From 9c2f38c543d2d3e7fb0a595876e9e27f505c2523 Mon Sep 17 00:00:00 2001
From: Inako <baptiste.signolle@free.fr>
Date: Wed, 31 Mar 2021 08:19:04 +0200
Subject: [PATCH] stuff
---
.../heldAndKarp-checkpoint.ipynb | 1569 +++++++----------
Column_Generation_Flowchart.png | Bin 0 -> 16576 bytes
example_graph.png | Bin 0 -> 26342 bytes
example_graph_solution.png | Bin 0 -> 30971 bytes
exercise_graph.png | Bin 0 -> 51737 bytes
heldAndKarp.ipynb | 1569 +++++++----------
heldAndKarp_fluff.ipynb | 6 +-
wheel_like_1-trees.png | Bin 0 -> 56918 bytes
8 files changed, 1247 insertions(+), 1897 deletions(-)
create mode 100644 Column_Generation_Flowchart.png
create mode 100644 example_graph.png
create mode 100644 example_graph_solution.png
create mode 100644 exercise_graph.png
create mode 100644 wheel_like_1-trees.png
diff --git a/.ipynb_checkpoints/heldAndKarp-checkpoint.ipynb b/.ipynb_checkpoints/heldAndKarp-checkpoint.ipynb
index a72966c..bb23119 100644
--- a/.ipynb_checkpoints/heldAndKarp-checkpoint.ipynb
+++ b/.ipynb_checkpoints/heldAndKarp-checkpoint.ipynb
@@ -13,16 +13,47 @@
"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",
+ "**A REDIGER : Note sur l'utilisation d'un notebook, nbviewer, conversion en .jl, jupyter et jupyter lab**\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",
+ "**Requirements : Version julia et des modules, un project.toml etc PKG blabla**"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "The Held and Karp algorithm as described in \"The traveling-salesman problem and minimum spanning trees\" (1969) tackles the STSP (Symetric Traveling Salesman 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 LP of the STSP are derived, and the authors show how they are related to one another. Then resolution methods are presented.\n",
+ "\n",
+ "The core idea behind this work is that instead of relaxing the connectivity constraint as seen in the polyhedral approach, Held and Karp chose to relax the degree constraints using the concept of 1-tree."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Here we present our attemps to solve the STSP with column generation, and more generally the techniques presented in Held and Karp's paper, as shown in John E. Mitchell's slides.\n",
"\n",
- "Here we present our attempts at implementing such methods"
+ "We ran into some troubles while working on this project, mostly because we hardly saw a link between column generation, JuMP and the algorithm described in J.E. Mitchell's slides. In these the only things which seems to be involved are a greedy algorithm (Kruskal's for example) to solve a spanning tree problem, basic minimum finding techniques to connect vertex 1 and a loop to start again until a cycle is found (updating the weights between each iteration).\n",
+ "\n",
+ "We will go through the following\n",
+ "* STSP using column generation\n",
+ "* Held and Karp algorithm - attempt 1\n",
+ "* Held and Karp algorithm - attempt 2\n",
+ "* The Held and Karp algorithm - dynamical programming"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "### Modules and useful functions"
]
},
{
"cell_type": "code",
- "execution_count": 103,
+ "execution_count": 2,
"metadata": {},
"outputs": [
{
@@ -34,137 +65,329 @@
}
],
"source": [
- "using LinearAlgebra\n",
- "using JuMP\n",
- "using GLPK\n",
- "using LightGraphs\n",
- "using GraphPlot\n",
