From 2360c26e35bf1ab731fea2e4387a5f364c4ee556 Mon Sep 17 00:00:00 2001
From: Inako <baptiste.signolle@free.fr>
Date: Wed, 31 Mar 2021 19:30:57 +0200
Subject: [PATCH] explicit commit msg go brrr
---
.ipynb_checkpoints/README-checkpoint.md | 3 +
.ipynb_checkpoints/tsp_hk-checkpoint.jl | 136 +
heldAndKarp.ipynb | 3091 ++++++++++++++---
...Column_Generation_Flowchart-checkpoint.png | Bin
.../wheel_like_1-trees-checkpoint.png | Bin
images/Column_Generation_Flowchart.png | Bin 0 -> 16576 bytes
example_graph.png => images/example_graph.png | Bin
.../example_graph_solution.png | Bin
.../exercise_graph.png | Bin
images/wheel_like_1-trees.png | Bin 0 -> 56918 bytes
10 files changed, 2706 insertions(+), 524 deletions(-)
create mode 100644 .ipynb_checkpoints/README-checkpoint.md
create mode 100644 .ipynb_checkpoints/tsp_hk-checkpoint.jl
rename Column_Generation_Flowchart.png => images/.ipynb_checkpoints/Column_Generation_Flowchart-checkpoint.png (100%)
rename wheel_like_1-trees.png => images/.ipynb_checkpoints/wheel_like_1-trees-checkpoint.png (100%)
create mode 100644 images/Column_Generation_Flowchart.png
rename example_graph.png => images/example_graph.png (100%)
rename example_graph_solution.png => images/example_graph_solution.png (100%)
rename exercise_graph.png => images/exercise_graph.png (100%)
create mode 100644 images/wheel_like_1-trees.png
diff --git a/.ipynb_checkpoints/README-checkpoint.md b/.ipynb_checkpoints/README-checkpoint.md
new file mode 100644
index 0000000..2d94c38
--- /dev/null
+++ b/.ipynb_checkpoints/README-checkpoint.md
@@ -0,0 +1,3 @@
+# TSPNotTheSchool
+
+Julia module for the TSP using the Held and Kard relaxation
\ No newline at end of file
diff --git a/.ipynb_checkpoints/tsp_hk-checkpoint.jl b/.ipynb_checkpoints/tsp_hk-checkpoint.jl
new file mode 100644
index 0000000..2732f94
--- /dev/null
+++ b/.ipynb_checkpoints/tsp_hk-checkpoint.jl
@@ -0,0 +1,136 @@
+#=using LinearAlgebra
+using JuMP
+using GLPK
+using LightGraphs
+using GraphPlot=#
+using Combinatorics
+#=function algorithm TSP (G, n) is
+ for k := 2 to n do
+ C({k}, k) := d1,k
+ end for
+
+ for s := 2 to n−1 do
+ for all S ⊆ {2, . . . , n}, |S| = s do
+ for all k ∈ S do
+ C(S, k) := minm≠k,m∈S [C(S\{k}, m) + dm,k]
+ end for
+ end for
+ end for
+
+ opt := mink≠1 [C({2, 3, . . . , n}, k) + dk, 1]
+ return (opt)
+end function=#
+ #C[ ([1],1) ] = 1
+ #dico[(array,entier)] = valeur
+function dp_held_karp(W,n)
+ C = Dict{Tuple{Array,Integer},Integer}()
+
+ for k in collect(2:n)
+ C[(Array([k]),k)]= collect(W[1,k])[1]
+ end
+
+ for s in 2:(n-1)
+ for S in collect(combinations(collect(2:n),s))
+ #println(S)
+ for k in S
+ #println(k)
+ Sp = [s for s in S if (s!=k)]
+ #print(Sp)
+ C[(S,k)] = collect(20000000)[1]
+ for m in S
+ if m != k
+ if C[(S,k)] >= C[(Sp,m)] + W[m,k]
+ C[(S,k)] = C[(Sp,m)] + W[m,k]
+ end
+ end
+ end
+ end
+ #println("")=#
+ #print(S)
+ #println("")
+ end
+ end
+ opt = 2000000000
+
+ for k in 2:n
+ if opt >= C[(collect(2:n),k)] + W[k,1]
+ opt = C[(collect(2:n),k)] + W[k,1]
+ end
+ end
+
+ return opt
+end
+
+function getPath(W,n,opt)
+ println("Valeur optimale : ",opt)
+ S = collect(permutations(collect(1:n),n))
+ idgood = Integer[]
+ Sp = [Integer[]]
+ M=Array{Integer,2}[]#unique(map(matriceAdj,S))
+ #Ss = []
+ if (true)
+ for i in 1:length(S)
+ if (sommeTour(W,S[i],opt))
+ append!(idgood,i)
+ end
+ end
+
+ end
+ for index in idgood
