From d935e91f7f81a47fd5708d2fa6d8741ea950680f Mon Sep 17 00:00:00 2001
From: Inako <baptiste.signolle@free.fr>
Date: Tue, 30 Mar 2021 12:38:08 +0200
Subject: [PATCH] iofjveoi
---
.../heldAndKarp-checkpoint.ipynb | 1165 ++++++++++++++-
.../heldAndKarp_fluff-checkpoint.ipynb | 366 +++++
heldAndKarp.ipynb | 1292 +++++++++--------
heldAndKarp_fluff.ipynb | 6 +-
4 files changed, 2222 insertions(+), 607 deletions(-)
create mode 100644 .ipynb_checkpoints/heldAndKarp_fluff-checkpoint.ipynb
diff --git a/.ipynb_checkpoints/heldAndKarp-checkpoint.ipynb b/.ipynb_checkpoints/heldAndKarp-checkpoint.ipynb
index 3907e96..a72966c 100644
--- a/.ipynb_checkpoints/heldAndKarp-checkpoint.ipynb
+++ b/.ipynb_checkpoints/heldAndKarp-checkpoint.ipynb
@@ -4,22 +4,27 @@
"cell_type": "markdown",
"metadata": {},
"source": [
- "# Held \\& Karp algorithm"
+ "# CORO Project - Held \\& Karp algorithm\n",
+ "\n",
+ "## Joseph ELANG, François LEFOULON, Théophile RAGEAU, Baptiste SIGNOLLE"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "The Held and Karp algorithm as described in \"The traveling-salesman problem and minimum spanning trees\" (1969) tackles the TSP problem using the concept of 1-tree, which consist for $n$ vertices numbered from $1$ to $n$ of a spanning tree for the vertices $2$ to $n$, to which is connected the vertex $1$ such that it have a degree equals to $2$.\n",
+ "\n",
+ "In this paper several linear programms modeling the TSP are derived, and the authors show how they are related to one another. Then resolution methods are presented\n",
+ "\n",
+ "Here we present our attempts at implementing such methods"
]
},
{
"cell_type": "code",
- "execution_count": 1,
+ "execution_count": 103,
"metadata": {},
"outputs": [
- {
- "name": "stderr",
- "output_type": "stream",
- "text": [
- "┌ Info: Precompiling Graphs [86223c79-3864-5bf0-83f7-82e725a168b6]\n",
- "â”” @ Base loading.jl:1260\n"
- ]
- },
{
"name": "stdout",
"output_type": "stream",
@@ -31,17 +36,197 @@
"source": [
"using LinearAlgebra\n",
"using JuMP\n",
- "using Cbc\n",
- "using Graphs\n",
+ "using GLPK\n",
+ "using LightGraphs\n",
+ "using GraphPlot\n",
"\n",
"println(\"Modules loaded\")"
]
},
{
- "cell_type": "code",
- "execution_count": null,
+ "cell_type": "markdown",
"metadata": {},
- "outputs": [],
+ "source": [
+ "## Column-generation technique\n",
+ "\n",
+ "The master problem is the following\n",
+ "\n",
+ "$$ \\min \\sum_k c_k y_k \\ ; \\quad y_k \\geq 0 \\ , \\quad \\sum_k y_k = 1 \\ , \\quad \\sum -v_{ik} y_k = 0 \\quad \\forall i \\in {2, \\ldots, n-1}$$\n",
+ "\n",
+ "The subproblem is the following\n",
+ "\n",
+ "$$ K(\\pi) = \\left\\{k \\ |\\ w(\\pi) = c_k + \\sum_{i=1}^n \\pi_n v_{ik} \\right\\} $$"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 116,
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "columnGeneration (generic function with 1 method)"
+ ]
+ },
+ "execution_count": 116,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "function columnGeneration(W)\n",
+ " \n",
+ " n = size(W, 1)\n",
+ " T_List = wheel_like(n)\n",
+ " m = length(T_List)\n",
+ " S = 0\n",
+ " g = complete_graph(n-1)\n",
+ " \n",
+ " cr = -1\n",
+ " while cr < 0\n",
+ " \n",
+ " m = length(T_List)\n",
+ " \n",
+ " C = [sum(W .* T) for T in T_List]\n",
+ " D = [sum(T[i,:]) for i in 2:n-1, T in T_List]\n",
+ " V = D .- 2\n",
+ " \n",
+ " ### Master problem\n",
+ " master = Model(GLPK.Optimizer)\n",
+ " set_silent(master)\n",
+ " \n",
+ " @variable(master, 1 >= Y[1:m] >= 0)\n",
+ " \n",
+ " @objective(master, Min, dot(C, Y))\n",
+ " \n",
+ " @constraint(master, convexCombination, sum(Y) == 1)\n",
+ " \n",
+ " @constraint(master, averageDegreeTwo[k in 1:n-2], dot(V[k,:], Y) == 0)\n",
+ " \n",
+ " optimize!(master)\n",
+ " \n",
+ " println(termination_status(master))\n",
+ " println(\"obj \", objective_value(master))\n",
+ " println(\"Y \", value.(Y))\n",
+ " #println(\"C \", C)\n",
+ " #println(\"D \", D)\n",
+ " #println(\"V \", V)\n",
+ " S = value.(Y)\n",
+ " \n",
+ " θ = shadow_price(convexCombination)\n",
+ " π = shadow_price.(averageDegreeTwo)\n",
+ " \n",
+ " println(\"θ \", θ)\n",
+ " println(\"π \", π)\n",
+ " \n",
+ " W_updated = convert.(Float64, W)\n",