+ "using LinearAlgebra # Basic algebra structures & functions\n",
+ "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",
"\n",
"println(\"Modules loaded\")"
]
},
{
- "cell_type": "markdown",
+ "cell_type": "code",
+ "execution_count": null,
"metadata": {},
+ "outputs": [],
"source": [
- "## Column-generation technique\n",
- "\n",
- "The master problem is the following\n",
+ "function checkTermination(m, verbose = true)\n",
+ " if !( \n",
+ " ( termination_status(m) == MOI.OPTIMAL ) || \n",
+ " ( termination_status(m) == MOI.TIME_LIMIT && has_values(m) ) \n",
+ " )\n",
+ " \n",
+ " if verbose\n",
+ " println(termination_status(m))\n",
+ " println(termination_status(m))\n",
+ " println(has_values(m))\n",
+ " println(m)\n",
+ " end\n",
+ " \n",
+ " error(\"Couldn't connect vertex 1 (PLNE failed).\")\n",
+ " end\n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "function display(S, W = Nothing)\n",
+ " gplot(Graph(S), nodelabel=1:size(S,1))\n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {
+ "tags": []
+ },
+ "source": [
+ "## Data to test our results\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",
+ "The data we will be using is extracted from [John E. Mitchell's slides](http://eaton.math.rpi.edu/faculty/Mitchell/courses/matp6620/notesMATP6620/lecture22/22A_heldkarpbeamer.pdf).\n",
"\n",
- "The subproblem is the following\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",
- "$$ K(\\pi) = \\left\\{k \\ |\\ w(\\pi) = c_k + \\sum_{i=1}^n \\pi_n v_{ik} \\right\\} $$"
+ "\n",
+ "\n",
+ ""
]
},
{
"cell_type": "code",
- "execution_count": 116,
+ "execution_count": 11,
"metadata": {
"tags": []
},
"outputs": [
{
- "data": {
- "text/plain": [
- "columnGeneration (generic function with 1 method)"
- ]
- },
- "execution_count": 116,
- "metadata": {},
- "output_type": "execute_result"
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "Weight matrices initialized\n"
+ ]
}
],
"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",
+ "example = [\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",
+ "]\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",
+ "exercise = [\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",
- " st = Matrix(adjacency_matrix(Graph(kruskal_mst(g, W_updated[2:n,2:n]))))\n",
- " T = zeros(n,n)\n",
+ "println(\"Weight matrices initialized\")"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## STSP using column generation\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",
+ "The STSP can be written as\n",
+ "\n",
+ "$$\\begin{array}{r l}\n",
+ "\\min & \\sum_{e\\in E} c_e x_e \\\\\n",
+ "\\text{s.t.}\n",
+ "& \\sum_{e\\in \\delta(i)} x_e = 2 \\quad (i\\in N) \\\\\n",
+ "& x \\in X^1\n",
+ "\\end{array}$$\n",
+ "\n",
+ "$N$ is the set of nodes \n",
+ "$\\delta(i)$ the set of edges incident to $i$ \n",
+ "$x_e$ indicator variable for edge $e$ \n",
+ "$X^1$ set of indicator vectors of 1-trees\n",
+ "\n",
+ "A 1-tree is a subset $E' \\subset E$ of edges of which exactly two are incident to node 1 : $| E' \\cap \\delta(1) | = 2$ and $E'\\backslash \\delta(1)$ is a spanning tree on the remaining nodes $2, \\ldots, n$ (hence of cardinality $n-2$).\n",
+ "\n",
