diff --git a/heldAndKarp.ipynb b/heldAndKarp.ipynb
index 41d38fc50284c6503e366a643c2276425cd6b267..3a86dcefc4f3f34b4e2f98b3ecee245254779e0d 100644
--- a/heldAndKarp.ipynb
+++ b/heldAndKarp.ipynb
@@ -53,11 +53,8 @@
},
{
"cell_type": "code",
- "execution_count": 9,
+ "execution_count": 1,
"metadata": {
- "jupyter": {
- "source_hidden": true
- },
"tags": []
},
"outputs": [
@@ -82,7 +79,7 @@
},
{
"cell_type": "code",
- "execution_count": 48,
+ "execution_count": 2,
"metadata": {
"tags": []
},
@@ -93,7 +90,7 @@
"checkTermination (generic function with 2 methods)"
]
},
- "execution_count": 48,
+ "execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
@@ -121,9 +118,6 @@
"cell_type": "code",
"execution_count": 3,
"metadata": {
- "jupyter": {
- "source_hidden": true
- },
"tags": []
},
"outputs": [
@@ -164,9 +158,6 @@
{
"cell_type": "markdown",
"metadata": {
- "jupyter": {
- "source_hidden": true
- },
"tags": []
},
"source": [
@@ -183,11 +174,8 @@
},
{
"cell_type": "code",
- "execution_count": 12,
+ "execution_count": 4,
"metadata": {
- "jupyter": {
- "source_hidden": true
- },
"tags": []
},
"outputs": [
@@ -278,11 +266,8 @@
},
{
"cell_type": "code",
- "execution_count": 7,
+ "execution_count": 5,
"metadata": {
- "jupyter": {
- "source_hidden": true
- },
"tags": []
},
"outputs": [
@@ -292,7 +277,7 @@
"IPM (generic function with 1 method)"
]
},
- "execution_count": 7,
+ "execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
@@ -340,9 +325,6 @@
"cell_type": "code",
"execution_count": 6,
"metadata": {
- "jupyter": {
- "source_hidden": true
- },
"tags": []
},
"outputs": [
@@ -395,11 +377,8 @@
},
{
"cell_type": "code",
- "execution_count": 8,
+ "execution_count": 7,
"metadata": {
- "jupyter": {
- "source_hidden": true
- },
"tags": []
},
"outputs": [
@@ -409,7 +388,7 @@
"DM (generic function with 1 method)"
]
},
- "execution_count": 8,
+ "execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
@@ -450,31 +429,38 @@
},
{
"cell_type": "code",
- "execution_count": 7,
+ "execution_count": 20,
"metadata": {},
"outputs": [
{
- "ename": "LoadError",
- "evalue": "syntax: missing comma or ) in argument list",
- "output_type": "error",
- "traceback": [
- "syntax: missing comma or ) in argument list",
- ""
- ]
+ "data": {
+ "text/plain": [
+ "SP (generic function with 1 method)"
+ ]
+ },
+ "execution_count": 20,
+ "metadata": {},
+ "output_type": "execute_result"
}
],
"source": [
- "function SP()\n",
+ "function SP(W, π, μ)\n",
+ " n = size(W,1)\n",
+ "\n",
" sp = Model(Cbc.Optimizer)\n",
" set_silent(sp)\n",
" \n",
- " @variable(sp, )\n",
+ " @variable(sp, X[u in 1:n, v in 1:n], Bin)\n",
" \n",
- " @objective(sp, Min, (c - π^T A ) x - μ)\n",
+ " @objective(sp, Min, sum(W .* X)/2 - dot(π, [sum(X[u,:]) for u ∈ 1:n]) - μ)\n",
" \n",
- " @constraint(sp, degree1)\n",
- " @constraint(sp, nbEdges, sum(x[2:]) == )\n",
- " @constraint(sp, spanningTree)\n",
+ " @constraint(sp, degree1, sum(X[1,:]) == 2)\n",
+ " @constraint(sp, nbEdges, sum(X) == 2*n)\n",
+ " for np in 2:n, X2N in collect(permutations(collect(2:np),np-1))\n",
+ " @constraint(sp, sum(X[X2N[i], X2N[i+1]] for i in 1:length(X2N)-1) + X[X2N[length(X2N)], X2N[1]] <= length(X2N) - 1)\n",
+ " end\n",
+ " @constraint(sp, symmetry[u in 1:n, v in 1:n], X[u,v] == X[v,u])\n",
+ " @constraint(sp, diagonal[u in 1:n], X[u,u] == 0)\n",
" \n",
" optimize!(sp)\n",
" checkTermination(sp)\n",
@@ -486,9 +472,6 @@
{
"cell_type": "markdown",
"metadata": {
- "jupyter": {
- "source_hidden": true
- },
"tags": []
},
"source": [
@@ -500,9 +483,6 @@
{
"cell_type": "markdown",
"metadata": {
- "jupyter": {
- "source_hidden": true
- },
"tags": []
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"source": [
@@ -513,11 +493,8 @@
},
{
"cell_type": "code",
- "execution_count": 4,
+ "execution_count": 9,
"metadata": {
- "jupyter": {
- "source_hidden": true
- },
"tags": []
},
"outputs": [
@@ -527,7 +504,7 @@
"wheel_like_1trees (generic function with 1 method)"
]
},
- "execution_count": 4,
+ "execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
@@ -560,15 +537,90 @@
},
{
"cell_type": "code",
- "execution_count": 5,
+ "execution_count": null,
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [],
+ "source": [
+ "wl = wheel_like_1trees(6)\n",
+ "\n",
+ "for t in wl\n",
+ " display_S(t)\n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 29,
"metadata": {
- "collapsed": true,
- "jupyter": {
- "outputs_hidden": true,
- "source_hidden": true
- },
"tags": []
},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "column_generation (generic function with 1 method)"
+ ]
+ },
+ "execution_count": 29,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "function column_generation(W; max_iter = 1000)\n",
+ " n = size(W, 1)\n",
+ " one_trees_init = wheel_like_1trees(n)\n",
+ " \n",
+ " columns = one_trees_init[1:n-2]\n",
+ " tree_degrees = [sum(columns[t][u,:]) for u ∈ 1:n, t ∈ 1:n-2]\n",
+ " tree_weights = [dot(columns[t], W)/2 for t ∈ 1:n-2]\n",
+ " \n",
+ " reduced_cost = -1\n",
+ " new_column = one_trees_init[n-1]\n",
+ " k = 1\n",
+ " \n",
+ " while reduced_cost < 0 && k <= max_iter\n",
+ " \n",
+ " push!(columns, new_column)\n",
+ " tree_degrees = hcat(tree_degrees, [sum(new_column[u,:]) for u ∈ 1:n]) \n",
+ " push!(tree_weights, dot(new_column, W)/2)\n",
+ " \n",
+ " dm = DM(tree_weights, tree_degrees)\n",
+ " π = value.(dm[:π])\n",
+ " μ = value(dm[:μ])\n",
+ " \n",
+ " # Solve subproblem\n",
+ " sp = SP(W, π, μ)\n",
+ " new_column = value.(sp[:X])\n",
+ " new_degree = [sum(new_column[u,:]) for u ∈ 1:n]\n",
+ " new_weight = dot(new_column, W)/2\n",
+ " reduced_cost = objective_value(sp)\n",
+ " \n",
+ " k += 1\n",
+ " end\n",
+ " \n",
+ " lpm = LPM(tree_weights, tree_degrees)\n",
+ " # ipm = IPM(tree_weights, tree_degrees)\n",
+ " \n",
+ " λ = value.(lpm[:λ])\n",
+ " #println(\"Lambda_t : \", λ)\n",
+ " println(\"Minimal 1-tree \")\n",
+ " println(\"Lambda : \", maximum(λ))\n",
+ " println(\"Index : \", argmax(λ))\n",
+ " #println(\"1-tree : \", columns[argmax(λ)])\n",
+ " println(\"Weight : \", tree_weights[argmax(λ)])\n",
+ " println(\"k : \", k)\n",
+ " \n",
+ " return columns[argmax(λ)]\n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 30,
+ "metadata": {},
"outputs": [
{
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@@ -588,77 +640,77 @@
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- " <path fill=\"none\" d=\"M21.99,10.81 L-21.99,-10.81 \" class=\"primitive\"/>\n",
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@@ -677,83 +729,83 @@
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- "Compose.Context(Measures.BoundingBox{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}},Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}((0.0w, 0.0h), (1.0w, 1.0h)), Compose.UnitBox{Float64,Float64,Float64,Float64}(-1.2, -1.2, 2.4, 2.4, 0.0mm, 0.0mm, 0.0mm, 0.0mm), nothing, nothing, nothing, List([Compose.Context(Measures.BoundingBox{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}},Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}((0.0w, 0.0h), (1.0w, 1.0h)), nothing, nothing, nothing, nothing, List([]), List([Compose.Form{Compose.LinePrimitive}(Compose.LinePrimitive[Compose.LinePrimitive{Tuple{Measures.Measure,Measures.Measure}}(Tuple{Measures.Measure,Measures.Measure}[(0.9161961853249513cx, 0.5101722949453402cy), (0.16977040845513006cx, -0.008690289791820786cy)]), Compose.LinePrimitive{Tuple{Measures.Measure,Measures.Measure}}(Tuple{Measures.Measure,Measures.Measure}[(0.9170842310522285cx, 0.6279384953117152cy), (0.4816184383887985cx, 0.9404883042884304cy)]), Compose.LinePrimitive{Tuple{Measures.Measure,Measures.Measure}}(Tuple{Measures.Measure,Measures.Measure}[(0.11467455686797243cx, 0.03099661417103121cy), (0.36999470635313586cx, 0.9020585913823425cy)]), Compose.LinePrimitive{Tuple{Measures.Measure,Measures.Measure}}(Tuple{Measures.Measure,Measures.Measure}[(-0.00967346678948637cx, -0.03131270945407927cy), (-0.9043599394304324cx, 0.3020156494466911cy)]), Compose.LinePrimitive{Tuple{Measures.Measure,Measures.Measure}}(Tuple{Measures.Measure,Measures.Measure}[(0.150420012213972cx, -0.14608027017560937cy), (0.7521874596531807cx, -0.8849263141625895cy)]), Compose.LinePrimitive{Tuple{Measures.Measure,Measures.Measure}}(Tuple{Measures.Measure,Measures.Measure}[(0.026851259529856328cx, -0.15014361609183255cy), (-0.5178826339609167cx, -0.9168011783547937cy)])], Symbol(\"\"))]), 