+ if !(matriceAdj(S[index]) in M)
+ append!(M,[matriceAdj(S[index])])
+ append!(Sp,[S[index]])
+ end
+ end
+ println("Circuits optimaux : ")
+ for i in 1:length(Sp)
+ if length(Sp[i]) != 0
+ println(Sp[i])
+ end
+ end
+ #print(idgood)
+ return
+end
+
+function sommeTour(W,L,opt)
+ somme = 0
+ n = length(L)
+ for i in 1:(n-1)
+ somme+= W[L[i] ,L[i+1]]
+ end
+ somme+= W[L[n],L[1]]
+ return (somme==opt)
+end
+function matriceAdj(L)
+ n = length(L)
+ Madj = zeros(Integer,n,n)
+ for i in 1:(n-1)
+ Madj[L[i],L[i+1]] = 1
+ Madj[L[i+1],L[i]] = 1
+ end
+ Madj[L[n],L[1]] = 1
+ Madj[L[1],L[n]] = 1
+ return Madj
+end
+W = [
+ 0 8 4 15 15 3 ;
+ 8 0 5 15 2 15 ;
+ 4 5 0 6 15 15 ;
+ 15 15 6 0 5 3 ;
+ 15 2 15 5 0 4 ;
+ 3 15 15 3 4 0
+ ]
+n1=size(W,1)
+W2 = [
+ 0 15 0 0 15 15 15 ;
+ 15 0 1 15 0 15 1 ;
+ 0 1 0 15 15 15 1 ;
+ 0 15 15 0 1 1 15 ;
+ 15 0 15 1 0 1 15 ;
+ 15 15 15 1 1 0 0 ;
+ 15 1 1 15 15 0 0 ;
+ ]
+n2 = size(W2,1)
+
+getPath(W,n1,dp_held_karp(W,n1))
+getPath(W2,n2,dp_held_karp(W2,n2))
\ No newline at end of file
diff --git a/heldAndKarp.ipynb b/heldAndKarp.ipynb
index 9ebd2e6..41d38fc 100644
--- a/heldAndKarp.ipynb
+++ b/heldAndKarp.ipynb
@@ -53,8 +53,13 @@
},
{
"cell_type": "code",
- "execution_count": 1,
- "metadata": {},
+ "execution_count": 9,
+ "metadata": {
+ "jupyter": {
+ "source_hidden": true
+ },
+ "tags": []
+ },
"outputs": [
{
"name": "stdout",
@@ -69,15 +74,18 @@
"using JuMP # Modeling language\n",
"using GLPK, Cbc # LP and MIP solvers\n",
"using LightGraphs # Efficient graph structures & functions\n",
- "using GraphPlot # Graph visualization \n",
+ "using GraphPlot # Graph visualization\n",
+ "using Combinatorics # Combinatorials\n",
"\n",
"println(\"Modules loaded\")"
]
},
{
"cell_type": "code",
- "execution_count": 2,
- "metadata": {},
+ "execution_count": 48,
+ "metadata": {
+ "tags": []
+ },
"outputs": [
{
"data": {
@@ -85,7 +93,7 @@
"checkTermination (generic function with 2 methods)"
]
},
- "execution_count": 2,
+ "execution_count": 48,
"metadata": {},
"output_type": "execute_result"
}
@@ -111,8 +119,13 @@
},
{
"cell_type": "code",
- "execution_count": 22,
- "metadata": {},
+ "execution_count": 3,
+ "metadata": {
+ "jupyter": {
+ "source_hidden": true
+ },
+ "tags": []
+ },
"outputs": [
{
"data": {
@@ -120,7 +133,7 @@
"display_S (generic function with 2 methods)"
]
},
- "execution_count": 22,
+ "execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
@@ -138,11 +151,9 @@
" if m != n\n",
" error(\"display_S : S and W don't match ; S is of size \", n, \", W of size \", m)\n",
" end\n",
- " for i in 1:n\n",
- " for j in i:n\n",
- " if X[i, j] == 1\n",
- " push!(edgelabel, W[i, j])\n",
- " end\n",
+ " for i in 1:n, j in i:n\n",
+ " if X[i, j] == 1\n",
+ " push!(edgelabel, W[i, j])\n",
" end\n",
" end\n",
" end\n",
@@ -153,6 +164,9 @@
{
"cell_type": "markdown",
"metadata": {
+ "jupyter": {
+ "source_hidden": true
+ },
"tags": []
},
"source": [
@@ -162,15 +176,18 @@
"\n",
"However on such small graphs column generation techniques will not show their full potential, as these methods aim to tackle large scale problems.\n",
"\n",
- "\n",
- "\n",
- ""
+ "\n",
+ "\n",
+ ""
]
},
{
"cell_type": "code",
- "execution_count": 6,
+ "execution_count": 12,
"metadata": {
+ "jupyter": {
+ "source_hidden": true
+ },
"tags": []
},
"outputs": [
@@ -262,7 +279,12 @@
{
"cell_type": "code",
"execution_count": 7,
- "metadata": {},
+ "metadata": {