+ "Every Hamiltonian tour is a 1-tree that satisfies the degree constraint $|E' \\cap \\delta(i)| = 2$ for all nodes $i \\in N$. Conversely, every 1-tree that satisfies these degree constraints is a Hamiltonian tour."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Given a conceptual enumeration of the set of 1-tree : $\\{ E^t : t = 1, \\ldots, T_1 \\}$\n",
+ "\n",
+ "$$ x_e = \\sum_{t: e\\in E^t} \\lambda_t $$\n",
+ "\n",
+ "Subject to $\\lambda_t \\in \\{0, 1\\}$ for all $t$ and $\\sum_{t=1}^{T_1} \\lambda_t = 1$. \n",
+ "This defines the indicator vector of the single 1-tree for which $\\lambda_t = 1$\n",
+ "\n",
+ "The degree constraints become\n",
+ "\n",
+ "$$ \\sum_{e \\in \\delta(i)} x_e = \\sum_{e\\in \\delta(i)} \\sum_{t: e \\in E^t} \\lambda_t = \\sum_t d_i^t \\lambda_t = 2 $$\n",
+ "\n",
+ "Where $d_i^t$ is the degree of node $i$ in the 1-tree $t$.\n",
+ "\n",
+ "The IP Master Problem being \n",
+ "\n",
+ "$$ \\text{(IPM)} \\quad \n",
+ "\\begin{array}{r l}\n",
+ "\\min & \\sum_{t = 1}^{T_1} \\left( \\sum_{e\\in E} c_e x_e^t \\right) \\lambda_t \\\\\n",
+ "\\text{s.t.}\n",
+ "& \\sum_{t = 1}^{T_1} d_i^t \\lambda_t = 2 \\quad (i \\in N) \\\\\n",
+ "& \\sum_{t = 1}^{T_1} \\lambda_t = 1 \\\\\n",
+ "& \\lambda_t \\in \\{0, 1\\}, \\quad (t = 1, \\ldots , T_1)\n",
+ "\\end{array}$$"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "function IPM(C, D)\n",
+ " ipm = Model(Cbc.Optimizer)\n",
+ " set_silent(ipm)\n",
" \n",
- " Tf = zeros(n,n)\n",
- " for i in 1:m\n",
- " Tf .+= S[i] .* T_List[i]\n",
- " end\n",
+ " @variable(ipm, λ[1:length(C)], Bin)\n",
" \n",
- " return Tf\n",
+ " @objective(ipm, Min, dot(C, λ))\n",
+ " \n",
+ " @constraint(ipm, degree[i in 1:size(D,1)], dot(D[i,:], λ) == 2)\n",
+ " @constraint(ipm, convexity, sum(λ) == 1)\n",
+ " \n",
+ " optimize!(ipm)\n",
+ " checkTermination(ipm)\n",
+ " \n",
+ " return ipm\n",
"end"
]
},
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Thus the LP relaxation of (IPM) called LP Master Problem\n",
+ "\n",
+ "$$ \\text{(LPM)} \\quad \n",
+ "\\begin{array}{r l}\n",
+ "\\min & \\sum_{t = 1}^{T_1} \\left( \\sum_{e\\in E} c_e x_e^t \\right) \\lambda_t \\\\\n",
+ "\\text{s.t.}\n",
+ "& \\sum_{t = 1}^{T_1} d_i^t \\lambda_t = 2 \\quad (i \\in N) \\\\\n",
+ "& \\sum_{t = 1}^{T_1} \\lambda_t = 1 \\\\\n",
+ "& \\lambda_t \\geq 0, \\quad (t = 1, \\ldots , T_1)\n",
+ "\\end{array}$$"
+ ]
+ },
{
"cell_type": "code",
- "execution_count": 33,
+ "execution_count": null,
"metadata": {
- "collapsed": true,
"jupyter": {
- "outputs_hidden": true,
"source_hidden": true
},
"tags": []
},
+ "outputs": [],
+ "source": [
+ "function LPM(C, D)\n",
+ " lpm = Model(Cbc.Optimizer)\n",
+ " set_silent(lpm)\n",
+ " \n",
+ " @variable(lpm, λ[1:length(C)] >= 0)\n",
+ " \n",
+ " @objective(lpm, Min, dot(C, λ))\n",
+ " \n",
+ " @constraint(lpm, degree[i in 1:size(D,1)], dot(D[i,:], λ) == 2)\n",
+ " @constraint(lpm, convexity, sum(λ) == 1)\n",
+ " \n",
+ " optimize!(lpm)\n",
+ " checkTermination(lpm)\n",
+ " \n",
+ " return lpm\n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "In (LPM) there is a column $ ( c^T x^t , d_1^t , \\ldots , d_n^t , 1 )^T$ for every $x^t \\in X^1 \\ (t = 1, \\ldots, T_1)$ \n",
+ "The second equation is a convexity constraint\n",
+ "\n",
+ "Associating the second equation with dual variable $\\mu$ and the degree constraints with dual variables $\\{\\pi_i\\}_{i=1}^m$ the dual of (LPM) is\n",
+ "\n",
+ "$$\\text{(DM)} \\quad\n",
+ "\\begin{array}{r l}\n",
+ "\\max_{\\mu, \\pi} & \\sum_{i=1}^m 2 \\pi_i + \\mu \\\\\n",
