Write a simple solver for the hitting set problem

I think we need this to find the dimension of the solution variety.

engine-proto/engine.jl

@@ -12,46 +12,68 @@ using Groebner
 abstract type Element{T} end
 
 mutable struct Point{T} <: Element{T}
-  coords::Union{Vector{MPolyRingElem{T}}, Nothing}
+  coords::Vector{MPolyRingElem{T}}
   vec::Union{Vector{MPolyRingElem{T}}, Nothing}
   rel::Nothing
   
   ## [to do] constructor argument never needed?
   Point{T}(
-    coords::Union{Vector{MPolyRingElem{T}}, Nothing} = nothing,
+    coords::Vector{MPolyRingElem{T}} = MPolyRingElem{T}[],
     vec::Union{Vector{MPolyRingElem{T}}, Nothing} = nothing
   ) where T = new(coords, vec, nothing)
 end
 
-coordnames(_::Point) = [:xₚ, :yₚ, :zₚ]
+##coordnames(_::Point) = [:xₚ, :yₚ, :zₚ]
 
-function buildvec(pt::Point, coordqueue)
-  coordring = parent(coordqueue[1])
-  pt.coords = splice!(coordqueue, 1:3)
+function buildvec!(pt::Point)
+  coordring = parent(pt.coords[1])
   pt.vec = [one(coordring), dot(pt.coords, pt.coords), pt.coords...]
 end
 
 mutable struct Sphere{T} <: Element{T}
-  coords::Union{Vector{MPolyRingElem{T}}, Nothing}
+  coords::Vector{MPolyRingElem{T}}
   vec::Union{Vector{MPolyRingElem{T}}, Nothing}
   rel::Union{MPolyRingElem{T}, Nothing}
   
+  ## [to do] constructor argument never needed?
   Sphere{T}(
-    coords::Union{Vector{MPolyRingElem{T}}, Nothing} = nothing,
+    coords::Vector{MPolyRingElem{T}} = MPolyRingElem{T}[],
     vec::Union{Vector{MPolyRingElem{T}}, Nothing} = nothing,
     rel::Union{MPolyRingElem{T}, Nothing} = nothing
   ) where T = new(coords, vec, rel)
 end
 
-coordnames(_::Sphere) = [:rₛ, :sₛ, :xₛ, :yₛ, :zₛ]
+##coordnames(_::Sphere) = [:rₛ, :sₛ, :xₛ, :yₛ, :zₛ]
 
-function buildvec(sph::Sphere, coordqueue)
-  coordring = parent(coordqueue[1])
-  sph.coords = splice!(coordqueue, 1:5)
+function buildvec!(sph::Sphere)
+  coordring = parent(sph.coords[1])
   sph.vec = sph.coords
   sph.rel = mprod(sph.coords, sph.coords) + one(coordring)
 end
 
+const coordnames = IdDict{Symbol, Vector{Union{Symbol, Nothing}}}(
+  nameof(Point) => [nothing, nothing, :xₚ, :yₚ, :zₚ],
+  nameof(Sphere) => [:rₛ, :sₛ, :xₛ, :yₛ, :zₛ]
+)
+
+coordname(elem::Element, index) = coordnames[nameof(typeof(elem))][index]
+
+function pushcoordname!(coordnamelist, indexed_elem::Tuple{Any, Element}, coordindex)
+  elemindex, elem = indexed_elem
+  name = coordname(elem, coordindex)
+  if !isnothing(name)
+    subscript = Subscripts.sub(string(elemindex))
+    push!(coordnamelist, Symbol(name, subscript))
+  end
+end
+
+function takecoord!(coordlist, indexed_elem::Tuple{Any, Element}, coordindex)
+  elem = indexed_elem[2]
+  if !isnothing(coordname(elem, coordindex))
+    push!(elem.coords, popfirst!(coordlist))
+  end
+end
+
 # --- primitive relations ---
 
 abstract type Relation{T} end
@@ -99,22 +121,38 @@ function Base.push!(ctx::Construction{T}, rel::Relation{T}) where T
 end
 
 function realize(ctx::Construction{T}) where T
-  # collect variable names
+  # collect coordinate names
   coordnamelist = Symbol[]
   elemenum = enumerate(ctx.elements)
-  for (index, elem) in elemenum
-    subscript = Subscripts.sub(string(index))
-    append!(coordnamelist,
-      [Symbol(name, subscript) for name in coordnames(elem)]
-    )
+  for coordindex in 1:5
+    for indexed_elem in elemenum
+      pushcoordname!(coordnamelist, indexed_elem, coordindex)
+    end
   end
   