+ "jupyter": {
+ "source_hidden": true
+ },
+ "tags": []
+ },
"outputs": [
{
"data": {
@@ -316,14 +338,25 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 6,
"metadata": {
"jupyter": {
"source_hidden": true
},
"tags": []
},
- "outputs": [],
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "LPM (generic function with 1 method)"
+ ]
+ },
+ "execution_count": 6,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"function LPM(C, D)\n",
" lpm = Model(Cbc.Optimizer)\n",
@@ -363,7 +396,12 @@
{
"cell_type": "code",
"execution_count": 8,
- "metadata": {},
+ "metadata": {
+ "jupyter": {
+ "source_hidden": true
+ },
+ "tags": []
+ },
"outputs": [
{
"data": {
@@ -412,9 +450,19 @@
},
{
"cell_type": "code",
- "execution_count": null,
+ "execution_count": 7,
"metadata": {},
- "outputs": [],
+ "outputs": [
+ {
+ "ename": "LoadError",
+ "evalue": "syntax: missing comma or ) in argument list",
+ "output_type": "error",
+ "traceback": [
+ "syntax: missing comma or ) in argument list",
+ ""
+ ]
+ }
+ ],
"source": [
"function SP()\n",
" sp = Model(Cbc.Optimizer)\n",
@@ -438,29 +486,38 @@
{
"cell_type": "markdown",
"metadata": {
+ "jupyter": {
+ "source_hidden": true
+ },
"tags": []
},
"source": [
"Our algorithm will follow this flowchart, classic for column generation algorithm\n",
"\n",
- ""
+ ""
]
},
{
"cell_type": "markdown",
"metadata": {
+ "jupyter": {
+ "source_hidden": true
+ },
"tags": []
},
"source": [
"Our first \"dummy\" basis will consist in a collection of \"wheel\"-like 1-trees, as described in Held and Karp's paper\n",
"\n",
- ""
+ ""
]
},
{
"cell_type": "code",
- "execution_count": 10,
+ "execution_count": 4,
"metadata": {
+ "jupyter": {
+ "source_hidden": true
+ },
"tags": []
},
"outputs": [
@@ -470,7 +527,7 @@
"wheel_like_1trees (generic function with 1 method)"
]
},
- "execution_count": 10,
+ "execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
@@ -503,8 +560,15 @@
},
{
"cell_type": "code",
- "execution_count": 25,
- "metadata": {},
+ "execution_count": 5,
+ "metadata": {
+ "collapsed": true,
+ "jupyter": {
+ "outputs_hidden": true,
+ "source_hidden": true
+ },
+ "tags": []
+ },
"outputs": [
{
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@@ -524,77 +588,77 @@
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- " <path fill=\"none\" d=\"M-17.17,12.54 L17.17,-12.54 \" class=\"primitive\"/>\n",
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- " <g transform=\"translate(63.57,49.31)\">\n",
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" </g>\n",
- " <g transform=\"translate(59.24,91.67)\">\n",
+ " <g transform=\"translate(94.2,91.67)\">\n",
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" <text text-anchor=\"middle\" dy=\"0.35em\">3</text>\n",
" </g>\n",
" </g>\n",
- " <g transform=\"translate(89.84,8.33)\">\n",
+ " <g transform=\"translate(11.79,64.07)\">\n",
" <g class=\"primitive\">\n",
" <text text-anchor=\"middle\" dy=\"0.35em\">4</text>\n",
" </g>\n",
" </g>\n",
- " <g transform=\"translate(11.79,21.31)\">\n",
+ " <g transform=\"translate(118.83,9.83)\">\n",
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" <text text-anchor=\"middle\" dy=\"0.35em\">5</text>\n",
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- " <g transform=\"translate(129.64,55.59)\">\n",
+ " <g transform=\"translate(36.71,8.33)\">\n",
" <g class=\"primitive\">\n",
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@@ -613,83 +677,83 @@