+ "\\text{s.t.} & \\pi^T d^t + \\mu \\leq c^T x^t \\quad (t = 1 \\ldots T_1)\n",
+ "\\end{array}$$"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "function DM(C, D)\n",
+ " dm = Model(Cbc.Optimizer)\n",
+ " set_silent(dm)\n",
+ " \n",
+ " @variable(dm, π[1:size(D,1)])\n",
+ " @variable(dm, μ)\n",
+ " \n",
+ " @objective(dm, Max, 2*sum(π) + μ)\n",
+ " \n",
+ " @constraint(dm, oneTree[t in 1:size(D,2)], dot(π, D[:,t]) + μ <= C[t])\n",
+ " \n",
+ " optimize!(dm)\n",
+ " checkTermination(dm)\n",
+ " \n",
+ " return dm\n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "The associated subsidiary problem is the following\n",
+ "\n"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "function SP()\n",
+ " sp = Model(Cbc.Optimizer)\n",
+ " set_silent(sp)\n",
+ " \n",
+ " @variable(sp, )\n",
+ " \n",
+ " @objective(sp, Min, (c - π^T A ) x - μ)\n",
+ " \n",
+ " @constraint(sp, degree1)\n",
+ " @constraint(sp, nbEdges, sum(x[2:]) == )\n",
+ " @constraint(sp, spanningTree)\n",
+ " \n",
+ " optimize!(sp)\n",
+ " checkTermination(sp)\n",
+ " \n",
+ " return sp\n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {
+ "tags": []
+ },
+ "source": [
+ "Our algorithm will follow this flowchart, classic for column generation algorithm\n",
+ "\n",
+ ""
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {
+ "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": 6,
+ "metadata": {
+ "tags": []
+ },
"outputs": [
{
"data": {
@@ -172,30 +395,23 @@
"wheel_like (generic function with 1 method)"
]
},
- "execution_count": 33,
+ "execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
- "function wheel_like(n)\n",
+ "function wheel_like_1trees(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",
+ " one_tree[k,v] = 1 ; 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",
+ " one_tree[1,k] = 1 ; one_tree[1, n==k ? 2 : k+1] = 1\n",
+ " one_tree[k,1] = 1 ; one_tree[n==k ? 2 : k+1, 1] = 1 \n",
" \n",
" push!(res, one_tree)\n",
" end\n",
@@ -204,13 +420,82 @@
"end"
]
},
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "function column_generation(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",
+ " \n",
+ " while reduced_cost < 0\n",
+ " \n",
+ " push!(columns, new_column)\n",
+ " tree_degrees = hcat(tree_degrees, [sum(new_column[u,:]) for u ∈ 1:n]) \n",
+ " push!(tree_weights, dot(new_column, W))\n",
+ " \n",
+ " dm = DM(tree_weights, tree_degrees)\n",
+ " π = value.(dm[:π])\n",
+ " μ = value(dm[:μ])\n",
+ " \n",
+ " # Solve subproblem\n",
+ " sp = SP()\n",
+ " new_column = value.(sp[:??????])\n",
+ " new_degree = [sum(new_column[u,:]) for u ∈ 1:n]\n",
+ " new_weight = dot(new_column, W)\n",
+ " reduced_cost = objective_value(sp)\n",
+ " 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(column_generation(W))\n",
+ "display(column_generation(W2))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Held and Karp algorithm - attempt 1\n",
+ "\n",
+ "This is the part where we implement the Held and Karp algorithm as described in J.E. Mitchell's slides using Kruskal's algorithm to solve the minimum spanning tree problem.\n",
+ "\n",
+ "However this approach seems to be flawed as the example is easily solved, however the exercise seems to alternate between two identical states. \n",
+ "This may be due to the update rule of $\\lambda$, results may vary by modifying it."