+ "\n",
+ " for i in 2:n-1, j in 2:n-1\n",
+ " W_updated[i,j] -= π[i-1] + π[j-1]\n",
+ " end\n",
+ "\n",
+ " st = Matrix(adjacency_matrix(Graph(kruskal_mst(g, W_updated[2:n,2:n]))))\n",
+ " T = zeros(n,n)\n",
+ "\n",
+ " w_tmp = W_updated[1,2:n]\n",
+ " min_idx_1 = argmin(w_tmp[1:n-1])\n",
+ " min_idx_2 = argmin(w_tmp[1:n-1 .!= min_idx_1])\n",
+ " T[1,min_idx_1+1] = 1 ; T[1,min_idx_2+1] = 1 ; T[:,1] .= T[1,:]\n",
+ " T[2:n,2:n] .= st[1:n-1,1:n-1]\n",
+ " \n",
+ " cr = sum(W_updated .* T) - θ - 2*sum(π)\n",
+ " \n",
+ " println(\"cr \", cr)\n",
+ " \n",
+ " push!(T_List, T)\n",
+ " \n",
+ " end\n",
+ " \n",
+ " Tf = zeros(n,n)\n",
+ " for i in 1:m\n",
+ " Tf .+= S[i] .* T_List[i]\n",
+ " end\n",
+ " \n",
+ " return Tf\n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 33,
+ "metadata": {
+ "collapsed": true,
+ "jupyter": {
+ "outputs_hidden": true,
+ "source_hidden": true
+ },
+ "tags": []
+ },
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "wheel_like (generic function with 1 method)"
+ ]
+ },
+ "execution_count": 33,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "function wheel_like(n)\n",
+ " res = []\n",
+ " for k in 2:n\n",
+ " one_tree = zeros(n,n)\n",
+ " \n",
+ " for v in union(2:k-1, k+1:n)\n",
+ " one_tree[k,v] = 1\n",
+ " one_tree[v,k] = 1\n",
+ " end\n",
+ " \n",
+ " one_tree[1,k] = 1\n",
+ " one_tree[k,1] = 1\n",
+ " one_tree[1, n==k ? 2 : k+1] = 1\n",
+ " one_tree[n==k ? 2 : k+1, 1] = 1\n",
+ " \n",
+ " for i in 1:n\n",
+ " one_tree[i:n, i] .= one_tree[i, i:n]\n",
+ " end\n",
+ " \n",
+ " push!(res, one_tree)\n",
+ " end\n",
+ " \n",
+ " return res\n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 88,
+ "metadata": {
+ "collapsed": true,
+ "jupyter": {
+ "outputs_hidden": true,
+ "source_hidden": true
+ },
+ "tags": []
+ },
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "hk (generic function with 1 method)"
+ ]
+ },
+ "execution_count": 88,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
"source": [
"function hk(W)\n",
" \n",
@@ -55,26 +240,51 @@
" \n",
" λ = zeros(n)\n",
"\n",
- " g = simple_graph(n-1, is_directed = false)\n",
- " W_reduced = W[2:n,2:n]\n",
- " z = 0\n",
- " while z!=n # While not a tour\n",
+ " W_updated = W\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",
" \n",
" # Minimum spanning tree (span_tree : st)\n",
- " st = kruskal_minimum_spantree(g, W_reduced)[1]\n",
+ " for u in 1:n\n",
+ " for v in 1:n\n",
+ " W_updated[u,v] -= λ[u] + λ[v]\n",
+ " end\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",
+ " m = Model(Cbc.Optimizer) # PL(NE) Model\n",
+ " set_silent(m)\n",
" \n",
- " @variable(m, X[1:n, 1:n], Bin)\n",
+ " @variable(m, X[1:n,1:n], Bin)\n",
" \n",
" @objective(m, Min, \n",
- " 2*sum(λ[2:n]) + \n",
- " sum((W[u,v] - λ[u] - λ[v])*X[u,v] for u ∈ 1:n for v ∈ 1:u)\n",
+ " sum(W .* X) + \n",
+ " sum(λ[j]*(2-sum(X[1,:])) for j in 2:n)\n",
" )\n",
" \n",
" @constraint(m, degreeRoot, sum(X[1,:]) == 2)\n",
- " @constraint(m, spanningTree[u ∈ 2:n, v ∈ 2:n], X[u,v] == st[u,v])\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",
+ " 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",
" \n",
@@ -83,22 +293,921 @@
" ( 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",
" \n",
" z = objective_value(m)\n",
- " D = [sum(value.(X)[u,:]) for u ∈ 1:n]\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 z!=n # If not a tour, update\n",
- " λ .= λ .+ 2.*(2.-D) # Experimenter d'autres règles\n",
+ " if D != fill(2, n) # z!=sum(W .* S) # If not a tour, update\n",
+ " λ .= λ .+ 2 .* (2 .- D) # Experimenter d'autres règles\n",
" end\n",
" \n",
+ " println(D)\n",
+ " \n",
" end\n",
" \n",
+ " return S\n",
" \n",
"end"
]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 6,
+ "metadata": {
+ "collapsed": true,
+ "jupyter": {
+ "outputs_hidden": true
+ },
+ "tags": []
+ },
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "6×6 Array{Int64,2}:\n",
+ " 0 8 4 15 15 3\n",
+ " 8 0 5 15 2 15\n",
+ " 4 5 0 6 15 15\n",
+ " 15 15 6 0 5 3\n",
+ " 15 2 15 5 0 4\n",
+ " 3 15 15 3 4 0"
+ ]
+ },
+ "execution_count": 6,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "W = [\n",
+ " 0 8 4 15 15 3 ;\n",
+ " 8 0 5 15 2 15 ;\n",