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"\n",
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- " <path fill=\"none\" d=\"M-17.17,12.54 L17.17,-12.54 \" class=\"primitive\"/>\n",
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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}[(0.8988399984324502cx, 0.0552287667851781cy), (0.11991084705147577cx, -0.0490257650301336cy)]), Compose.LinePrimitive{Tuple{Measures.Measure,Measures.Measure}}(Tuple{Measures.Measure,Measures.Measure}[(0.9797713066360427cx, -0.031268960094218604cy), (0.8876070998575483cx, -0.4870502926179921cy)]), Compose.LinePrimitive{Tuple{Measures.Measure,Measures.Measure}}(Tuple{Measures.Measure,Measures.Measure}[(-0.4033070241627086cx, -0.9086700102736021cy), (-0.02680680637877119cx, -0.15389535406818045cy)]), Compose.LinePrimitive{Tuple{Measures.Measure,Measures.Measure}}(Tuple{Measures.Measure,Measures.Measure}[(-0.04040357897883748cx, 0.8981308592134905cy), (0.012478300379666556cx, 0.03930377644472703cy)]), Compose.LinePrimitive{Tuple{Measures.Measure,Measures.Measure}}(Tuple{Measures.Measure,Measures.Measure}[(-0.9001355548713583cx, 0.13126626561929647cy), (-0.08111359964471565cx, -0.04149976997999948cy)]), Compose.LinePrimitive{Tuple{Measures.Measure,Measures.Measure}}(Tuple{Measures.Measure,Measures.Measure}[(0.10556812775320468cx, -0.1162256491532058cy), (0.7805611242243123cx, -0.5334273339976144cy)])], 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)]), 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(\"\")), 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, 0.06876836609682702cy), 0.04082482904638631w), Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}((-0.4488646760254058cx, -1.0cy), 0.04082482904638631w), Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}((-0.04667612408309696cx, 1.0cy), 0.04082482904638631w), Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}((-1.0cx, 0.15233185998107945cy), 0.04082482904638631w), Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}((0.018750845483926026cx, -0.0625653643417825cy), 0.04082482904638631w), Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}((0.867378406493591cx, -0.5870876188090377cy), 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, 0.06876836609682702cy), \"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}}}((-0.4488646760254058cx, -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}}}((-0.04667612408309696cx, 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, 0.15233185998107945cy), \"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}}}((0.018750845483926026cx, -0.0625653643417825cy), \"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}}}((0.867378406493591cx, -0.5870876188090377cy), \"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}[(0.89990424304399cx, -0.15391948490322985cy), (0.1274084595896624cx, -5.005259251717603e-5cy)]), Compose.LinePrimitive{Tuple{Measures.Measure,Measures.Measure}}(Tuple{Measures.Measure,Measures.Measure}[(0.9815880828792808cx, -0.07346945132320928cy), (0.9010501059786039cx, 0.36564904191562886cy)]), Compose.LinePrimitive{Tuple{Measures.Measure,Measures.Measure}}(Tuple{Measures.Measure,Measures.Measure}[(-0.8986263652372324cx, -0.08820693659423305cy), (-0.07406093212911521cx, 0.008053110252434397cy)]), Compose.LinePrimitive{Tuple{Measures.Measure,Measures.Measure}}(Tuple{Measures.Measure,Measures.Measure}[(-0.24629382386101645cx, 