+ ]
+ },
{
"cell_type": "code",
"execution_count": 88,
"metadata": {
- "collapsed": true,
"jupyter": {
- "outputs_hidden": true,
"source_hidden": true
},
"tags": []
@@ -320,7 +605,7 @@
},
{
"cell_type": "code",
- "execution_count": 6,
+ "execution_count": 9,
"metadata": {
"collapsed": true,
"jupyter": {
@@ -330,883 +615,273 @@
},
"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"
+ "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": [
- "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",
- "]"
+ "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": 117,
+ "execution_count": null,
"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"
- }
- ],
+ "outputs": [],
"source": [
- "columnGeneration(W)"
+ "function held_and_karp_algo()\n",
+ " \n",
+ "end"
]
},
{
"cell_type": "code",
- "execution_count": 118,
+ "execution_count": null,
"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"
- }
- ],
+ "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 graph, 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": [
- "columnGeneration(W2)"
+ "function bellman_held_karp()\n",
+ " \n",
+ "end"
]
},
{
"cell_type": "code",
- "execution_count": 99,
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "display(bellman_held_karp(W))\n",
+ "display(bellman_held_karp(W2))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
"metadata": {
- "collapsed": true,
"jupyter": {
- "outputs_hidden": true
+ "source_hidden": true
},
"tags": []
},
- "outputs": [
- {
- "name": "stdout",
- "output_type": "stream",
- "text": [
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- "["
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- },
- {
- "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)"
+ "## 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": 90,
- "metadata": {},
+ "execution_count": 3,
+ "metadata": {
+ "collapsed": true,
+ "jupyter": {
+ "outputs_hidden": true,
+ "source_hidden": true
+ },
+ "tags": []
+ },
"outputs": [
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+ "columnGeneration (generic function with 1 method)"
]
},
- "execution_count": 90,
+ "execution_count": 3,
"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",
+ "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, Bin)\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",
- "S2 = hk(W2)\n",
+ " st = Matrix(adjacency_matrix(Graph(kruskal_mst(g, W_updated[2:n,2:n]))))\n",
+ " T = zeros(n,n)\n",
"\n",
- "gplot(Graph(S2), nodelabel=1:size(W2,1))"
+ " 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"
]
}
],
diff --git a/Column_Generation_Flowchart.png b/Column_Generation_Flowchart.png
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diff --git a/example_graph.png b/example_graph.png
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diff --git a/heldAndKarp.ipynb b/heldAndKarp.ipynb
index a72966c..bb23119 100644
--- a/heldAndKarp.ipynb
+++ b/heldAndKarp.ipynb
@@ -13,16 +13,47 @@
"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",
+ "**A REDIGER : Note sur l'utilisation d'un notebook, nbviewer, conversion en .jl, jupyter et jupyter lab**\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",
+ "**Requirements : Version julia et des modules, un project.toml etc PKG blabla**"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "The Held and Karp algorithm as described in \"The traveling-salesman problem and minimum spanning trees\" (1969) tackles the STSP (Symetric Traveling Salesman 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 LP of the STSP are derived, and the authors show how they are related to one another. Then resolution methods are presented.\n",
+ "\n",