+ " 4 5 0 6 15 15 ;\n",
+ " 15 15 6 0 5 3 ;\n",
+ " 15 2 15 5 0 4 ;\n",
+ " 3 15 15 3 4 0\n",
+ "]"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 117,
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "OPTIMAL\n",
+ "obj 104.00000000000001\n",
+ "Y [0.20000000000000004, 0.20000000000000004, 0.2, 0.19999999999999998, 0.2]\n",
+ "θ -104.0\n",
+ "Ï€ [-0.780487804878029, -4.341463414634163, -5.85365853658539, -2.195121951219477]\n",
+ "cr 188.5365853658535\n"
+ ]
+ },
+ {
+ "data": {
+ "text/plain": [
+ "6×6 Array{Float64,2}:\n",
+ " 0.0 0.4 0.4 0.4 0.4 0.4\n",
+ " 0.4 0.0 0.4 0.4 0.4 0.4\n",
+ " 0.4 0.4 0.0 0.4 0.4 0.4\n",
+ " 0.4 0.4 0.4 0.0 0.4 0.4\n",
+ " 0.4 0.4 0.4 0.4 0.0 0.4\n",
+ " 0.4 0.4 0.4 0.4 0.4 0.0"
+ ]
+ },
+ "execution_count": 117,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "columnGeneration(W)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 118,
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "OPTIMAL\n",
+ "obj 113.99999999999997\n",
+ "Y [0.16666666666666666, 0.16666666666666663, 0.16666666666666666, 0.16666666666666666, 0.16666666666666666, 0.16666666666666663]\n",
+ "θ -113.99999999999999\n",
+ "Ï€ [-5.760368663594462, -6.958525345622108, -0.9677419354838632, -0.9216589861751061, -1.1520737327188897]\n",
+ "cr 161.44700460829478\n"
+ ]
+ },
+ {
+ "data": {
+ "text/plain": [
+ "7×7 Array{Float64,2}:\n",
+ " 0.0 0.333333 0.333333 0.333333 0.333333 0.333333 0.333333\n",
+ " 0.333333 0.0 0.333333 0.333333 0.333333 0.333333 0.333333\n",
+ " 0.333333 0.333333 0.0 0.333333 0.333333 0.333333 0.333333\n",
+ " 0.333333 0.333333 0.333333 0.0 0.333333 0.333333 0.333333\n",
+ " 0.333333 0.333333 0.333333 0.333333 0.0 0.333333 0.333333\n",
+ " 0.333333 0.333333 0.333333 0.333333 0.333333 0.0 0.333333\n",
+ " 0.333333 0.333333 0.333333 0.333333 0.333333 0.333333 0.0"
+ ]
+ },
+ "execution_count": 118,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "columnGeneration(W2)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 99,
+ "metadata": {
+ "collapsed": true,
+ "jupyter": {
+ "outputs_hidden": true
+ },
+ "tags": []
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
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+ "[2.0, 2.0, 5.0, 1.0, 1.0, 1.0]\n",
+ "[2.0, 3.0, 4.0, 1.0, 1.0, 1.0]\n",
+ "[2.0, 2.0, 1.0, 1.0, 5.0, 1.0]\n",
+ "[2.0, 2.0, 1.0, 1.0, 5.0, 1.0]\n",
+ "[2.0, 2.0, 1.0, 1.0, 5.0, 1.0]\n",
+ "[2.0, 1.0, 1.0, 2.0, 5.0, 1.0]\n",
+ "[2.0, 1.0, 1.0, 4.0, 3.0, 1.0]\n",
+ "[2.0, 1.0, 1.0, 5.0, 2.0, 1.0]\n",
+ "[2.0, 1.0, 1.0, 5.0, 2.0, 1.0]\n",
+ "[2.0, 1.0, 1.0, 5.0, 2.0, 1.0]\n",
+ "[2.0, 1.0, 1.0, 5.0, 2.0, 1.0]\n",
+ "[2.0, 1.0, 1.0, 5.0, 2.0, 1.0]\n",
+ "[2.0, 2.0, 1.0, 5.0, 1.0, 1.0]\n",
+ "[2.0, 2.0, 1.0, 5.0, 1.0, 1.0]\n",
+ "[2.0, 3.0, 1.0, 4.0, 1.0, 1.0]\n",
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+ "[2.0, 1.0, 4.0, 1.0, 1.0, 3.0]\n",
+ "[2.0, 1.0, 2.0, 1.0, 1.0, 5.0]\n",
+ "[2.0, 1.0, 1.0, 1.0, 2.0, 5.0]\n",
+ "[2.0, 1.0, 1.0, 1.0, 3.0, 4.0]\n",
+ "[2.0, 1.0, 1.0, 2.0, 3.0, 3.0]\n",
+ "[2.0, 1.0, 1.0, 5.0, 1.0, 2.0]\n",
+ "[2.0, 1.0, 1.0, 5.0, 2.0, 1.0]\n",
+ "[2.0, 2.0, 1.0, 5.0, 1.0, 1.0]\n",
+ "[2.0, 2.0, 1.0, 5.0, 1.0, 1.0]\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",
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+ " [10] write at .\\io.jl:647 [inlined]",
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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, -0.4192705032974433cy), 0.04082482904638631w), Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}((-0.0806820976966316cx, 1.0cy), 0.04082482904638631w), Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}((0.907229559561997cx, 0.5733430413496701cy), 0.04082482904638631w), Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}((-0.9032091926856136cx, -0.5654744994173764cy), 0.04082482904638631w), Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}((-1.0cx, 0.4215291750842578cy), 0.04082482904638631w), Compose.CirclePrimitive{Tuple{Measures.Measure,Measures.Measure},Measures.Measure}((0.08041069418848856cx, -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, -0.4192705032974433cy), \"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}}}((-0.0806820976966316cx, 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}}}((0.907229559561997cx, 0.5733430413496701cy), \"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}}}((-0.9032091926856136cx, -0.5654744994173764cy), \"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, 0.4215291750842578cy), \"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}}}((0.08041069418848856cx, -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(\"\"))"