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Compose.Context(Measures.BoundingBox{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}},Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}((0.0w, 0.0h), (1.0w, 1.0h)), nothing, nothing, nothing, nothing, List([]), List([Compose.Form{Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}}(Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}[Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}((1.0cx, 0.5684267996001457cy), 0.04082482904638631w), Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}((0.08596659378008131cx, -0.06694479444662627cy), 0.04082482904638631w), Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}((0.39870266944102695cx, 1.0cy), 0.04082482904638631w), Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}((-1.0cx, 0.3376477344392381cy), 0.04082482904638631w), Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}((0.8166408780870713cx, -0.9640617898915725cy), 0.04082482904638631w), Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}((-0.5769979682111417cx, -1.0cy), 0.04082482904638631w)], Symbol(\"\"))]), List([Compose.Property{Compose.LineWidthPrimitive}(Compose.LineWidthPrimitive[Compose.LineWidthPrimitive(0.0mm)]), Compose.Property{Compose.StrokePrimitive}(Compose.StrokePrimitive[Compose.StrokePrimitive(RGBA{Float64}(0.0,0.0,0.0,0.0))]), Compose.Property{Compose.FillPrimitive}(Compose.FillPrimitive[Compose.FillPrimitive(RGBA{Float64}(0.25098039215686274,0.8784313725490196,0.8156862745098039,1.0))])]), 0, false, false, false, false, nothing, nothing, 0.0, Symbol(\"\")), Compose.Context(Measures.BoundingBox{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}},Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}((0.0w, 0.0h), (1.0w, 1.0h)), nothing, nothing, nothing, nothing, List([]), List([Compose.Form{Compose.TextPrimitive{Tuple{Measures.Length{:cx,Float64},Measures.Length{:cy,Float64}},Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}},Tuple{Measures.Length{:mm,Float64},Measures.Length{:mm,Float64}}}}(Compose.TextPrimitive{Tuple{Measures.Length{:cx,Float64},Measures.Length{:cy,Float64}},Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}},Tuple{Measures.Length{:mm,Float64},Measures.Length{:mm,Float64}}}[Compose.TextPrimitive{Tuple{Measures.Length{:cx,Float64},Measures.Length{:cy,Float64}},Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}},Tuple{Measures.Length{:mm,Float64},Measures.Length{:mm,Float64}}}((1.0cx, 0.5684267996001457cy), \"1\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm)), Compose.TextPrimitive{Tuple{Measures.Length{:cx,Float64},Measures.Length{:cy,Float64}},Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}},Tuple{Measures.Length{:mm,Float64},Measures.Length{:mm,Float64}}}((0.08596659378008131cx, -0.06694479444662627cy), \"2\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm)), Compose.TextPrimitive{Tuple{Measures.Length{:cx,Float64},Measures.Length{:cy,Float64}},Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}},Tuple{Measures.Length{:mm,Float64},Measures.Length{:mm,Float64}}}((0.39870266944102695cx, 1.0cy), \"3\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm)), Compose.TextPrimitive{Tuple{Measures.Length{:cx,Float64},Measures.Length{:cy,Float64}},Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}},Tuple{Measures.Length{:mm,Float64},Measures.Length{:mm,Float64}}}((-1.0cx, 0.3376477344392381cy), \"4\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm)), Compose.TextPrimitive{Tuple{Measures.Length{:cx,Float64},Measures.Length{:cy,Float64}},Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}},Tuple{Measures.Length{:mm,Float64},Measures.Length{:mm,Float64}}}((0.8166408780870713cx, -0.9640617898915725cy), \"5\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm)), Compose.TextPrimitive{Tuple{Measures.Length{:cx,Float64},Measures.Length{:cy,Float64}},Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}},Tuple{Measures.Length{:mm,Float64},Measures.Length{:mm,Float64}}}((-0.5769979682111417cx, -1.0cy), \"6\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm))], Symbol(\"\"))]), List([Compose.Property{Compose.FontSizePrimitive}(Compose.FontSizePrimitive[Compose.FontSizePrimitive(4.0mm)]), Compose.Property{Compose.StrokePrimitive}(Compose.StrokePrimitive[Compose.StrokePrimitive(RGBA{Float64}(0.0,0.0,0.0,0.0))]), Compose.Property{Compose.FillPrimitive}(Compose.FillPrimitive[Compose.FillPrimitive(RGBA{Float64}(0.0,0.0,0.0,1.0))])]), 0, false, false, false, false, nothing, nothing, 0.0, Symbol(\"\"))]), List([]), List([]), 0, false, false, false, false, nothing, nothing, 0.0, Symbol(\"\"))"
+ "Compose.Context(Measures.BoundingBox{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}},Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}((0.0w, 0.0h), (1.0w, 1.0h)), Compose.UnitBox{Float64,Float64,Float64,Float64}(-1.2, -1.2, 2.4, 2.4, 0.0mm, 0.0mm, 0.0mm, 0.0mm), nothing, nothing, nothing, List([Compose.Context(Measures.BoundingBox{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}},Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}((0.0w, 0.0h), (1.0w, 1.0h)), nothing, nothing, nothing, nothing, List([]), List([Compose.Form{Compose.LinePrimitive}(Compose.LinePrimitive[Compose.LinePrimitive{Tuple{Measures.Measure,Measures.Measure}}(Tuple{Measures.Measure,Measures.Measure}[(0.9135192758072158cx, 0.48896789789630135cy), (0.1846200872106957cx, 0.9457989759344138cy)]), Compose.LinePrimitive{Tuple{Measures.Measure,Measures.Measure}}(Tuple{Measures.Measure,Measures.Measure}[(0.9908121359972968cx, 0.3331191988856653cy), (0.9181856072854541cx, -0.47036654934881905cy)]), Compose.LinePrimitive{Tuple{Measures.Measure,Measures.Measure}}(Tuple{Measures.Measure,Measures.Measure}[(0.004396831248990096cx, 0.959639073298307cy), (-0.8161783253363644cx, 0.6063397424266846cy)]), Compose.LinePrimitive{Tuple{Measures.Measure,Measures.Measure}}(Tuple{Measures.Measure,Measures.Measure}[(0.8152750949959612cx, -0.6124213012084072cy), (0.010023978188983887cx, -0.9595929230854616cy)]), Compose.LinePrimitive{Tuple{Measures.Measure,Measures.Measure}}(Tuple{Measures.Measure,Measures.Measure}[(-0.1702230927371292cx, -0.9458687617581273cy), (-0.9134755773606764cx, -0.48087661166042883cy)]), Compose.LinePrimitive{Tuple{Measures.Measure,Measures.Measure}}(Tuple{Measures.Measure,Measures.Measure}[(-0.9191440105717664cx, 0.4643343367465424cy), (-0.9907768465335194cx, -0.32510089444010687cy)])], Symbol(\"\"))]), List([Compose.Property{Compose.LineWidthPrimitive}(Compose.LineWidthPrimitive[Compose.LineWidthPrimitive(1.2247448713915892mm)]), Compose.Property{Compose.FillPrimitive}(Compose.FillPrimitive[Compose.FillPrimitive(RGBA{Float64}(0.0,0.0,0.0,0.0))]), Compose.Property{Compose.StrokePrimitive}(Compose.StrokePrimitive[Compose.StrokePrimitive(RGBA{Float64}(0.8274509803921568,0.8274509803921568,0.8274509803921568,1.0))])]), 0, false, false, false, false, nothing, nothing, 0.0, Symbol(\"\")), Compose.Context(Measures.BoundingBox{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}},Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}((0.0w, 0.0h), (1.0w, 1.0h)), nothing, nothing, nothing, nothing, List([]), List([]), List([Compose.Property{Compose.LineWidthPrimitive}(Compose.LineWidthPrimitive[Compose.LineWidthPrimitive(1.2247448713915892mm)]), 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nothing, nothing, List([]), List([Compose.Form{Compose.TextPrimitive{Tuple{Measures.Length{:cx,Float64},Measures.Length{:cy,Float64}},Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}},Tuple{Measures.Length{:mm,Float64},Measures.Length{:mm,Float64}}}}(Compose.TextPrimitive{Tuple{Measures.Length{:cx,Float64},Measures.Length{:cy,Float64}},Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}},Tuple{Measures.Length{:mm,Float64},Measures.Length{:mm,Float64}}}[Compose.TextPrimitive{Tuple{Measures.Length{:cx,Float64},Measures.Length{:cy,Float64}},Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}},Tuple{Measures.Length{:mm,Float64},Measures.Length{:mm,Float64}}}((1.0cx, 0.4347668738307151cy), \"1\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm)), Compose.TextPrimitive{Tuple{Measures.Length{:cx,Float64},Measures.Length{:cy,Float64}},Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}},Tuple{Measures.Length{:mm,Float64},Measures.Length{:mm,Float64}}}((0.09813936301791149cx, 