+  display(collect(elemenum))
+  display(coordnamelist)
+  println()
+  
   # construct coordinate ring
   coordring, coordqueue = polynomial_ring(parent_type(T)(), coordnamelist, ordering = :degrevlex)
   
+  # retrieve coordinates
+  for (_, elem) in elemenum
+    empty!(elem.coords)
+  end
+  for coordindex in 1:5
+    for indexed_elem in elemenum
+      takecoord!(coordqueue, indexed_elem, coordindex)
+    end
+  end
+  
   # construct coordinate vectors
   for (_, elem) in elemenum
-    buildvec(elem, coordqueue)
+    buildvec!(elem)
+    display(elem.coords)
+    display(elem.vec)
+    println()
   end
   
   # turn relations into equations

engine-proto/hitting-set.jl

@@ -0,0 +1,110 @@
+module HittingSet
+
+HittingSetProblem{T} = Pair{Set{T}, Vector{Pair{T, Set{Set{T}}}}}
+
+# `subsets` should be a collection of Set objects
+function HittingSetProblem(subsets, chosen = Set())
+  wholeset = union(subsets...)
+  T = eltype(wholeset)
+  unsorted_moves = [
+    elt => Set(filter(s -> elt ∉ s, subsets))
+    for elt in wholeset
+  ]
+  moves = sort(unsorted_moves, by = pair -> length(pair.second))
+  Set{T}(chosen) => moves
+end
+
+function Base.display(problem::HittingSetProblem{T}) where T
+  println("HittingSetProblem{$T}")
+  
+  chosen = problem.first
+  println(" {", join(string.(chosen), ", "), "}")
+  
+  moves = problem.second
+  for (choice, missed) in moves
+    println(" | ", choice)
+    for s in missed
+      println(" | | {", join(string.(s), ", "), "}")
+    end
+  end
+  println()
+end
+
+function solve(pblm::HittingSetProblem{T}, maxdepth = Inf) where T
+  problems = Dict(pblm)
+  println(typeof(problems))
+  while length(first(problems).first) < maxdepth
+    subproblems = typeof(problems)()
+    for (chosen, moves) in problems
+      if isempty(moves)
+        return chosen
+      else
+        for (choice, missed) in moves
+          to_be_chosen = union(chosen, Set([choice]))
+          if isempty(missed)
+            return to_be_chosen
+          elseif !haskey(subproblems, to_be_chosen)
+            push!(subproblems, HittingSetProblem(missed, to_be_chosen))
+          end
+        end
+      end
+    end
+    problems = subproblems
+  end
+  problems
+end
+
+function test(n = 1)
+  T = [Int64, Int64, Symbol, Symbol][n]
+  subsets = Set{T}.([
+    [
+      [1, 3, 5],
+      [2, 3, 4],
+      [1, 4],
+      [2, 3, 4, 5],
+      [4, 5]
+    ],
+    # example from Amit Chakrabarti's graduate-level algorithms class (CS 105)
+    # notes by Valika K. Wan and Khanh Do Ba, Winter 2005
+    # https://www.cs.dartmouth.edu/~ac/Teach/CS105-Winter05/
+    [
+      [1, 3], [1,    4], [1,    5],
+      [1, 3], [1, 2, 4], [1, 2, 5],
+      [4, 3], [   2, 4], [   2, 5],
+      [6, 3], [6,    4], [      5]
+    ],
+    [
+      [:w, :x, :y],
+      [:x, :y, :z],
+      [:w, :z],
+      [:x, :y]
+    ],
+    # Wikipedia showcases this as an example of a problem where the greedy
+    # algorithm performs especially poorly
+    [
+      [:a, :x, :t1],
+      [:a, :y, :t2],
+      [:a, :y, :t3],
+      [:a, :z, :t4],
+      [:a, :z, :t5],
+      [:a, :z, :t6],
+      [:a, :z, :t7],
+      [:b, :x, :t8],
+      [:b, :y, :t9],
+      [:b, :y, :t10],
+      [:b, :z, :t11],
+      [:b, :z, :t12],
+      [:b, :z, :t13],
+      [:b, :z, :t14]
+    ]
+  ][n])
+  problem = HittingSetProblem(subsets)
+  if isa(problem, HittingSetProblem{T})
+    println("Correct type")
+  else
+    println("Wrong type: ", typeof(problem))
+  end
+  problem
+end
+
+end