+ ]
+ },
+ "execution_count": 90,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "gplot(Graph(S), nodelabel=1:6)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 114,
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [
+ {
+ "ename": "UndefVarError",
+ "evalue": "UndefVarError: hk not defined",
+ "output_type": "error",
+ "traceback": [
+ "UndefVarError: hk not defined",
+ "",
+ "Stacktrace:",
+ " [1] top-level scope at In[114]:10"
+ ]
+ }
+ ],
+ "source": [
+ "W2 = [\n",
+ " 0 15 0 0 15 15 15 ;\n",
+ " 15 0 1 15 0 15 1 ;\n",
+ " 0 1 0 15 15 15 1 ;\n",
+ " 0 15 15 0 1 1 15 ;\n",
+ " 15 0 15 1 0 1 15 ;\n",
+ " 15 15 15 1 1 0 0 ;\n",
+ " 15 1 1 15 15 0 0 ;\n",
+ "]\n",
+ "\n",
+ "S2 = hk(W2)\n",
+ "\n",
+ "gplot(Graph(S2), nodelabel=1:size(W2,1))"
+ ]
}
],
"metadata": {
diff --git a/.ipynb_checkpoints/heldAndKarp_fluff-checkpoint.ipynb b/.ipynb_checkpoints/heldAndKarp_fluff-checkpoint.ipynb
new file mode 100644
index 0000000..39a8b8c
--- /dev/null
+++ b/.ipynb_checkpoints/heldAndKarp_fluff-checkpoint.ipynb
@@ -0,0 +1,366 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Held \\& Karp algorithm"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 1,
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "Modules loaded\n"
+ ]
+ }
+ ],
+ "source": [
+ "using LinearAlgebra\n",
+ "using JuMP\n",
+ "using Cbc\n",
+ "using LightGraphs\n",
+ "using GraphPlot\n",
+ "\n",
+ "println(\"Modules loaded\")"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 2,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "connect_v1! (generic function with 1 method)"
+ ]
+ },
+ "execution_count": 2,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "function connect_v1!(X, W1, n) \n",
+ " argmin1, argmin2 = 2, 3\n",
+ " min1, min2 = W1[2], W1[3]\n",
+ " \n",
+ " for i in 4:n\n",
+ " if W1[i] < min1\n",
+ " min2 = min1\n",
+ " argmin2 = argmin1\n",
+ " \n",
+ " min1 = W1[i]\n",
+ " argmin1 = i\n",
+ " end\n",
+ " end\n",
+ " \n",
+ " X[1, argmin1] = 1\n",
+ " X[1, argmin2] = 1\n",
+ " \n",
+ " for u ∈ 2:n\n",
+ " X[u, 1] = X[1, u]\n",
+ " end\n",
+ " \n",
+ "end\n",
+ " "
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 3,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "adj_to_graph (generic function with 1 method)"
+ ]
+ },
+ "execution_count": 3,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "function adj_to_graph(adj, n)\n",
+ " \n",
+ " t = SimpleGraph(n)\n",
+ " \n",
+ " for i in 1:n\n",
+ " for j in i:n\n",
+ " if adj[i, j] == 1\n",
+ " add_edge!(t, i, j)\n",
+ " end\n",
+ " end\n",
+ " end\n",
+ " \n",
+ " return t\n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 11,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "hk2 (generic function with 1 method)"
+ ]
+ },
+ "execution_count": 11,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "function hk2(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",
+ " \n",
+ " λ = zeros(n)\n",
+ "\n",
+ " g = complete_graph(n-1)\n",
+ " z = -1\n",
+ " S = zeros(n,n)\n",
+ " \n",
+ " k = 0\n",
+ " X = falses(n, n)\n",
+ " \n",
+ " converge = false\n",
+ "\n",
+ " while !converge && k < 100 # While not a tour\n",
+ " k+=1\n",
+ " print(z, \" \")\n",
+ " \n",
+ " X = falses(n, n)\n",
+ " W_updated = deepcopy(W)\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",
+ " end\n",
+ " st = kruskal_mst(g, W_updated[2:n,2:n])\n",
+ "\n",
+ " println(st)\n",
+ " \n",
+ " for i ∈ 1:size(st, 1)\n",
+ " X[src(st[i])+1, dst(st[i])+1] = 1\n",
+ " end\n",
+ " \n",
+ " for u ∈ 1:n\n",
+ " for v ∈ 1:(u-1)\n",
+ " X[u,v] = X[v,u]\n",
+ " end\n",
+ " end\n",
+ " \n",
+ " # Connect vertex 1\n",
+ " W1 = deepcopy(W[1, :])\n",
+ " for v in 2:n\n",
+ " W1[v] -= λ[v]\n",
+ " end\n",