1.0cy), \"2\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm)), Compose.TextPrimitive{Tuple{Measures.Length{:cx,Float64},Measures.Length{:cy,Float64}},Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}},Tuple{Measures.Length{:mm,Float64},Measures.Length{:mm,Float64}}}((0.9089977432827507cx, -0.5720142242938688cy), \"3\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm)), Compose.TextPrimitive{Tuple{Measures.Length{:cx,Float64},Measures.Length{:cy,Float64}},Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}},Tuple{Measures.Length{:mm,Float64},Measures.Length{:mm,Float64}}}((-0.08369867009780563cx, -1.0cy), \"4\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm)), Compose.TextPrimitive{Tuple{Measures.Length{:cx,Float64},Measures.Length{:cy,Float64}},Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}},Tuple{Measures.Length{:mm,Float64},Measures.Length{:mm,Float64}}}((-0.9099208571052858cx, 0.5659788157249916cy), \"5\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm)), Compose.TextPrimitive{Tuple{Measures.Length{:cx,Float64},Measures.Length{:cy,Float64}},Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}},Tuple{Measures.Length{:mm,Float64},Measures.Length{:mm,Float64}}}((-1.0cx, -0.4267453734185561cy), \"6\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm))], Symbol(\"\"))]), List([Compose.Property{Compose.FontSizePrimitive}(Compose.FontSizePrimitive[Compose.FontSizePrimitive(4.0mm)]), Compose.Property{Compose.StrokePrimitive}(Compose.StrokePrimitive[Compose.StrokePrimitive(RGBA{Float64}(0.0,0.0,0.0,0.0))]), Compose.Property{Compose.FillPrimitive}(Compose.FillPrimitive[Compose.FillPrimitive(RGBA{Float64}(0.0,0.0,0.0,1.0))])]), 0, false, false, false, false, nothing, nothing, 0.0, Symbol(\"\"))]), List([]), List([]), 0, false, false, false, false, nothing, nothing, 0.0, Symbol(\"\"))"
]
},
"metadata": {},
"output_type": "display_data"
},
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "Minimal 1-tree \n",
+ "Lambda : 1.0\n",
+ "Index : 14\n",
+ "Weight : 27.0\n",
+ "k : 11\n"
+ ]
+ }
+ ],
+ "source": [
+ "display_S(column_generation(example))"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 31,
+ "metadata": {},
+ "outputs": [
{
"data": {
"image/svg+xml": [
@@ -932,81 +1005,92 @@
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+ "Minimal 1-tree \n",
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+ "display_S(column_generation(exercise))"
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+ "source": [
+ "## Held and Karp algorithm - attempt 1\n",
+ "\n",
+ "This is the part where we implement the Held and Karp algorithm as described in J.E. Mitchell's slides using Kruskal's algorithm to solve the minimum spanning tree problem. Since the goal of this algorithm is allegedly to provide a new column to the master problem, there is no guarantee that it gives an optimal solution.\n",
+ "\n",
+ "Even though this approach seems to be flawed, the example is easily solved ; however the exercise seems to alternate between two identical states. \n",
+ "This may be due to the update rule of $\\lambda$, results may vary by modifying it."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 32,
+ "metadata": {},
+ "outputs": [
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+ "update_rule_05! (generic function with 1 method)"
]
},
+ "execution_count": 32,
"metadata": {},
- "output_type": "display_data"
+ "output_type": "execute_result"
}
],
"source": [
- "wl = wheel_like_1trees(6)\n",
+ "function update_rule!(λ, D, Ï)\n",
+ " λ .= λ .+ Ï .* (2 .- D)\n",
+ "end\n",
"\n",
- "for t in wl\n",
- " display_S(t)\n",
- "end"
+ "update_rule_1!(λ, D) = update_rule!(λ, D, 1)\n",
+ "update_rule_2!(λ, D) = update_rule!(λ, D, 2)\n",
+ "update_rule_3!(λ, D) = update_rule!(λ, D, 3)\n",
+ "update_rule_05!(λ, D) = update_rule!(λ, D, 0.5)"
]
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 33,
"metadata": {},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "held_and_karp_iter! (generic function with 1 method)"
+ ]
+ },
+ "execution_count": 33,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
- "function column_generation(W)\n",
+ "function held_and_karp_iter!(W, W_updated, π, X, D, g, update_rule)\n",
" n = size(W, 1)\n",
- " one_trees_init = wheel_like_1trees(n)\n",
- " \n",
- " columns = one_trees_init[1:n-2]\n",
- " tree_degrees = [sum(columns[t][u,:]) for u ∈ 1:n, t ∈ 1:n-2]\n",
- " tree_weights = [dot(columns[t], W) for t ∈ 1:n-2]\n",
+ " X .= falses(n, n) #X is reset every iteration because a different 1-tree is computed\n",
+ " W_updated .= deepcopy(W) #we need to modify the weights according to λ\n",
" \n",
- " reduced_cost = -1\n",
- " new_column = one_trees_init[n-1]\n",
+ " # Minimum spanning tree (span_tree : st)\n",
+ " for u in 2:n, v in 2:n\n",
+ " W_updated[u,v] -= π[u] + π[v]\n",
+ " end\n",
+ " st = kruskal_mst(g, W_updated[2:n,2:n])\n",
" \n",
- " while reduced_cost < 0\n",
- " \n",
- " push!(columns, new_column)\n",
- " tree_degrees = hcat(tree_degrees, [sum(new_column[u,:]) for u ∈ 1:n]) \n",
- " push!(tree_weights, dot(new_column, W))\n",
- " \n",
- " dm = DM(tree_weights, tree_degrees)\n",
- " π = value.(dm[:π])\n",
- " μ = value(dm[:μ])\n",
- " \n",
- " # Solve subproblem\n",
- " sp = SP()\n",
- " new_column = value.(sp[:??????])\n",
- " new_degree = [sum(new_column[u,:]) for u ∈ 1:n]\n",
- " new_weight = dot(new_column, W)\n",
- " reduced_cost = objective_value(sp)\n",
+ " # Conversion of the spanning tree to an adjacency matrix\n",
+ " for i ∈ 1:size(st, 1)\n",
+ " X[src(st[i])+1, dst(st[i])+1] = 1\n",
+ " X[dst(st[i])+1, src(st[i])+1] = 1\n",
" end\n",
" \n",
- " lpm = LPM(tree_weights, tree_degrees)\n",
- " # ipm = IPM(tree_weights, tree_degrees)\n",
+ " # We connect vertex 1 to two vertices to obtain a 1-tree\n",
+ " W_updated[1,2:n] .-= π[2:n]\n",
+ " connect_v1!(X, W_updated[1,:], n)\n",
" \n",
- " λ = value.(lpm[:λ])\n",
- " println(\"Lambda_t : \", λ)\n",
- " println(\"Minimal 1-tree \")\n",
- " println(\"Lambda : \", maximum(λ))\n",
- " println(\"Index : \", argmax(λ))\n",
- " println(\"1-tree : \", columns[argmax(λ)])\n",
+ " #then we compute the objective value\n",
+ " z = 2*sum(π) + sum((W[u,v] - π[u] - π[v])*X[u,v] for u ∈ 1:n for v ∈ 1:u)\n",
" \n",
- " return columns[argmax(λ)]\n",
+ " #the degrees of all vertices in graph X\n",
+ " D = [sum(X[u,:]) for u ∈ 1:n]\n",
+ " \n",
+ " if all(D .== 2) # If not a tour, update\n",
+ " return true, z\n",
+ " else\n",
+ " update_rule(Ï€, D)\n",
+ " end\n",
+ " \n",
+ " return false, z\n",
"end"
]
},
{
"cell_type": "code",
- "execution_count": null,
- "metadata": {},
- "outputs": [],
+ "execution_count": 34,
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "multiargmin (generic function with 1 method)"
+ ]
+ },
+ "execution_count": 34,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
- "display_S(column_generation(W))\n",
- "display_S(column_generation(W2))"
+ "#returns the nbmin indexes for smallest values of itr\n",
+ "#size is the length of itr\n",
+ "function multiargmin(itr, nbmin, size)\n",
+ " if nbmin > size\n",
+ " error(\"multiargmin : size of array is \", size, \" < nbmin = \", nbmin)\n",
+ " end\n",
+ " \n",
+ " argmins = [i for i in 1:nbmin]\n",
+ " itr_copy = deepcopy(itr)\n",
+ " maxitr = maximum(itr)\n",
+ " \n",
+ " #we find the nbmin smallest values of itr\n",
+ " for i in 1:nbmin\n",
+ " argmins[i] = argmin(itr_copy)\n",
+ " #once a min is found, it is set to maxitr to find a new minimum\n",
+ " itr_copy[argmins[i]] = maxitr\n",
+ " end\n",
+ " \n",
+ " return argmins\n",
+ "end"
]
},
{
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "## Held and Karp algorithm - attempt 1\n",
- "\n",
- "This is the part where we implement the Held and Karp algorithm as described in J.E. Mitchell's slides using Kruskal's algorithm to solve the minimum spanning tree problem. Since the goal of this algorithm is allegedly to provide a new column to the master problem, there is no guarantee that it gives an optimal solution.\n",
- "\n",
- "Even though this approach seems to be flawed, the example is easily solved ; however the exercise seems to alternate between two identical states. \n",
- "This may be due to the update rule of $\\lambda$, results may vary by modifying it."