0.9025144194141625cy), (-0.002906963885170581cx, 0.11737308316114455cy)]), Compose.LinePrimitive{Tuple{Measures.Measure,Measures.Measure}}(Tuple{Measures.Measure,Measures.Measure}[(-0.09866371329569534cx, -0.8988840041128118cy), (0.013448286222969471cx, -0.08122849331188105cy)]), Compose.LinePrimitive{Tuple{Measures.Measure,Measures.Measure}}(Tuple{Measures.Measure,Measures.Measure}[(0.11780405974368525cx, 0.06708899571960471cy), (0.7921468317478518cx, 0.4188351375191762cy)])], 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)]), 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(\"\")), 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, -0.17385704007105418cy), 0.04082482904638631w), Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}((-1.0cx, -0.10004132891710582cy), 0.04082482904638631w), Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}((-0.2765134903798394cx, 1.0cy), 0.04082482904638631w), Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}((-0.11252812970637827cx, -1.0cy), 0.04082482904638631w), Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}((0.027312702633652375cx, 0.019887502575307137cy), 0.04082482904638631w), Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}((0.8826381888578847cx, 0.4660366306634738cy), 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, -0.17385704007105418cy), \"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}}}((-1.0cx, -0.10004132891710582cy), \"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}}}((-0.2765134903798394cx, 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}}}((-0.11252812970637827cx, -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}}}((0.027312702633652375cx, 0.019887502575307137cy), \"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}}}((0.8826381888578847cx, 0.4660366306634738cy), \"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": {},
@@ -1900,77 +1964,77 @@
" <path d=\"M0,0 L15,3.5 L0,7 z\" stroke=\"context-stroke\" fill=\"context-stroke\"/>\n",
" </marker>\n",
"</defs>\n",
- "<g stroke-width=\"1.22\" fill=\"#000000\" fill-opacity=\"0.000\" stroke=\"#D3D3D3\" id=\"img-ed1aa8e5-1\">\n",
- " <g transform=\"translate(17.7,34.21)\">\n",
- " <path fill=\"none\" d=\"M-4.16,9.63 L4.16,-9.63 \" class=\"primitive\"/>\n",
+ "<g stroke-width=\"1.22\" fill=\"#000000\" fill-opacity=\"0.000\" stroke=\"#D3D3D3\" id=\"img-de7d6ec0-1\">\n",
+ " <g transform=\"translate(87.09,88.53)\">\n",
+ " <path fill=\"none\" d=\"M-13.33,2.18 L13.33,-2.18 \" class=\"primitive\"/>\n",
" </g>\n",
- " <g transform=\"translate(41.39,47.32)\">\n",
- " <path fill=\"none\" d=\"M-23.59,0.47 L23.59,-0.47 \" class=\"primitive\"/>\n",
+ " <g transform=\"translate(71.05,71.05)\">\n",
+ " <path fill=\"none\" d=\"M-2.5,16.39 L2.5,-16.39 \" class=\"primitive\"/>\n",
" </g>\n",
- " <g transform=\"translate(47.3,33.62)\">\n",
- " <path fill=\"none\" d=\"M-18.95,-10.49 L18.95,10.49 \" class=\"primitive\"/>\n",
+ " <g transform=\"translate(90.23,67.91)\">\n",
+ " <path fill=\"none\" d=\"M12.77,13.91 L-12.77,-13.91 \" class=\"primitive\"/>\n",
" </g>\n",
- " <g transform=\"translate(69.86,69.2)\">\n",
- " <path fill=\"none\" d=\"M-0.92,18.22 L0.92,-18.22 \" class=\"primitive\"/>\n",
+ " <g transform=\"translate(101.91,39.94)\">\n",
+ " <path fill=\"none\" d=\"M22.42,-8.48 L-22.42,8.48 \" class=\"primitive\"/>\n",