+ " connect_v1!(X, W1, n)\n",
+ " \n",
+ " one_tree = adj_to_graph(X, n)\n",
+ " \n",
+ " z = 2*sum(λ[2:n]) + sum((W[u,v] - λ[u] - λ[v])*X[u,v] for u ∈ 1:n for v ∈ 1:u)\n",
+ " \n",
+ " #@constraint(m, spanningTree[i ∈ 1:size(st,1)], X[src(st[i]), dst(st[i])] == 1)\n",
+ " \n",
+ " D = [sum(X[u,:]) for u ∈ 1:n]\n",
+ "\n",
+ " if is_cyclic(one_tree) # If not a tour, update\n",
+ " converge = true\n",
+ " λ .= λ .+ 2 .* (2 .- D) # Experimenter d'autres règles\n",
+ " end\n",
+ " \n",
+ " println(\"sommes des poids des arêtes :\", sum(W .* X)/2)\n",
+ " \n",
+ " end\n",
+ " \n",
+ " return (X, λ)\n",
+ " \n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 5,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "6×6 Array{Int64,2}:\n",
+ " 0 8 4 15 15 3\n",
+ " 8 0 5 15 2 15\n",
+ " 4 5 0 6 15 15\n",
+ " 15 15 6 0 5 3\n",
+ " 15 2 15 5 0 4\n",
+ " 3 15 15 3 4 0"
+ ]
+ },
+ "execution_count": 5,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "W = [\n",
+ " 0 8 4 15 15 3 ;\n",
+ " 8 0 5 15 2 15 ;\n",
+ " 4 5 0 6 15 15 ;\n",
+ " 15 15 6 0 5 3 ;\n",
+ " 15 2 15 5 0 4 ;\n",
+ " 3 15 15 3 4 0\n",
+ "]"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 12,
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "-1 LightGraphs.SimpleGraphs.SimpleEdge{Int64}[Edge 1 => 4, Edge 3 => 5, Edge 4 => 5, Edge 1 => 2]\n",
+ "sommes des poids des arêtes :25.0\n"
+ ]
+ },
+ {
+ "data": {
+ "text/plain": [
+ "(Bool[0 1 … 0 1; 1 0 … 1 0; … ; 0 1 … 0 1; 1 0 … 1 0], [0.0, -2.0, 2.0, 2.0, 0.0, -2.0])"
+ ]
+ },
+ "execution_count": 12,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "X, λ = hk2(W)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 13,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "6×6 BitArray{2}:\n",
+ " 0 1 0 0 0 1\n",
+ " 1 0 1 0 1 0\n",
+ " 0 1 0 0 0 0\n",
+ " 0 0 0 0 0 1\n",
+ " 0 1 0 0 0 1\n",
+ " 1 0 0 1 1 0"
+ ]
+ },
+ "execution_count": 13,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "X"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 14,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "6-element Array{Float64,1}:\n",
+ " 0.0\n",
+ " -2.0\n",
+ " 2.0\n",
+ " 2.0\n",
+ " 0.0\n",
+ " -2.0"
+ ]
+ },
+ "execution_count": 14,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "λ"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "g = complete_graph(5)\n",
+ "gplot(g)\n",
+ "k = kruskal_mst(g, rand(5, 5))"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "for e in k\n",
+ " println(src(e), dst(e))\n",
+ "end\n",
+ "typeof(k)\n",
+ "for i ∈ 1:size(k,1)\n",
+ " println(src(k[i]), \" \", dst(k[i]))\n",
+ "end\n"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": []
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "Julia 1.5.3",
+ "language": "julia",
+ "name": "julia-1.5"
+ },
+ "language_info": {
+ "file_extension": ".jl",
+ "mimetype": "application/julia",
+ "name": "julia",
+ "version": "1.5.3"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}
diff --git a/heldAndKarp.ipynb b/heldAndKarp.ipynb
index c4f1851..a72966c 100644
--- a/heldAndKarp.ipynb
+++ b/heldAndKarp.ipynb
@@ -4,12 +4,25 @@
"cell_type": "markdown",
"metadata": {},
"source": [
- "# Held \\& Karp algorithm"
+ "# CORO Project - Held \\& Karp algorithm\n",
+ "\n",
+ "## Joseph ELANG, François LEFOULON, Théophile RAGEAU, Baptiste SIGNOLLE"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "The Held and Karp algorithm as described in \"The traveling-salesman problem and minimum spanning trees\" (1969) tackles the TSP problem using the concept of 1-tree, which consist for $n$ vertices numbered from $1$ to $n$ of a spanning tree for the vertices $2$ to $n$, to which is connected the vertex $1$ such that it have a degree equals to $2$.\n",
+ "\n",
+ "In this paper several linear programms modeling the TSP are derived, and the authors show how they are related to one another. Then resolution methods are presented\n",
+ "\n",
+ "Here we present our attempts at implementing such methods"
]
},
{
"cell_type": "code",
- "execution_count": 1,
+ "execution_count": 103,
"metadata": {},
"outputs": [
{
@@ -23,7 +36,7 @@
"source": [
"using LinearAlgebra\n",
"using JuMP\n",
- "using Cbc\n",
+ "using GLPK\n",