+ "cell_type": "code",
+ "execution_count": 35,
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "connect_v1! (generic function with 1 method)"
+ ]
+ },
+ "execution_count": 35,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "#connects the vertex 1 (of graph X) to the two vertices v1, v2 with smallest W1[v1], W1[v2]\n",
+ "#X is an adjacency matrix, W1 the weights of edges starting from 1, n the number of vertices\n",
+ "function connect_v1!(X, W1, n) \n",
+ " argmin1, argmin2 = multiargmin(W1[2:n], 2, n-1)\n",
+ " \n",
+ " X[1, argmin1+1] = 1\n",
+ " X[1, argmin2+1] = 1\n",
+ " \n",
+ " for u ∈ 2:n\n",
+ " X[u, 1] = X[1, u]\n",
+ " end \n",
+ "end"
]
},
{
"cell_type": "code",
- "execution_count": 54,
- "metadata": {
- "tags": []
- },
+ "execution_count": 36,
+ "metadata": {},
"outputs": [
{
"data": {
"text/plain": [
- "hk (generic function with 1 method)"
+ "hk_mitchell (generic function with 1 method)"
]
},
- "execution_count": 54,
+ "execution_count": 36,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
- "function hk(W)\n",
+ "function hk_mitchell(W; disp=true, maxiter=10000, update_rule=update_rule_2!)\n",
+ " \n",
" # Initialization\n",
" n = size(W, 1)\n",
" \n",
" if size(W, 2)!=n ; error(\"Weight matrice must be square\") ; end\n",
" \n",
- " λ = zeros(n) ; W_updated = W ; z = -1\n",
- " g = complete_graph(n-1)\n",
- " S = zeros(n,n) ; D = zeros(n)\n",
- " while D != fill(2, n) # While not a tour\n",
- " \n",
- " # Minimum spanning tree (span_tree : st)\n",
- " for u in 1:n, v in 1:n\n",
- " W_updated[u,v] -= λ[u] + λ[v]\n",
- " end\n",
- " st = kruskal_mst(g, W_updated[2:n,2:n])\n",
- " \n",
- " # Connect vertex 1\n",
- " m = Model(Cbc.Optimizer) # PL(NE) Model\n",
- " set_silent(m)\n",
- " \n",
- " @variable(m, X[1:n,1:n], Bin)\n",
+ " λ = zeros(n)\n",
+ "\n",
+ " g = complete_graph(n-1) #in the TSP the graph is complete\n",
+ " z = -1 #the objective function\n",
+ " \n",
+ " k = 0 \n",
+ " X = Matrix{Bool}(undef, (n,n)) #the adjacency matrix of the 1-tree solutions\n",
+ " D = Array{Integer}(undef, n)\n",
+ " W_updated = Matrix{Integer}(undef, (n,n))\n",
+ " \n",
+ " converge = false #whether the algorithm has converged or not\n",
+ "\n",
+ " while !converge && k <= maxiter # While not a tour\n",
" \n",
- " @objective(m, Min, \n",
- " sum(W .* X) + \n",
- " sum(λ[j]*(2-sum(X[1,:])) for j in 2:n)\n",
- " )\n",
+ " converge, z = held_and_karp_iter!(W, W_updated, λ, X, D, g, update_rule)\n",
" \n",
- " @constraint(m, degreeRoot, sum(X[1,:]) == 2)\n",
- " for i ∈ 1:size(st, 1)\n",
- " @constraint(m, X[src(st[i])+1, dst(st[i])+1] == 1)\n",
- " end\n",
- " #@constraint(m, spanningTree[i ∈ 1:size(st,1)], X[src(st[i]), dst(st[i])] == 1)\n",
- " for u ∈ 1:n\n",
- " for v ∈ 1:n\n",
- " @constraint(m, X[u,v] == X[v,u])\n",
+ " # We display the current 1-Tree\n",
+ " if disp && (k == 0 || k == maxiter || converge)\n",
+ " println(\"Iteration\", k)\n",
+ " \n",
+ " one_tree = SimpleGraph(X)\n",
+ " edgelabel = []\n",
+ " for i in 1:n, j in i:n\n",
+ " if X[i, j] == 1\n",
+ " push!(edgelabel, W_updated[i, j])\n",
+ " end\n",
" end\n",
- " @constraint(m, X[u,u] == 0)\n",
- " end\n",
- " \n",
- " #@constraint(m, diagonalMatrix[u ∈ 1:n, v ∈ 1:n], X[u,v] == X[v,u])\n",
- " \n",
- " #@constraint(m, diagonalNulle[i ∈ 1:n], X[i,i] == 0)\n",
- " \n",
- " optimize!(m)\n",
- " checkTermination(m)\n",
- " \n",
- " z = objective_value(m)\n",
- " S = value.(X)\n",
- " D = [sum(S[u,:]) for u ∈ 1:n]\n",
- " # For the degree, we suppose a null diagonal, to be checked later on\n",
- " \n",
- " if D != fill(2, size(D)) # If not a tour, update\n",
- " λ .= λ .+ 2 .* (2 .- D) # Experimenter d'autres règles\n",
+ " display(gplot(one_tree, nodelabel = 1:n, edgelabel = edgelabel, edgelabelc=\"red\"))\n",
" end\n",
" \n",
- " println(D)\n",
- " \n",
+ " k+=1\n",
+ "\n",
" end\n",
" \n",
- " return S\n",
+ " return (z, X, λ)\n",
" \n",
"end"
]
},
{
"cell_type": "code",
- "execution_count": 57,
- "metadata": {
- "tags": []
- },
+ "execution_count": 37,
+ "metadata": {},
"outputs": [
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List([Compose.Property{Compose.LineWidthPrimitive}(Compose.LineWidthPrimitive[Compose.LineWidthPrimitive(1.2247448713915892mm)]), Compose.Property{Compose.FillPrimitive}(Compose.FillPrimitive[Compose.FillPrimitive(RGBA{Float64}(0.0,0.0,0.0,0.0))]), Compose.Property{Compose.StrokePrimitive}(Compose.StrokePrimitive[Compose.StrokePrimitive(RGBA{Float64}(0.8274509803921568,0.8274509803921568,0.8274509803921568,1.0))])]), 0, false, false, false, false, nothing, nothing, 0.0, Symbol(\"\")), Compose.Context(Measures.BoundingBox{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}},Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}((0.0w, 0.0h), (1.0w, 1.0h)), nothing, nothing, nothing, nothing, List([]), List([]), List([Compose.Property{Compose.LineWidthPrimitive}(Compose.LineWidthPrimitive[Compose.LineWidthPrimitive(1.2247448713915892mm)]), Compose.Property{Compose.StrokePrimitive}(Compose.StrokePrimitive[Compose.StrokePrimitive(RGBA{Float64}(0.8274509803921568,0.8274509803921568,0.8274509803921568,1.0))])]), 0, false, false, false, false, nothing, nothing, 0.0, Symbol(\"\")), Compose.Context(Measures.BoundingBox{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}},Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}((0.0w, 0.0h), (1.0w, 1.0h)), nothing, nothing, nothing, nothing, List([]), List([]), List([Compose.Property{Compose.FontSizePrimitive}(Compose.FontSizePrimitive[Compose.FontSizePrimitive(4.0mm)]), Compose.Property{Compose.StrokePrimitive}(Compose.StrokePrimitive[Compose.StrokePrimitive(RGBA{Float64}(0.0,0.0,0.0,0.0))]), Compose.Property{Compose.FillPrimitive}(Compose.FillPrimitive[Compose.FillPrimitive(RGBA{Float64}(0.0,0.0,0.0,1.0))])]), 0, false, false, false, false, nothing, nothing, 0.0, Symbol(\"\")), 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0.04082482904638631w), Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}((-1.0cx, 0.4741823705304855cy), 0.04082482904638631w), Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}((0.04086380992427707cx, -1.0cy), 0.04082482904638631w)], Symbol(\"\"))]), List([Compose.Property{Compose.LineWidthPrimitive}(Compose.LineWidthPrimitive[Compose.LineWidthPrimitive(0.0mm)]), Compose.Property{Compose.StrokePrimitive}(Compose.StrokePrimitive[Compose.StrokePrimitive(RGBA{Float64}(0.0,0.0,0.0,0.0))]), Compose.Property{Compose.FillPrimitive}(Compose.FillPrimitive[Compose.FillPrimitive(RGBA{Float64}(0.25098039215686274,0.8784313725490196,0.8156862745098039,1.0))])]), 0, false, false, false, false, nothing, nothing, 0.0, Symbol(\"\")), Compose.Context(Measures.BoundingBox{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}},Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}((0.0w, 0.0h), (1.0w, 1.0h)), nothing, nothing, nothing, nothing, List([]), List([Compose.Form{Compose.TextPrimitive{Tuple{Measures.Length{:cx,Float64},Measures.Length{:cy,Float64}},Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}},Tuple{Measures.Length{:mm,Float64},Measures.Length{:mm,Float64}}}}(Compose.TextPrimitive{Tuple{Measures.Length{:cx,Float64},Measures.Length{:cy,Float64}},Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}},Tuple{Measures.Length{:mm,Float64},Measures.Length{:mm,Float64}}}[Compose.TextPrimitive{Tuple{Measures.Length{:cx,Float64},Measures.Length{:cy,Float64}},Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}},Tuple{Measures.Length{:mm,Float64},Measures.Length{:mm,Float64}}}((1.0cx, -0.47527601432374444cy), \"1\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm)), Compose.TextPrimitive{Tuple{Measures.Length{:cx,Float64},Measures.Length{:cy,Float64}},Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}},Tuple{Measures.Length{:mm,Float64},Measures.Length{:mm,Float64}}}((-0.038411276525742166cx, 