+ "The core idea behind this work is that instead of relaxing the connectivity constraint as seen in the polyhedral approach, Held and Karp chose to relax the degree constraints using the concept of 1-tree."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Here we present our attemps to solve the STSP with column generation, and more generally the techniques presented in Held and Karp's paper, as shown in John E. Mitchell's slides.\n",
"\n",
- "Here we present our attempts at implementing such methods"
+ "We ran into some troubles while working on this project, mostly because we hardly saw a link between column generation, JuMP and the algorithm described in J.E. Mitchell's slides. In these the only things which seems to be involved are a greedy algorithm (Kruskal's for example) to solve a spanning tree problem, basic minimum finding techniques to connect vertex 1 and a loop to start again until a cycle is found (updating the weights between each iteration).\n",
+ "\n",
+ "We will go through the following\n",
+ "* STSP using column generation\n",
+ "* Held and Karp algorithm - attempt 1\n",
+ "* Held and Karp algorithm - attempt 2\n",
+ "* The Held and Karp algorithm - dynamical programming"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "### Modules and useful functions"
]
},
{
"cell_type": "code",
- "execution_count": 103,
+ "execution_count": 2,
"metadata": {},
"outputs": [
{
@@ -34,137 +65,329 @@
}
],
"source": [
- "using LinearAlgebra\n",
- "using JuMP\n",
- "using GLPK\n",
- "using LightGraphs\n",
- "using GraphPlot\n",
+ "using LinearAlgebra # Basic algebra structures & functions\n",
+ "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",
"\n",
"println(\"Modules loaded\")"
]
},
{
- "cell_type": "markdown",
+ "cell_type": "code",
+ "execution_count": null,
"metadata": {},
+ "outputs": [],
"source": [
- "## Column-generation technique\n",
- "\n",
- "The master problem is the following\n",
+ "function checkTermination(m, verbose = true)\n",
+ " if !( \n",
+ " ( termination_status(m) == MOI.OPTIMAL ) || \n",
+ " ( termination_status(m) == MOI.TIME_LIMIT && has_values(m) ) \n",
+ " )\n",
+ " \n",
+ " if verbose\n",
+ " println(termination_status(m))\n",
+ " println(termination_status(m))\n",
+ " println(has_values(m))\n",
+ " println(m)\n",
+ " end\n",
+ " \n",
+ " error(\"Couldn't connect vertex 1 (PLNE failed).\")\n",
+ " end\n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "function display(S, W = Nothing)\n",
+ " gplot(Graph(S), nodelabel=1:size(S,1))\n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {
+ "tags": []
+ },
+ "source": [
+ "## Data to test our results\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",
+ "The data we will be using is extracted from [John E. Mitchell's slides](http://eaton.math.rpi.edu/faculty/Mitchell/courses/matp6620/notesMATP6620/lecture22/22A_heldkarpbeamer.pdf).\n",
"\n",
- "The subproblem is the following\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",
- "$$ K(\\pi) = \\left\\{k \\ |\\ w(\\pi) = c_k + \\sum_{i=1}^n \\pi_n v_{ik} \\right\\} $$"
+ "\n",
+ "\n",
+ ""
]
},
{
"cell_type": "code",
- "execution_count": 116,
+ "execution_count": 11,
"metadata": {
"tags": []
},
"outputs": [
{
- "data": {
- "text/plain": [
- "columnGeneration (generic function with 1 method)"
- ]
- },
- "execution_count": 116,
- "metadata": {},
- "output_type": "execute_result"
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "Weight matrices initialized\n"
+ ]
}
],
"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",
+ "example = [\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",
+ "]\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",