"using LightGraphs\n",
"using GraphPlot\n",
"\n",
@@ -31,9 +44,177 @@
]
},
{
- "cell_type": "code",
- "execution_count": 41,
+ "cell_type": "markdown",
"metadata": {},
+ "source": [
+ "## Column-generation technique\n",
+ "\n",
+ "The master problem is the following\n",
+ "\n",
+ "$$ \\min \\sum_k c_k y_k \\ ; \\quad y_k \\geq 0 \\ , \\quad \\sum_k y_k = 1 \\ , \\quad \\sum -v_{ik} y_k = 0 \\quad \\forall i \\in {2, \\ldots, n-1}$$\n",
+ "\n",
+ "The subproblem is the following\n",
+ "\n",
+ "$$ K(\\pi) = \\left\\{k \\ |\\ w(\\pi) = c_k + \\sum_{i=1}^n \\pi_n v_{ik} \\right\\} $$"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 116,
+ "metadata": {
+ "tags": []
+ },
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "columnGeneration (generic function with 1 method)"
+ ]
+ },
+ "execution_count": 116,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "function columnGeneration(W)\n",
+ " \n",
+ " n = size(W, 1)\n",
+ " T_List = wheel_like(n)\n",
+ " m = length(T_List)\n",
+ " S = 0\n",
+ " g = complete_graph(n-1)\n",
+ " \n",
+ " cr = -1\n",
+ " while cr < 0\n",
+ " \n",
+ " m = length(T_List)\n",
+ " \n",
+ " C = [sum(W .* T) for T in T_List]\n",
+ " D = [sum(T[i,:]) for i in 2:n-1, T in T_List]\n",
+ " V = D .- 2\n",
+ " \n",
+ " ### Master problem\n",
+ " master = Model(GLPK.Optimizer)\n",
+ " set_silent(master)\n",
+ " \n",
+ " @variable(master, 1 >= Y[1:m] >= 0)\n",
+ " \n",
+ " @objective(master, Min, dot(C, Y))\n",
+ " \n",
+ " @constraint(master, convexCombination, sum(Y) == 1)\n",
+ " \n",
+ " @constraint(master, averageDegreeTwo[k in 1:n-2], dot(V[k,:], Y) == 0)\n",
+ " \n",
+ " optimize!(master)\n",
+ " \n",
+ " println(termination_status(master))\n",
+ " println(\"obj \", objective_value(master))\n",
+ " println(\"Y \", value.(Y))\n",
+ " #println(\"C \", C)\n",
+ " #println(\"D \", D)\n",
+ " #println(\"V \", V)\n",
+ " S = value.(Y)\n",
+ " \n",
+ " θ = shadow_price(convexCombination)\n",
+ " π = shadow_price.(averageDegreeTwo)\n",
+ " \n",
+ " println(\"θ \", θ)\n",
+ " println(\"π \", π)\n",
+ " \n",
+ " W_updated = convert.(Float64, W)\n",
+ "\n",
+ " for i in 2:n-1, j in 2:n-1\n",
+ " W_updated[i,j] -= π[i-1] + π[j-1]\n",
+ " end\n",
+ "\n",
+ " st = Matrix(adjacency_matrix(Graph(kruskal_mst(g, W_updated[2:n,2:n]))))\n",
+ " T = zeros(n,n)\n",
+ "\n",
+ " w_tmp = W_updated[1,2:n]\n",
+ " min_idx_1 = argmin(w_tmp[1:n-1])\n",
+ " min_idx_2 = argmin(w_tmp[1:n-1 .!= min_idx_1])\n",
+ " T[1,min_idx_1+1] = 1 ; T[1,min_idx_2+1] = 1 ; T[:,1] .= T[1,:]\n",
+ " T[2:n,2:n] .= st[1:n-1,1:n-1]\n",
+ " \n",
+ " cr = sum(W_updated .* T) - θ - 2*sum(π)\n",
+ " \n",
+ " println(\"cr \", cr)\n",
+ " \n",
+ " push!(T_List, T)\n",
+ " \n",
+ " end\n",
+ " \n",
+ " Tf = zeros(n,n)\n",
+ " for i in 1:m\n",
+ " Tf .+= S[i] .* T_List[i]\n",
+ " end\n",
+ " \n",
+ " return Tf\n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 33,
+ "metadata": {
+ "collapsed": true,
+ "jupyter": {
+ "outputs_hidden": true,
+ "source_hidden": true
+ },
+ "tags": []
+ },
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "wheel_like (generic function with 1 method)"
+ ]
+ },
+ "execution_count": 33,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "function wheel_like(n)\n",
+ " res = []\n",
+ " for k in 2:n\n",
+ " one_tree = zeros(n,n)\n",
+ " \n",
+ " for v in union(2:k-1, k+1:n)\n",
+ " one_tree[k,v] = 1\n",
+ " one_tree[v,k] = 1\n",
+ " end\n",
+ " \n",
+ " one_tree[1,k] = 1\n",
+ " one_tree[k,1] = 1\n",
+ " one_tree[1, n==k ? 2 : k+1] = 1\n",
+ " one_tree[n==k ? 2 : k+1, 1] = 1\n",
+ " \n",
+ " for i in 1:n\n",
+ " one_tree[i:n, i] .= one_tree[i, i:n]\n",
+ " end\n",
+ " \n",
+ " push!(res, one_tree)\n",
+ " end\n",
+ " \n",
+ " return res\n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 88,
+ "metadata": {
+ "collapsed": true,
+ "jupyter": {
+ "outputs_hidden": true,
+ "source_hidden": true
+ },
+ "tags": []
+ },
"outputs": [
{
"data": {
@@ -41,7 +222,7 @@
"hk (generic function with 1 method)"
]
},
- "execution_count": 41,
+ "execution_count": 88,
"metadata": {},
"output_type": "execute_result"
}
@@ -63,9 +244,10 @@
" g = complete_graph(n-1)\n",
" z = -1\n",
" S = zeros(n,n)\n",