1.0cy), \"2\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm)), Compose.TextPrimitive{Tuple{Measures.Length{:cx,Float64},Measures.Length{:cy,Float64}},Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}},Tuple{Measures.Length{:mm,Float64},Measures.Length{:mm,Float64}}}((0.9479873385735835cx, 0.5315817344232745cy), \"3\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm)), Compose.TextPrimitive{Tuple{Measures.Length{:cx,Float64},Measures.Length{:cy,Float64}},Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}},Tuple{Measures.Length{:mm,Float64},Measures.Length{:mm,Float64}}}((-0.9447889366832646cx, -0.5314965701039989cy), \"4\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm)), Compose.TextPrimitive{Tuple{Measures.Length{:cx,Float64},Measures.Length{:cy,Float64}},Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}},Tuple{Measures.Length{:mm,Float64},Measures.Length{:mm,Float64}}}((-1.0cx, 0.4741823705304855cy), \"5\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm)), Compose.TextPrimitive{Tuple{Measures.Length{:cx,Float64},Measures.Length{:cy,Float64}},Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}},Tuple{Measures.Length{:mm,Float64},Measures.Length{:mm,Float64}}}((0.04086380992427707cx, -1.0cy), \"6\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm))], Symbol(\"\"))]), List([Compose.Property{Compose.FontSizePrimitive}(Compose.FontSizePrimitive[Compose.FontSizePrimitive(4.0mm)]), Compose.Property{Compose.StrokePrimitive}(Compose.StrokePrimitive[Compose.StrokePrimitive(RGBA{Float64}(0.0,0.0,0.0,0.0))]), Compose.Property{Compose.FillPrimitive}(Compose.FillPrimitive[Compose.FillPrimitive(RGBA{Float64}(0.0,0.0,0.0,1.0))])]), 0, false, false, false, false, nothing, nothing, 0.0, Symbol(\"\"))]), List([]), List([]), 0, false, false, false, false, nothing, nothing, 0.0, Symbol(\"\"))"
- ]
- },
- "metadata": {},
- "output_type": "display_data"
- },
- {
- "name": "stdout",
- "output_type": "stream",
- "text": [
- "[2.0, 1.0, 2.0, 3.0, 2.0, 2.0]\n",
- "[2.0, 2.0, 2.0, 2.0, 2.0, 2.0]\n"
- ]
- }
- ],
- "source": [
- "display_S(hk(example))"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 56,
- "metadata": {
- "collapsed": true,
- "jupyter": {
- "outputs_hidden": true
- },
- "tags": []
- },
- "outputs": [
- {
- "name": "stdout",
- "output_type": "stream",
- "text": [
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- "["
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- },
- {
- "ename": "InterruptException",
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- "traceback": [
- "InterruptException:",
- "",
- "Stacktrace:",
- " [1] try_yieldto(::typeof(Base.ensure_rescheduled), ::Base.RefValue{Task}) at .\\task.jl:654",
- " [2] wait at .\\task.jl:710 [inlined]",
- " [3] uv_write(::Base.PipeEndpoint, ::Ptr{UInt8}, ::UInt64) at .\\stream.jl:935",
- " [4] unsafe_write(::Base.PipeEndpoint, ::Ptr{UInt8}, ::UInt64) at .\\stream.jl:1007",
- " [5] unsafe_write at .\\io.jl:593 [inlined]",
- " [6] unsafe_write(::Base.PipeEndpoint, ::Base.RefValue{UInt8}, ::Int64) at .\\io.jl:591",
- " [7] write at .\\io.jl:594 [inlined]",
- " [8] write(::Base.PipeEndpoint, ::UInt8) at .\\stream.jl:1045",
- " [9] write at .\\io.jl:309 [inlined]",
- " [10] write at .\\io.jl:647 [inlined]",
- " [11] print at .\\char.jl:229 [inlined]",
- " [12] show_delim_array(::IOContext{Base.PipeEndpoint}, ::Array{Float64,1}, ::Char, ::String, ::Char, ::Bool, ::Int64, ::Int64) at .\\show.jl:705 (repeats 2 times)",
- " [13] show_vector(::IJulia.IJuliaStdio{Base.PipeEndpoint}, ::Array{Float64,1}, ::Char, ::Char) at .\\arrayshow.jl:458",
- " [14] show_vector at .\\arrayshow.jl:447 [inlined]",
- " [15] show at .\\arrayshow.jl:420 [inlined]",
- " [16] print(::IJulia.IJuliaStdio{Base.PipeEndpoint}, ::Array{Float64,1}) at .\\strings\\io.jl:35",
- " [17] print(::IJulia.IJuliaStdio{Base.PipeEndpoint}, ::Array{Float64,1}, ::Char) at .\\strings\\io.jl:46",
- " [18] println(::IJulia.IJuliaStdio{Base.PipeEndpoint}, ::Array{Float64,1}) at .\\strings\\io.jl:73",
- " [19] println(::Array{Float64,1}) at .\\coreio.jl:4",
- " [20] hk(::Array{Int64,2}) at .\\In[54]:57",
- " [21] top-level scope at In[56]:1"
- ]
- }
- ],
- "source": [
- "display_S(hk(exercise))"
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "## Held and Karp algorithm - attempt 2\n",
- "\n",
- "Here we will work on the same Master Problem as seen in \"STSP using column generation\". \n",
- "However the subsidiary problem will be solved using the algorithm presented by Held and Karp in the section 4 \"An ascent method\" of their paper (which correspond to the one J.E. Mitchell presents in his slides)."
- ]
- },
- {
- "cell_type": "code",
- "execution_count": null,
- "metadata": {},
- "outputs": [],
- "source": [
- "function held_and_karp_algo()\n",
- " \n",
- "end"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": null,
- "metadata": {},
- "outputs": [],
- "source": [
- "function update_rule!(λ, D, Ï)\n",
- " λ .= λ .+ Ï .* (2 .- D)\n",
- "end\n",
- "\n",
- "update_rule_1!(λ, D) = update_rule!(λ, D, 1)\n",
- "update_rule_2!(λ, D) = update_rule!(λ, D, 2)\n",
- "update_rule_3!(λ, D) = update_rule!(λ, D, 3)\n",
- "update_rule_05!(λ, D) = update_rule!(λ, D, 0.5)"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": null,
- "metadata": {},
- "outputs": [],
- "source": [
- "function held_and_karp_2(W)\n",
- " n = size(W, 1)\n",
- " one_trees_init = wheel_like_1trees(n)\n",
- " \n",
- " columns = one_trees_init[1:n-2]\n",
- " tree_degrees = [sum(columns[t][u,:]) for u ∈ 1:n, t ∈ 1:n-2]\n",
- " tree_weights = [dot(columns[t], W) for t ∈ 1:n-2]\n",
- " \n",
- " reduced_cost = -1\n",
- " new_column = one_trees_init[n-1]\n",
- " new_degree = [sum(new_column[u,:]) for u ∈ 1:n]\n",
- " new_weight = dot(new_column, W)\n",
- " \n",
- " while reduced_cost < 0\n",
- " \n",
- " push!(columns, new_column)\n",
- " tree_degrees = hcat(tree_degrees, new_degree)\n",
- " push!(tree_weights, new_weights)\n",
- " \n",
- " dm = DM(tree_weights, tree_degrees)\n",
- " π = value.(dm[:π])\n",
- " μ = value.(dm[:μ])\n",
- " \n",
- " new_column = held_and_karp_algo(W, , , update_rule_2!)\n",
- " new_degree = [sum(new_column[u,:]) for u ∈ 1:n]\n",
- " new_weight = dot(new_column, W)\n",
- " \n",
- " reduced_cost = new_weight - dot(π, new_degree) - μ\n",
- " end\n",
- " \n",
- " lpm = LPM(tree_weights, tree_degrees)\n",
- " # ipm = IPM(tree_weights, tree_degrees)\n",
- " \n",
- " λ = value.(lpm[:λ])\n",
- " println(\"Lambda_t : \", λ)\n",
- " println(\"Minimal 1-tree \")\n",
- " println(\"Lambda : \", maximum(λ))\n",
- " println(\"Index : \", argmax(λ))\n",
- " println(\"1-tree : \", columns[argmax(λ)])\n",
- " \n",
- " return columns[argmax(λ)] \n",
- "end"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": null,
- "metadata": {},
- "outputs": [],
- "source": [
- "display_S(held_and_karp_2(W))\n",
- "display_S(held_and_karp_2(W2))"
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "## The Held and Karp algorithm - dynamical programming\n",
- "\n",
- "Whilst column generation based techniques work well on wide graphs, on smaller ones we may want to use a different method.\n",
- "\n",
- "Here we use dynamical programming to implement the Bellman-Held-Karp algorithm. \n",
- "It works well on small graphs, but scales terribly because of its exponential complexity $O(2^n n^2)$ in time (and $O(2^n n)$ in space)."