+ "exercise = [\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",
- " st = Matrix(adjacency_matrix(Graph(kruskal_mst(g, W_updated[2:n,2:n]))))\n",
- " T = zeros(n,n)\n",
+ "println(\"Weight matrices initialized\")"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## STSP using column generation\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",
+ "The STSP can be written as\n",
+ "\n",
+ "$$\\begin{array}{r l}\n",
+ "\\min & \\sum_{e\\in E} c_e x_e \\\\\n",
+ "\\text{s.t.}\n",
+ "& \\sum_{e\\in \\delta(i)} x_e = 2 \\quad (i\\in N) \\\\\n",
+ "& x \\in X^1\n",
+ "\\end{array}$$\n",
+ "\n",
+ "$N$ is the set of nodes \n",
+ "$\\delta(i)$ the set of edges incident to $i$ \n",
+ "$x_e$ indicator variable for edge $e$ \n",
+ "$X^1$ set of indicator vectors of 1-trees\n",
+ "\n",
+ "A 1-tree is a subset $E' \\subset E$ of edges of which exactly two are incident to node 1 : $| E' \\cap \\delta(1) | = 2$ and $E'\\backslash \\delta(1)$ is a spanning tree on the remaining nodes $2, \\ldots, n$ (hence of cardinality $n-2$).\n",
+ "\n",
+ "Every Hamiltonian tour is a 1-tree that satisfies the degree constraint $|E' \\cap \\delta(i)| = 2$ for all nodes $i \\in N$. Conversely, every 1-tree that satisfies these degree constraints is a Hamiltonian tour."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Given a conceptual enumeration of the set of 1-tree : $\\{ E^t : t = 1, \\ldots, T_1 \\}$\n",
+ "\n",
+ "$$ x_e = \\sum_{t: e\\in E^t} \\lambda_t $$\n",
+ "\n",
+ "Subject to $\\lambda_t \\in \\{0, 1\\}$ for all $t$ and $\\sum_{t=1}^{T_1} \\lambda_t = 1$. \n",
+ "This defines the indicator vector of the single 1-tree for which $\\lambda_t = 1$\n",
+ "\n",
+ "The degree constraints become\n",
+ "\n",
+ "$$ \\sum_{e \\in \\delta(i)} x_e = \\sum_{e\\in \\delta(i)} \\sum_{t: e \\in E^t} \\lambda_t = \\sum_t d_i^t \\lambda_t = 2 $$\n",
+ "\n",
+ "Where $d_i^t$ is the degree of node $i$ in the 1-tree $t$.\n",
+ "\n",
+ "The IP Master Problem being \n",
+ "\n",
+ "$$ \\text{(IPM)} \\quad \n",
+ "\\begin{array}{r l}\n",
+ "\\min & \\sum_{t = 1}^{T_1} \\left( \\sum_{e\\in E} c_e x_e^t \\right) \\lambda_t \\\\\n",
+ "\\text{s.t.}\n",
+ "& \\sum_{t = 1}^{T_1} d_i^t \\lambda_t = 2 \\quad (i \\in N) \\\\\n",
+ "& \\sum_{t = 1}^{T_1} \\lambda_t = 1 \\\\\n",
+ "& \\lambda_t \\in \\{0, 1\\}, \\quad (t = 1, \\ldots , T_1)\n",
+ "\\end{array}$$"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "function IPM(C, D)\n",
+ " ipm = Model(Cbc.Optimizer)\n",
+ " set_silent(ipm)\n",
" \n",
- " Tf = zeros(n,n)\n",
- " for i in 1:m\n",
- " Tf .+= S[i] .* T_List[i]\n",
- " end\n",
+ " @variable(ipm, λ[1:length(C)], Bin)\n",
" \n",
- " return Tf\n",
+ " @objective(ipm, Min, dot(C, λ))\n",
+ " \n",
+ " @constraint(ipm, degree[i in 1:size(D,1)], dot(D[i,:], λ) == 2)\n",
+ " @constraint(ipm, convexity, sum(λ) == 1)\n",
+ " \n",
+ " optimize!(ipm)\n",
+ " checkTermination(ipm)\n",
+ " \n",
+ " return ipm\n",
"end"
]
},
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Thus the LP relaxation of (IPM) called LP Master Problem\n",
+ "\n",
+ "$$ \\text{(LPM)} \\quad \n",
+ "\\begin{array}{r l}\n",
+ "\\min & \\sum_{t = 1}^{T_1} \\left( \\sum_{e\\in E} c_e x_e^t \\right) \\lambda_t \\\\\n",
+ "\\text{s.t.}\n",
+ "& \\sum_{t = 1}^{T_1} d_i^t \\lambda_t = 2 \\quad (i \\in N) \\\\\n",
+ "& \\sum_{t = 1}^{T_1} \\lambda_t = 1 \\\\\n",
+ "& \\lambda_t \\geq 0, \\quad (t = 1, \\ldots , T_1)\n",
+ "\\end{array}$$"
+ ]
+ },
{
"cell_type": "code",
- "execution_count": 33,
+ "execution_count": null,
"metadata": {
- "collapsed": true,
"jupyter": {
- "outputs_hidden": true,
"source_hidden": true
},
"tags": []