- " while z!=sum(W .* S) # While not a tour\n",
+ " D = zeros(n)\n",
+ " while D != fill(2, n) # z!=sum(W .* S) # While not a tour\n",
" \n",
- " print(z, \" \")\n",
+ " # print(z, \" \")\n",
" \n",
" # Minimum spanning tree (span_tree : st)\n",
" for u in 1:n\n",
@@ -75,7 +257,7 @@
" end\n",
" st = kruskal_mst(g, W_updated[2:n,2:n])\n",
"\n",
- " println(st)\n",
+ " # println(st)\n",
" \n",
" # Connect vertex 1\n",
" m = Model(Cbc.Optimizer) # PL(NE) Model\n",
@@ -123,75 +305,11 @@
" 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 z!=sum(W .* S) # If not a tour, update\n",
+ " if D != fill(2, n) # z!=sum(W .* S) # If not a tour, update\n",
" λ .= λ .+ 2 .* (2 .- D) # Experimenter d'autres règles\n",
" end\n",
" \n",
- " end\n",
- " \n",
- " return S\n",
- " \n",
- "end"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 41,
- "metadata": {},
- "outputs": [
- {
- "data": {
- "text/plain": [
- "hk (generic function with 1 method)"
- ]
- },
- "execution_count": 41,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
- "source": [
- "function hk2(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",
- " \n",
- " λ = zeros(n)\n",
- "\n",
- " W_updated = W\n",
- " g = complete_graph(n-1)\n",
- " z = -1\n",
- " S = zeros(n,n)\n",
- " while z!=sum(W .* S) # While not a tour\n",
- " \n",
- " print(z, \" \")\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",
- " end\n",
- " st = kruskal_mst(g, W_updated[2:n,2:n])\n",
- "\n",
- " println(st)\n",
- " \n",
- " # Connect vertex 1\n",
- " \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 z!=sum(W .* S) # If not a tour, update\n",
- " λ .= λ .+ 2 .* (2 .- D) # Experimenter d'autres règles\n",
- " end\n",
+ " println(D)\n",
" \n",
" end\n",
" \n",
@@ -202,8 +320,14 @@
},
{
"cell_type": "code",
- "execution_count": 5,
- "metadata": {},
+ "execution_count": 6,
+ "metadata": {
+ "collapsed": true,
+ "jupyter": {
+ "outputs_hidden": true
+ },
+ "tags": []
+ },
"outputs": [
{
"data": {
@@ -217,7 +341,7 @@
" 3 15 15 3 4 0"
]
},
- "execution_count": 5,
+ "execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
@@ -235,42 +359,469 @@
},
{
"cell_type": "code",
- "execution_count": 42,
- "metadata": {
- "tags": []
- },
+ "execution_count": 117,
+ "metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
- "-1 LightGraphs.SimpleGraphs.SimpleEdge{Int64}[Edge 1 => 4, Edge 1 => 2, Edge 3 => 4, Edge 3 => 5]\n"
+ "OPTIMAL\n",
+ "obj 104.00000000000001\n",
+ "Y [0.20000000000000004, 0.20000000000000004, 0.2, 0.19999999999999998, 0.2]\n",
+ "θ -104.0\n",
+ "Ï€ [-0.780487804878029, -4.341463414634163, -5.85365853658539, -2.195121951219477]\n",
+ "cr 188.5365853658535\n"
]
},
{
"data": {
"text/plain": [
"6×6 Array{Float64,2}:\n",
- " 0.0 0.0 1.0 0.0 0.0 1.0\n",
- " 0.0 0.0 1.0 0.0 1.0 0.0\n",
- " 1.0 1.0 0.0 0.0 0.0 0.0\n",
- " 0.0 0.0 0.0 0.0 1.0 1.0\n",
- " 0.0 1.0 0.0 1.0 0.0 0.0\n",
- " 1.0 0.0 0.0 1.0 0.0 0.0"
+ " 0.0 0.4 0.4 0.4 0.4 0.4\n",
+ " 0.4 0.0 0.4 0.4 0.4 0.4\n",
+ " 0.4 0.4 0.0 0.4 0.4 0.4\n",
+ " 0.4 0.4 0.4 0.0 0.4 0.4\n",
+ " 0.4 0.4 0.4 0.4 0.0 0.4\n",
+ " 0.4 0.4 0.4 0.4 0.4 0.0"
+ ]
+ },
+ "execution_count": 117,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "columnGeneration(W)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 118,
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "OPTIMAL\n",
+ "obj 113.99999999999997\n",
+ "Y [0.16666666666666666, 0.16666666666666663, 0.16666666666666666, 0.16666666666666666, 0.16666666666666666, 0.16666666666666663]\n",
+ "θ -113.99999999999999\n",
+ "Ï€ [-5.760368663594462, -6.958525345622108, -0.9677419354838632, -0.9216589861751061, -1.1520737327188897]\n",
+ "cr 161.44700460829478\n"
+ ]
+ },
+ {
+ "data": {
+ "text/plain": [
+ "7×7 Array{Float64,2}:\n",
+ " 0.0 0.333333 0.333333 0.333333 0.333333 0.333333 0.333333\n",
+ " 0.333333 0.0 0.333333 0.333333 0.333333 0.333333 0.333333\n",
+ " 0.333333 0.333333 0.0 0.333333 0.333333 0.333333 0.333333\n",
+ " 0.333333 0.333333 0.333333 0.0 0.333333 0.333333 0.333333\n",
+ " 0.333333 0.333333 0.333333 0.333333 0.0 0.333333 0.333333\n",
+ " 0.333333 0.333333 0.333333 0.333333 0.333333 0.0 0.333333\n",
+ " 0.333333 0.333333 0.333333 0.333333 0.333333 0.333333 0.0"