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 40,
- "metadata": {},
- "outputs": [
- {
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},
- "execution_count": 40,
"metadata": {},
- "output_type": "execute_result"
- }
- ],
- "source": [
- "function dp_held_karp(W,n)\n",
- " C = Dict{Tuple{Array,Integer},Integer}()\n",
- "\n",
- " for k in collect(2:n)\n",
- " C[(Array([k]),k)]= collect(W[1,k])[1]\n",
- " end\n",
- " \n",
- " for s in 2:(n-1), S in collect(combinations(collect(2:n),s)), k in S\n",
- " Sp = [s for s in S if (s!=k)]\n",
- " C[(S,k)] = collect(typemax(Int64))[1]\n",
- " for m in S\n",
- " if ( m != k ) && ( C[(S,k)] >= C[(Sp,m)] + W[m,k] )\n",
- " C[(S,k)] = C[(Sp,m)] + W[m,k]\n",
- " end\n",
- " end\n",
- " end\n",
- " \n",
- " opt = Inf\n",
- " \n",
- " for k in 2:n\n",
- " if opt >= C[(collect(2:n),k)] + W[k,1]\n",
- " opt = C[(collect(2:n),k)] + W[k,1]\n",
- " end\n",
- " end\n",
- " \n",
- " return opt\n",
- "end\n",
- "\n",
- "function getPath(W,n,opt)\n",
- " println(\"Valeur optimale : \",opt)\n",
- " S = collect(permutations(collect(1:n),n))\n",
- " idgood = Integer[]\n",
- " Sp = [Integer[]]\n",
- " M=Array{Integer,2}[]\n",
- "\n",
- " for i in 1:length(S)\n",
- " if (sommeTour(W,S[i],opt))\n",
- " append!(idgood,i)\n",
- " end\n",
- " end\n",
- "\n",
- " for index in idgood\n",
- " if !(matriceAdj(S[index]) in M)\n",
- " append!(M,[matriceAdj(S[index])])\n",
- " append!(Sp,[S[index]])\n",
- " end\n",
- " end\n",
- " println(\"Circuits optimaux : \")\n",
- " for i in 1:length(Sp)\n",
- " if length(Sp[i]) != 0\n",
- " println(Sp[i])\n",
- " end \n",
- " end\n",
- " \n",
- " return M\n",
- "end\n",
- "\n",
- "function sommeTour(W,L,opt)\n",
- " somme = 0\n",
- " n = length(L)\n",
- " for i in 1:(n-1)\n",
- " somme+= W[L[i] ,L[i+1]]\n",
- " end\n",
- " somme+= W[L[n],L[1]]\n",
- " return (somme==opt)\n",
- "end\n",
- "function matriceAdj(L)\n",
- " n = length(L)\n",
- " Madj = zeros(Integer,n,n)\n",
- " for i in 1:(n-1)\n",
- " Madj[L[i],L[i+1]] = 1\n",
- " Madj[L[i+1],L[i]] = 1\n",
- " end\n",
- " Madj[L[n],L[1]] = 1\n",
- " Madj[L[1],L[n]] = 1\n",
- " return Madj\n",
- "end\n",
- "\n",
- "bellman_held_karp(W) = getPath(W, size(W,1), dp_held_karp(W, size(W,1)))"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 44,
- "metadata": {},
- "outputs": [
+ "output_type": "display_data"
+ },
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" <text text-anchor=\"middle\" dy=\"0.35em\">6</text>\n",
" </g>\n",
@@ -3845,7 +2406,7 @@
"</svg>\n"
],
"text/plain": [
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(0.9357713630269383cx, 0.4654592372262474cy)]), Compose.LinePrimitive{Tuple{Measures.Measure,Measures.Measure}}(Tuple{Measures.Measure,Measures.Measure}[(-0.8333544840282768cx, -0.614346110596245cy), (-0.028233742974402065cx, -0.9597601280006668cy)]), Compose.LinePrimitive{Tuple{Measures.Measure,Measures.Measure}}(Tuple{Measures.Measure,Measures.Measure}[(-0.9345049927174921cx, -0.4723095948588338cy), (-0.9926440506846841cx, 0.33225834293691253cy)]), Compose.LinePrimitive{Tuple{Measures.Measure,Measures.Measure}}(Tuple{Measures.Measure,Measures.Measure}[(-0.1668476973223338cx, 0.946527442010796cy), (-0.9130669671024996cx, 0.48752754466419457cy)]), Compose.LinePrimitive{Tuple{Measures.Measure,Measures.Measure}}(Tuple{Measures.Measure,Measures.Measure}[(0.01387512934470543cx, 0.9597490217344468cy), (0.8346407113781176cx, 0.6075079486309175cy)])], Symbol(\"\"))]), List([Compose.Property{Compose.LineWidthPrimitive}(Compose.LineWidthPrimitive[Compose.LineWidthPrimitive(1.2247448713915892mm)]), Compose.Property{Compose.FillPrimitive}(Compose.FillPrimitive[Compose.FillPrimitive(RGBA{Float64}(0.0,0.0,0.0,0.0))]), Compose.Property{Compose.StrokePrimitive}(Compose.StrokePrimitive[Compose.StrokePrimitive(RGBA{Float64}(0.8274509803921568,0.8274509803921568,0.8274509803921568,1.0))])]), 0, false, false, false, false, nothing, nothing, 0.0, Symbol(\"\")), Compose.Context(Measures.BoundingBox{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}},Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}((0.0w, 0.0h), (1.0w, 1.0h)), nothing, nothing, nothing, nothing, List([]), List([]), List([Compose.Property{Compose.LineWidthPrimitive}(Compose.LineWidthPrimitive[Compose.LineWidthPrimitive(1.2247448713915892mm)]), 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Compose.Context(Measures.BoundingBox{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}},Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}((0.0w, 0.0h), (1.0w, 1.0h)), nothing, nothing, nothing, nothing, List([]), List([Compose.Form{Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}}(Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}[Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}((1.0cx, -0.4252172687072213cy), 0.04082482904638631w), Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}((-0.9271490434021763cx, -0.5741062385969118cy), 0.04082482904638631w), Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}((0.0655608163994974cx, -1.0cy), 0.04082482904638631w), Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}((-0.07991466442483341cx, 1.0cy), 0.04082482904638631w), Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}((-1.0cx, 0.43405498667499054cy), 0.04082482904638631w), Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}((0.9284305051476565cx, 0.5672569703653643cy), 0.04082482904638631w)], Symbol(\"\"))]), List([Compose.Property{Compose.LineWidthPrimitive}(Compose.LineWidthPrimitive[Compose.LineWidthPrimitive(0.0mm)]), Compose.Property{Compose.StrokePrimitive}(Compose.StrokePrimitive[Compose.StrokePrimitive(RGBA{Float64}(0.0,0.0,0.0,0.0))]), Compose.Property{Compose.FillPrimitive}(Compose.FillPrimitive[Compose.FillPrimitive(RGBA{Float64}(0.25098039215686274,0.8784313725490196,0.8156862745098039,1.0))])]), 0, false, false, false, false, nothing, nothing, 0.0, Symbol(\"\")), Compose.Context(Measures.BoundingBox{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}},Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}((0.0w, 0.0h), (1.0w, 1.0h)), nothing, nothing, nothing, nothing, List([]), List([Compose.Form{Compose.TextPrimitive{Tuple{Measures.Length{:cx,Float64},Measures.Length{:cy,Float64}},Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}},Tuple{Measures.Length{:mm,Float64},Measures.Length{:mm,Float64}}}}(Compose.TextPrimitive{Tuple{Measures.Length{:cx,Float64},Measures.Length{:cy,Float64}},Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}},Tuple{Measures.Length{:mm,Float64},Measures.Length{:mm,Float64}}}[Compose.TextPrimitive{Tuple{Measures.Length{:cx,Float64},Measures.Length{:cy,Float64}},Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}},Tuple{Measures.Length{:mm,Float64},Measures.Length{:mm,Float64}}}((1.0cx, -0.4252172687072213cy), \"1\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm)), 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Compose.TextPrimitive{Tuple{Measures.Length{:cx,Float64},Measures.Length{:cy,Float64}},Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}},Tuple{Measures.Length{:mm,Float64},Measures.Length{:mm,Float64}}}((-0.07991466442483341cx, 1.0cy), \"4\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm)), Compose.TextPrimitive{Tuple{Measures.Length{:cx,Float64},Measures.Length{:cy,Float64}},Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}},Tuple{Measures.Length{:mm,Float64},Measures.Length{:mm,Float64}}}((-1.0cx, 0.43405498667499054cy), \"5\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm)), Compose.TextPrimitive{Tuple{Measures.Length{:cx,Float64},Measures.Length{:cy,Float64}},Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}},Tuple{Measures.Length{:mm,Float64},Measures.Length{:mm,Float64}}}((0.9284305051476565cx, 0.5672569703653643cy), \"6\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm))], Symbol(\"\"))]), List([Compose.Property{Compose.FontSizePrimitive}(Compose.FontSizePrimitive[Compose.FontSizePrimitive(4.0mm)]), Compose.Property{Compose.StrokePrimitive}(Compose.StrokePrimitive[Compose.StrokePrimitive(RGBA{Float64}(0.0,0.0,0.0,0.0))]), Compose.Property{Compose.FillPrimitive}(Compose.FillPrimitive[Compose.FillPrimitive(RGBA{Float64}(0.0,0.0,0.0,1.0))])]), 0, false, false, false, false, nothing, nothing, 0.0, Symbol(\"\"))]), List([]), List([]), 0, false, false, false, false, nothing, nothing, 0.0, Symbol(\"\"))"