" </g>\n",
- " <g transform=\"translate(100.32,53.3)\">\n",
- " <path fill=\"none\" d=\"M23.59,5.28 L-23.59,-5.28 \" class=\"primitive\"/>\n",
+ " <g transform=\"translate(67.27,29.38)\">\n",
+ " <path fill=\"none\" d=\"M-5.55,-16.91 L5.55,16.91 \" class=\"primitive\"/>\n",
" </g>\n",
- " <g transform=\"translate(87.9,27.53)\">\n",
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},
"metadata": {},
@@ -2290,8 +2354,8 @@
"metadata": {},
"outputs": [],
"source": [
- "display(column_generation(W))\n",
- "display(column_generation(W2))"
+ "display_S(column_generation(W))\n",
+ "display_S(column_generation(W2))"
]
},
{
@@ -2308,11 +2372,8 @@
},
{
"cell_type": "code",
- "execution_count": 88,
+ "execution_count": 54,
"metadata": {
- "jupyter": {
- "source_hidden": true
- },
"tags": []
},
"outputs": [
@@ -2322,42 +2383,28 @@
"hk (generic function with 1 method)"
]
},
- "execution_count": 88,
+ "execution_count": 54,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"function hk(W)\n",
- " \n",
" # Initialization\n",
" n = size(W, 1)\n",
" \n",
- " if size(W, 2)!=n\n",
- " error(\"Weight matrice must be square\")\n",
- " end\n",
- " # \n",
+ " if size(W, 2)!=n ; error(\"Weight matrice must be square\") ; end\n",
" \n",
- " λ = zeros(n)\n",
- "\n",
- " W_updated = W\n",
+ " λ = zeros(n) ; W_updated = W ; z = -1\n",
" g = complete_graph(n-1)\n",
- " z = -1\n",
- " S = zeros(n,n)\n",
- " D = zeros(n)\n",
- " while D != fill(2, n) # z!=sum(W .* S) # While not a tour\n",
- " \n",
- " # print(z, \" \")\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\n",
- " for v in 1:n\n",
- " W_updated[u,v] -= λ[u] + λ[v]\n",
- " end\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",
- " # println(st)\n",
" \n",
" # Connect vertex 1\n",
" m = Model(Cbc.Optimizer) # PL(NE) Model\n",
@@ -2387,25 +2434,14 @@
" #@constraint(m, diagonalNulle[i ∈ 1:n], X[i,i] == 0)\n",
" \n",
" optimize!(m)\n",
- " \n",
- " if !( \n",
- " ( termination_status(m) == MOI.OPTIMAL ) || \n",
- " ( termination_status(m) == MOI.TIME_LIMIT \n",
- " && has_values(m) ) \n",
- " )\n",
- " println(termination_status(m))\n",
- " println(termination_status(m))\n",
- " println(has_values(m))\n",
- " println(m)\n",
- " error(\"Couldn't connect vertex 1 (PLNE failed).\")\n",
- " end\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, n) # z!=sum(W .* S) # If not a tour, update\n",
+ " if D != fill(2, size(D)) # If not a tour, update\n",
" λ .= λ .+ 2 .* (2 .- D) # Experimenter d'autres règles\n",
" end\n",
" \n",
@@ -2420,162 +2456,2169 @@
},
{
"cell_type": "code",
- "execution_count": 9,
+ "execution_count": 57,
"metadata": {
- "collapsed": true,
- "jupyter": {
- "outputs_hidden": true
- },
"tags": []
},
"outputs": [
{
- "ename": "UndefVarError",
- "evalue": "UndefVarError: hk not defined",
- "output_type": "error",
- "traceback": [
- "UndefVarError: hk not defined",
- "",
- "Stacktrace:",
- " [1] top-level scope at In[9]:1"
- ]
- }
- ],
- "source": [
- "columnGeneration(W)\n",
- "columnGeneration(W2)\n",
- "display(hk(W))\n",
- "display(hk(W2))"
- ]
- },
- {
- "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(held_and_karp_2(W))\n",
- "display(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": null,
- "metadata": {},
- "outputs": [],
- "source": [
- "function bellman_held_karp()\n",
- " \n",
- "end"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": null,