]
},
- "execution_count": 42,
+ "execution_count": 118,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
- "S = hk(W)"
+ "columnGeneration(W2)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 99,
+ "metadata": {
+ "collapsed": true,
+ "jupyter": {
+ "outputs_hidden": true
+ },
+ "tags": []
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
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+ "[2.0, 1.0, 2.0, 1.0, 1.0, 5.0]\n",
+ "[2.0, 1.0, 2.0, 1.0, 1.0, 5.0]\n",
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+ "[2.0, 1.0, 2.0, 1.0, 5.0, 1.0]\n",
+ "[2.0, 1.0, 4.0, 1.0, 3.0, 1.0]\n",
+ "[2.0, 1.0, 5.0, 1.0, 2.0, 1.0]\n",
+ "[2.0, 1.0, 5.0, 1.0, 2.0, 1.0]\n",
+ "[2.0, 1.0, 5.0, 1.0, 2.0, 1.0]\n",
+ "[2.0, 1.0, 5.0, 1.0, 1.0, 2.0]\n",
+ "[2.0, 1.0, 4.0, 1.0, 1.0, 3.0]\n",
+ "[2.0, 1.0, 2.0, 1.0, 1.0, 5.0]\n",
+ "[2.0, 1.0, 1.0, 1.0, 2.0, 5.0]\n",
+ "[2.0, 1.0, 1.0, 1.0, 3.0, 4.0]\n",
+ "[2.0, 1.0, 1.0, 2.0, 3.0, 3.0]\n",
+ "[2.0, 1.0, 1.0, 5.0, 1.0, 2.0]\n",
+ "[2.0, 1.0, 1.0, 5.0, 2.0, 1.0]\n",
+ "[2.0, 2.0, 1.0, 5.0, 1.0, 1.0]\n",
+ "[2.0, 2.0, 1.0, 5.0, 1.0, 1.0]\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] hk2(::Array{Int64,2}) at .\\In[98]:76",
+ " [21] top-level scope at In[99]:1"
+ ]
+ }
+ ],
+ "source": [
+ "S = hk2(W)"
]
},
{
"cell_type": "code",
- "execution_count": 46,
+ "execution_count": 90,
"metadata": {},
"outputs": [
{
@@ -291,77 +842,77 @@
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- " <path fill=\"none\" d=\"M-23.7,-5.4 L23.7,5.4 \" class=\"primitive\"/>\n",
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- " <g transform=\"translate(34.95,22.35)\">\n",
- " <path fill=\"none\" d=\"M18.59,-11.25 L-18.59,11.25 \" class=\"primitive\"/>\n",
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" </g>\n",
- " <g transform=\"translate(106.48,77.65)\">\n",
- " <path fill=\"none\" d=\"M18.59,-11.25 L-18.59,11.25 \" class=\"primitive\"/>\n",
+ " <g transform=\"translate(38.87,79.62)\">\n",
+ " <path fill=\"none\" d=\"M22,9.79 L-22,-9.79 \" class=\"primitive\"/>\n",
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- " <g transform=\"translate(53.89,84.96)\">\n",
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]
},
- "execution_count": 46,
+ "execution_count": 90,
"metadata": {},
"output_type": "execute_result"
}
@@ -625,382 +1176,21 @@
},
{
"cell_type": "code",
- "execution_count": 81,
- "metadata": {},
+ "execution_count": 114,
+ "metadata": {
+ "tags": []
+ },
"outputs": [
{
- "name": "stdout",
- "output_type": "stream",
- "text": [
- "-1 LightGraphs.SimpleGraphs.SimpleEdge{Int64}[Edge 1 => 4, Edge 5 => 6, Edge 1 => 2, Edge 1 => 6, Edge 3 => 4]\n"
+ "ename": "UndefVarError",
+ "evalue": "UndefVarError: hk not defined",
+ "output_type": "error",
+ "traceback": [
+ "UndefVarError: hk not defined",
+ "",
+ "Stacktrace:",
+ " [1] top-level scope at In[114]:10"
]
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- ]
- },
- "execution_count": 81,
- "metadata": {},
- "output_type": "execute_result"
}
],
"source": [
@@ -1018,56 +1208,6 @@
"\n",
"gplot(Graph(S2), nodelabel=1:size(W2,1))"
]
- },
- {
- "cell_type": "code",
- "execution_count": 50,
- "metadata": {},
- "outputs": [
- {
- "name": "stdout",
- "output_type": "stream",
- "text": [
- "8.0\n"
- ]
- }
- ],
- "source": [
- "println(sum(W2 .* S2))"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": null,
- "metadata": {},
- "outputs": [],
- "source": [
- "g = complete_graph(5)\n",
- "gplot(g)\n",
- "k = kruskal_mst(g, rand(5, 5))"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": null,
- "metadata": {},
- "outputs": [],
- "source": [
- "for e in k\n",
- " println(src(e), dst(e))\n",
- "end\n",
- "typeof(k)\n",
- "for i ∈ 1:size(k,1)\n",
- " println(src(k[i]), \" \", dst(k[i]))\n",
- "end\n"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": null,
- "metadata": {},
- "outputs": [],
- "source": []
}
],
"metadata": {
diff --git a/heldAndKarp_fluff.ipynb b/heldAndKarp_fluff.ipynb
index 39a8b8c..555a0e9 100644
--- a/heldAndKarp_fluff.ipynb
+++ b/heldAndKarp_fluff.ipynb
@@ -350,15 +350,15 @@
],
"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"
}
},
"nbformat": 4,
--
GitLab