentation: Independent Set -> Set Packing

docs/make.jl

@@ -41,6 +41,7 @@ makedocs(;
         ],
         "Rules" => [
             "rules/spinglass_sat.md",
+            "rules/independentset_setpacking.md"
         ],
         "Examples" => [
             "generated/material_compute.md",

docs/src/rules/independentset_setpacking.md

@@ -0,0 +1,47 @@
+# Independent Set -> Set Packing
+
+In this tutorial, we will demonstrate how to reduce the [`IndependentSet`] (@ref) problem to the [`SetPacking`] (@ref) problem and how to extract solutions back to the original problem.
+
+## Reduction Framework
+Given an undirected graph $G=(V,E)$ and parameter $k$, we can have an instance of the [`IndependentSet`] (@ref) problem $(G, k)$. we aim to generate a corresponding [`SetPacking`] (@ref) instance $(U, S, k)$ (where $U$ is the union set, $S$ is the set of subsets and parameter $k$ is the required set packing size).
+
+- Step-0: $k$ are the same;
+- Step-1: Each edge $(u,v)\in E$ -> Create an element $x_{u,v}$ in $U$;
+- Step-2: Each vertex $v \in V$ -> Create a subset $\{S_{u,v}|(u,v)\in E \}$.
+
+It can be proven that:
+
+- The instance $(G,k)$ is an yes-instance if and only if generated $(U,S,k)$ is an yes-instance;
+- This transformation is within polynomial time.
+
+## Construct Reduction
+We can firstly define a [`IndependentSet`] (@ref) problem over a simple graph with $4$ vertices.
+```@repl independentset_setpacking
+using ProblemReductions, Graphs
+graph = SimpleGraph(4)
+add_edge!(graph, 1, 2) 
+add_edge!(graph, 1, 3)
+add_edge!(graph, 3, 4)
+add_edge!(graph, 2, 3)
+IS = IndependentSet(graph)
+```
+Then the reduction [`ReductionIndependentSetToSetPacking`] (@ref) can be easily constructed by the [`reduceto`](@ref) function.
+```@repl independentset_setpacking
+result = reduceto(SetPacking, IS)
+```
+The target [`SetPacking`] (@ref) problem can be accessed by the `target` field:
+```@repl independentset_setpacking
+SP = result.target
+```
+
+## Extract Solutions
+We can extract solutions from the target [`SetPacking`] (@ref) problem either by extracting individual solutions via [`extract_solution`] (@ref) or extracting mutilple solutions via [`extract_multiple_solutions`] (@ref).
+```@repl independentset_setpacking
+sol_SP = findbest(SP, BruteForce())
+sol_extract_single = Set( unique( extract_solution.(Ref(result), sol_SP) ) )
+sol_extract_mutilple = Set( extract_multiple_solutions(result, sol_SP) )
+```
+We can find that these extracted solutions indeed match with the solutions to the original [`IndependentSet`] (@ref) problem.
+```@repl independentset_setpacking
+sol_IS = findbest(IS, BruteForce())
+```