- "metadata": {},
- "outputs": [],
- "source": [
- "display(bellman_held_karp(W))\n",
- "display(bellman_held_karp(W2))"
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+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ },
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
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+ "[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",
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+ "[2.0, 1.0, 2.0, 2.0, 2.0, 3.0, 2.0]\n",
+ "[2.0, 1.0, 2.0, 2.0, 2.0, 3.0, 2.0]\n",
+ "[2.0, 3.0, 2.0, 2.0, 2.0, 1.0, 2.0]\n",
+ "[2.0, 3.0, 2.0, 2.0, 2.0, 1.0, 2.0]\n",
+ "[2.0, 1.0, 2.0, 2.0, 2.0, 3.0, 2.0]\n",
+ "[2.0, 1.0, 2.0, 2.0, 2.0, 3.0, 2.0]\n",
+ "["
+ ]
+ },
+ {
+ "ename": "InterruptException",
+ "evalue": "InterruptException:",
+ "output_type": "error",
+ "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": [
+ {
+ "data": {
+ "text/plain": [
+ "bellman_held_karp (generic function with 1 method)"
+ ]
+ },
+ "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)))"
+ ]
+ },
+ {
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+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ },
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "Valeur optimale : 22\n",
+ "Circuits optimaux : \n",
+ "Integer[1, 3, 2, 5, 4, 6]\n"
+ ]
+ }
+ ],
+ "source": [
+ "for M in bellman_held_karp(example)\n",
+ " display_S(M)\n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 45,
+ "metadata": {},
+ "outputs": [
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+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ },
+ {
+ "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"
+ ]
+ }
+ ],
+ "source": [
+ "for M in bellman_held_karp(exercise)\n",
+ " display_S(M)\n",
+ "end"
]
},
{
@@ -2702,15 +4745,15 @@
],
"metadata": {
"kernelspec": {
- "display_name": "Julia 1.5.3",
+ "display_name": "Julia 1.4.2",
"language": "julia",
- "name": "julia-1.5"
+ "name": "julia-1.4"
},
"language_info": {
"file_extension": ".jl",
"mimetype": "application/julia",
"name": "julia",
- "version": "1.5.3"
+ "version": "1.4.2"
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diff --git a/Column_Generation_Flowchart.png b/images/.ipynb_checkpoints/Column_Generation_Flowchart-checkpoint.png
similarity index 100%
rename from Column_Generation_Flowchart.png
rename to images/.ipynb_checkpoints/Column_Generation_Flowchart-checkpoint.png
diff --git a/wheel_like_1-trees.png b/images/.ipynb_checkpoints/wheel_like_1-trees-checkpoint.png
similarity index 100%
rename from wheel_like_1-trees.png
rename to images/.ipynb_checkpoints/wheel_like_1-trees-checkpoint.png
diff --git a/images/Column_Generation_Flowchart.png b/images/Column_Generation_Flowchart.png
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diff --git a/example_graph.png b/images/example_graph.png
similarity index 100%
rename from example_graph.png
rename to images/example_graph.png
diff --git a/example_graph_solution.png b/images/example_graph_solution.png
similarity index 100%
rename from example_graph_solution.png
rename to images/example_graph_solution.png
diff --git a/exercise_graph.png b/images/exercise_graph.png
similarity index 100%
rename from exercise_graph.png
rename to images/exercise_graph.png
diff --git a/images/wheel_like_1-trees.png b/images/wheel_like_1-trees.png
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--
GitLab