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(0.862185718536918cx, 0.5192690346857449cy)]), Compose.LinePrimitive{Tuple{Measures.Measure,Measures.Measure}}(Tuple{Measures.Measure,Measures.Measure}[(-0.751163776893359cx, -0.6562606738470509cy), (0.06317711527517773cx, -0.9639283544682902cy)]), Compose.LinePrimitive{Tuple{Measures.Measure,Measures.Measure}}(Tuple{Measures.Measure,Measures.Measure}[(-0.8621403518525819cx, -0.5193110213262904cy), (-0.9844985686924176cx, 0.2769543562778442cy)]), Compose.LinePrimitive{Tuple{Measures.Measure,Measures.Measure}}(Tuple{Measures.Measure,Measures.Measure}[(-0.2407353218862397cx, 0.9393151476309368cy), (-0.9179389534577357cx, 0.43851721563595814cy)]), Compose.LinePrimitive{Tuple{Measures.Measure,Measures.Measure}}(Tuple{Measures.Measure,Measures.Measure}[(-0.06319957942107789cx, 0.9639271694332573cy), (0.7512138600667299cx, 0.6562205281086612cy)])], Symbol(\"\"))]), List([Compose.Property{Compose.LineWidthPrimitive}(Compose.LineWidthPrimitive[Compose.LineWidthPrimitive(1.2247448713915892mm)]), Compose.Property{Compose.FillPrimitive}(Compose.FillPrimitive[Compose.FillPrimitive(RGBA{Float64}(0.0,0.0,0.0,0.0))]), Compose.Property{Compose.StrokePrimitive}(Compose.StrokePrimitive[Compose.StrokePrimitive(RGBA{Float64}(0.8274509803921568,0.8274509803921568,0.8274509803921568,1.0))])]), 0, false, false, false, false, nothing, nothing, 0.0, Symbol(\"\")), Compose.Context(Measures.BoundingBox{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}},Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}((0.0w, 0.0h), (1.0w, 1.0h)), nothing, nothing, nothing, nothing, List([]), List([]), List([Compose.Property{Compose.LineWidthPrimitive}(Compose.LineWidthPrimitive[Compose.LineWidthPrimitive(1.2247448713915892mm)]), 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Compose.TextPrimitive{Tuple{Measures.Length{:cx,Float64},Measures.Length{:cy,Float64}},Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}},Tuple{Measures.Length{:mm,Float64},Measures.Length{:mm,Float64}}}((-0.9233194602724997cx, -0.12117833252422305cy), \"0\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm)), Compose.TextPrimitive{Tuple{Measures.Length{:cx,Float64},Measures.Length{:cy,Float64}},Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}},Tuple{Measures.Length{:mm,Float64},Measures.Length{:mm,Float64}}}((-0.5793371376719877cx, 0.6889161816334475cy), \"5\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm)), Compose.TextPrimitive{Tuple{Measures.Length{:cx,Float64},Measures.Length{:cy,Float64}},Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}},Tuple{Measures.Length{:mm,Float64},Measures.Length{:mm,Float64}}}((0.344007140322826cx, 0.8100738487709592cy), \"5\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm))], Symbol(\"\"))]), List([Compose.Property{Compose.FontSizePrimitive}(Compose.FontSizePrimitive[Compose.FontSizePrimitive(4.0mm)]), Compose.Property{Compose.StrokePrimitive}(Compose.StrokePrimitive[Compose.StrokePrimitive(RGBA{Float64}(0.0,0.0,0.0,0.0))]), Compose.Property{Compose.FillPrimitive}(Compose.FillPrimitive[Compose.FillPrimitive(RGBA{Float64}(1.0,0.0,0.0,1.0))])]), 0, false, false, false, false, nothing, nothing, 0.0, Symbol(\"\")), Compose.Context(Measures.BoundingBox{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}},Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}((0.0w, 0.0h), (1.0w, 1.0h)), nothing, nothing, nothing, nothing, List([]), List([Compose.Form{Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}}(Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}[Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}((1.0cx, -0.3778319922961574cy), 0.04082482904638631w), Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}((-0.8466389205449993cx, -0.6201890283153411cy), 0.04082482904638631w), Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}((0.158652258926818cx, -1.0cy), 0.04082482904638631w), Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}((-0.15867427534397538cx, 1.0cy), 0.04082482904638631w), Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}((-1.0cx, 0.37783236326689495cy), 0.04082482904638631w), Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}((0.8466885559896273cx, 0.6201476975419185cy), 0.04082482904638631w)], Symbol(\"\"))]), List([Compose.Property{Compose.LineWidthPrimitive}(Compose.LineWidthPrimitive[Compose.LineWidthPrimitive(0.0mm)]), Compose.Property{Compose.StrokePrimitive}(Compose.StrokePrimitive[Compose.StrokePrimitive(RGBA{Float64}(0.0,0.0,0.0,0.0))]), Compose.Property{Compose.FillPrimitive}(Compose.FillPrimitive[Compose.FillPrimitive(RGBA{Float64}(0.25098039215686274,0.8784313725490196,0.8156862745098039,1.0))])]), 0, false, false, false, false, nothing, nothing, 0.0, Symbol(\"\")), Compose.Context(Measures.BoundingBox{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}},Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}((0.0w, 0.0h), (1.0w, 1.0h)), nothing, nothing, 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},
"metadata": {},
@@ -3855,21 +2416,28 @@
"name": "stdout",
"output_type": "stream",
"text": [
- "Valeur optimale : 22\n",
- "Circuits optimaux : \n",
- "Integer[1, 3, 2, 5, 4, 6]\n"
+ "Iteration0\n",
+ "Iteration2\n"
]
+ },
+ {
+ "data": {
+ "text/plain": [
+ "(22.0, Bool[0 0 … 0 1; 0 0 … 1 0; … ; 0 1 … 0 0; 1 0 … 0 0], [0.0, 2.0, 0.0, 0.0, 0.0, -2.0])"
+ ]
+ },
+ "execution_count": 37,
+ "metadata": {},
+ "output_type": "execute_result"
}
],
"source": [
- "for M in bellman_held_karp(example)\n",
- " display_S(M)\n",
- "end"
+ "z, X, λ = hk_mitchell(example; disp=true, maxiter=100000)"
]
},
{
"cell_type": "code",
- "execution_count": 45,
+ "execution_count": 38,
"metadata": {},
"outputs": [
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(-0.9906669204788564cx, -0.28586234753154277cy)]), Compose.LinePrimitive{Tuple{Measures.Measure,Measures.Measure}}(Tuple{Measures.Measure,Measures.Measure}[(0.591439282824431cx, 0.7551965202109556cy), (-0.06650014380205842cx, 0.9705999768482185cy)]), Compose.LinePrimitive{Tuple{Measures.Measure,Measures.Measure}}(Tuple{Measures.Measure,Measures.Measure}[(0.7143643959035942cx, 0.6373015195482804cy), (0.9668758338381501cx, -0.03731150805446408cy)]), Compose.LinePrimitive{Tuple{Measures.Measure,Measures.Measure}}(Tuple{Measures.Measure,Measures.Measure}[(-0.9318462432934226cx, -0.4453409111446276cy), (-0.42242337422504794cx, -0.9345505013361288cy)]), Compose.LinePrimitive{Tuple{Measures.Measure,Measures.Measure}}(Tuple{Measures.Measure,Measures.Measure}[(-0.26052313330936977cx, -0.9881607379028823cy), (0.43037222063801533cx, -0.9009074265272865cy)]), Compose.LinePrimitive{Tuple{Measures.Measure,Measures.Measure}}(Tuple{Measures.Measure,Measures.Measure}[(0.5741114442909788cx, 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@@ -4608,13 +3323,259 @@
"name": "stdout",
"output_type": "stream",
"text": [
- "Valeur optimale : 4\n",
- "Circuits optimaux : \n",
- "Integer[1, 3, 2, 7, 6, 5, 4]\n",
- "Integer[1, 3, 7, 2, 5, 6, 4]\n"
+ "\n",
+ "Iteration100000\n"
+ ]
+ },
+ {
+ "data": {
+ "text/plain": [
+ "(3.0, Bool[0 0 … 0 0; 0 0 … 0 1; … ; 0 0 … 0 1; 0 1 … 1 0], [0.0, -1.0, 0.0, 0.0, 0.0, 1.0, 0.0])"
+ ]
+ },
+ "execution_count": 38,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "z, X, λ = hk_mitchell(exercise; disp=true, maxiter=100000, update_rule=update_rule_1!)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Linear changes in the update rule do not seem to impact the results"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Held and Karp algorithm - attempt 2\n",
+ "\n",
+ "Here we will work on the same Master Problem as seen in \"STSP using column generation\". \n",
+ "However the subsidiary problem will be solved using the algorithm presented by Held and Karp in the section 4 \"An ascent method\" of their paper (which correspond to the one J.E. Mitchell presents in his slides)."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 46,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "held_and_karp_2 (generic function with 1 method)"
+ ]
+ },
+ "execution_count": 46,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "function held_and_karp_2(W; update_rule = update_rule_1!)\n",
+ " n = size(W, 1)\n",
+ " one_trees_init = wheel_like_1trees(n)\n",
+ " \n",
+ " columns = one_trees_init[1:n-2]\n",
+ " tree_degrees = [sum(columns[t][u,:]) for u ∈ 1:n, t ∈ 1:n-2]\n",
+ " tree_weights = [dot(columns[t], W)/2 for t ∈ 1:n-2]\n",
+ " \n",
+ " reduced_cost = -1\n",
+ " new_column = one_trees_init[n-1]\n",
+ " new_degree = [sum(new_column[u,:]) for u ∈ 1:n]\n",
+ " new_weight = dot(new_column, W)/2\n",
+ " \n",
+ " g = complete_graph(n-1)\n",
+ " converge = false\n",
+ " \n",
+ " while reduced_cost < 0\n",
+ " \n",
+ " push!(columns, new_column)\n",
+ " tree_degrees = hcat(tree_degrees, new_degree)\n",
+ " push!(tree_weights, new_weight)\n",
+ " \n",
+ " dm = DM(tree_weights, tree_degrees)\n",
+ " π = value.(dm[:π])\n",
+ " μ = value.(dm[:μ])\n",
+ " \n",
+ " \n",
+ " held_and_karp_iter!(W, deepcopy(W), vcat([μ], π), new_column, new_degree, g, update_rule)\n",
+ " #new_column = held_and_karp_algo(W, , , update_rule_2!)\n",
+ " #reduced_cost, new_column, _ = hk_mitchell(W; disp=false, maxiter=100000, update_rule=update_rule_1!)\n",
+ " #new_degree = [sum(new_column[u,:]) for u ∈ 1:n]\n",
+ " new_weight = dot(new_column, W)/2\n",
+ " reduced_cost = new_weight - dot(vcat([μ], π), new_degree)\n",
+ " end\n",
+ " \n",
+ " lpm = LPM(tree_weights, tree_degrees)\n",
+ " # ipm = IPM(tree_weights, tree_degrees)\n",
+ " \n",
+ " λ = value.(lpm[:λ])\n",
+ " println(\"Lambda_t : \", λ)\n",
+ " println(\"Minimal 1-tree \")\n",
+ " println(\"Lambda : \", maximum(λ))\n",
+ " println(\"Index : \", argmax(λ))\n",
+ " println(\"1-tree : \", columns[argmax(λ)])\n",
+ " \n",
+ " return columns[argmax(λ)] \n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 47,
+ "metadata": {},
+ "outputs": [
+ {
+ "ename": "InexactError",
+ "evalue": "InexactError: Int64(12.439024390243906)",
+ "output_type": "error",
+ "traceback": [
+ "InexactError: Int64(12.439024390243906)",
+ "",
+ "Stacktrace:",
+ " [1] Int64 at .\\float.jl:710 [inlined]",
+ " [2] convert at .\\number.jl:7 [inlined]",
+ " [3] setindex! at .\\array.jl:828 [inlined]",
+ " [4] held_and_karp_iter!(::Array{Int64,2}, ::Array{Int64,2}, ::Array{Float64,1}, ::Array{Float64,2}, ::Array{Float64,1}, ::SimpleGraph{Int64}, ::typeof(update_rule_1!)) at .\\In[33]:8",
+ " [5] held_and_karp_2(::Array{Int64,2}; update_rule::Function) at .\\In[46]:28",
+ " [6] held_and_karp_2(::Array{Int64,2}) at .\\In[46]:2",
+ " [7] top-level scope at In[47]:1"
]
}
],
+ "source": [
+ "display_S(held_and_karp_2(example))"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "display_S(held_and_karp_2(exercise))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## The Held and Karp algorithm - dynamical programming\n",
+ "\n",
+ "Whilst column generation based techniques work well on wide graphs, on smaller ones we may want to use a different method.\n",
+ "\n",
+ "Here we use dynamical programming to implement the Bellman-Held-Karp algorithm. \n",
+ "It works well on small graphs, but scales terribly because of its exponential complexity $O(2^n n^2)$ in time (and $O(2^n n)$ in space)."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "function dp_held_karp(W,n)\n",
+ " C = Dict{Tuple{Array,Integer},Integer}()\n",
+ "\n",
+ " for k in collect(2:n)\n",
+ " C[(Array([k]),k)]= collect(W[1,k])[1]\n",
+ " end\n",
+ " \n",
+ " for s in 2:(n-1), S in collect(combinations(collect(2:n),s)), k in S\n",
+ " Sp = [s for s in S if (s!=k)]\n",
+ " C[(S,k)] = collect(typemax(Int64))[1]\n",
+ " for m in S\n",
+ " if ( m != k ) && ( C[(S,k)] >= C[(Sp,m)] + W[m,k] )\n",
+ " C[(S,k)] = C[(Sp,m)] + W[m,k]\n",
+ " end\n",
+ " end\n",
+ " end\n",
+ " \n",
+ " opt = Inf\n",
+ " \n",
+ " for k in 2:n\n",
+ " if opt >= C[(collect(2:n),k)] + W[k,1]\n",
+ " opt = C[(collect(2:n),k)] + W[k,1]\n",
+ " end\n",
+ " end\n",
+ " \n",
+ " return opt\n",
+ "end\n",
+ "\n",
+ "function getPath(W,n,opt)\n",
+ " println(\"Valeur optimale : \",opt)\n",
+ " S = collect(permutations(collect(1:n),n))\n",
+ " idgood = Integer[]\n",
+ " Sp = [Integer[]]\n",
+ " M=Array{Integer,2}[]\n",
+ "\n",
+ " for i in 1:length(S)\n",
+ " if (sommeTour(W,S[i],opt))\n",
+ " append!(idgood,i)\n",
+ " end\n",
+ " end\n",
+ "\n",
+ " for index in idgood\n",
+ " if !(matriceAdj(S[index]) in M)\n",
+ " append!(M,[matriceAdj(S[index])])\n",
+ " append!(Sp,[S[index]])\n",
+ " end\n",
+ " end\n",
+ " println(\"Circuits optimaux : \")\n",
+ " for i in 1:length(Sp)\n",
+ " if length(Sp[i]) != 0\n",
+ " println(Sp[i])\n",
+ " end \n",
+ " end\n",
+ " \n",
+ " return M\n",
+ "end\n",
+ "\n",
+ "function sommeTour(W,L,opt)\n",
+ " somme = 0\n",
+ " n = length(L)\n",
+ " for i in 1:(n-1)\n",
+ " somme+= W[L[i] ,L[i+1]]\n",
+ " end\n",
+ " somme+= W[L[n],L[1]]\n",
+ " return (somme==opt)\n",
+ "end\n",
+ "function matriceAdj(L)\n",
+ " n = length(L)\n",
+ " Madj = zeros(Integer,n,n)\n",
+ " for i in 1:(n-1)\n",
+ " Madj[L[i],L[i+1]] = 1\n",
+ " Madj[L[i+1],L[i]] = 1\n",
+ " end\n",
+ " Madj[L[n],L[1]] = 1\n",
+ " Madj[L[1],L[n]] = 1\n",
+ " return Madj\n",
+ "end\n",
+ "\n",
+ "bellman_held_karp(W) = getPath(W, size(W,1), dp_held_karp(W, size(W,1)))"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "for M in bellman_held_karp(example)\n",
+ " display_S(M)\n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
"source": [
"for M in bellman_held_karp(exercise)\n",
" display_S(M)\n",
@@ -4643,27 +3604,14 @@
},
{
"cell_type": "code",
- "execution_count": 3,
+ "execution_count": null,
"metadata": {
- "collapsed": true,
"jupyter": {
- "outputs_hidden": true,
"source_hidden": true
},
"tags": []
},
- "outputs": [
- {
- "data": {
- "text/plain": [
- "columnGeneration (generic function with 1 method)"
- ]
- },
- "execution_count": 3,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
+ "outputs": [],
"source": [
"function columnGeneration(W)\n",
" \n",