},
+ "outputs": [],
+ "source": [
+ "function LPM(C, D)\n",
+ " lpm = Model(Cbc.Optimizer)\n",
+ " set_silent(lpm)\n",
+ " \n",
+ " @variable(lpm, λ[1:length(C)] >= 0)\n",
+ " \n",
+ " @objective(lpm, Min, dot(C, λ))\n",
+ " \n",
+ " @constraint(lpm, degree[i in 1:size(D,1)], dot(D[i,:], λ) == 2)\n",
+ " @constraint(lpm, convexity, sum(λ) == 1)\n",
+ " \n",
+ " optimize!(lpm)\n",
+ " checkTermination(lpm)\n",
+ " \n",
+ " return lpm\n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "In (LPM) there is a column $ ( c^T x^t , d_1^t , \\ldots , d_n^t , 1 )^T$ for every $x^t \\in X^1 \\ (t = 1, \\ldots, T_1)$ \n",
+ "The second equation is a convexity constraint\n",
+ "\n",
+ "Associating the second equation with dual variable $\\mu$ and the degree constraints with dual variables $\\{\\pi_i\\}_{i=1}^m$ the dual of (LPM) is\n",
+ "\n",
+ "$$\\text{(DM)} \\quad\n",
+ "\\begin{array}{r l}\n",
+ "\\max_{\\mu, \\pi} & \\sum_{i=1}^m 2 \\pi_i + \\mu \\\\\n",
+ "\\text{s.t.} & \\pi^T d^t + \\mu \\leq c^T x^t \\quad (t = 1 \\ldots T_1)\n",
+ "\\end{array}$$"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "function DM(C, D)\n",
+ " dm = Model(Cbc.Optimizer)\n",
+ " set_silent(dm)\n",
+ " \n",
+ " @variable(dm, π[1:size(D,1)])\n",
+ " @variable(dm, μ)\n",
+ " \n",
+ " @objective(dm, Max, 2*sum(π) + μ)\n",
+ " \n",
+ " @constraint(dm, oneTree[t in 1:size(D,2)], dot(π, D[:,t]) + μ <= C[t])\n",
+ " \n",
+ " optimize!(dm)\n",
+ " checkTermination(dm)\n",
+ " \n",
+ " return dm\n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "The associated subsidiary problem is the following\n",
+ "\n"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "function SP()\n",
+ " sp = Model(Cbc.Optimizer)\n",
+ " set_silent(sp)\n",
+ " \n",
+ " @variable(sp, )\n",
+ " \n",
+ " @objective(sp, Min, (c - π^T A ) x - μ)\n",
+ " \n",
+ " @constraint(sp, degree1)\n",
+ " @constraint(sp, nbEdges, sum(x[2:]) == )\n",
+ " @constraint(sp, spanningTree)\n",
+ " \n",
+ " optimize!(sp)\n",
+ " checkTermination(sp)\n",
+ " \n",
+ " return sp\n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {
+ "tags": []
+ },
+ "source": [
+ "Our algorithm will follow this flowchart, classic for column generation algorithm\n",
+ "\n",
+ ""
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {
+ "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": 6,
+ "metadata": {
+ "tags": []
+ },
"outputs": [
{
"data": {
@@ -172,30 +395,23 @@
"wheel_like (generic function with 1 method)"
]
},
- "execution_count": 33,
+ "execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
- "function wheel_like(n)\n",
+ "function wheel_like_1trees(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",
+ " one_tree[k,v] = 1 ; 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",
+ " one_tree[1,k] = 1 ; one_tree[1, n==k ? 2 : k+1] = 1\n",
+ " one_tree[k,1] = 1 ; one_tree[n==k ? 2 : k+1, 1] = 1 \n",
" \n",
" push!(res, one_tree)\n",
" end\n",
@@ -204,13 +420,82 @@
"end"
]
},
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "function column_generation(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",
+ " \n",
+ " while reduced_cost < 0\n",
+ " \n",
+ " push!(columns, new_column)\n",
+ " tree_degrees = hcat(tree_degrees, [sum(new_column[u,:]) for u ∈ 1:n]) \n",
+ " push!(tree_weights, dot(new_column, W))\n",
+ " \n",
+ " dm = DM(tree_weights, tree_degrees)\n",
+ " π = value.(dm[:π])\n",
+ " μ = value(dm[:μ])\n",
+ " \n",
+ " # Solve subproblem\n",
+ " sp = SP()\n",
+ " new_column = value.(sp[:??????])\n",
+ " new_degree = [sum(new_column[u,:]) for u ∈ 1:n]\n",
+ " new_weight = dot(new_column, W)\n",
+ " reduced_cost = objective_value(sp)\n",
+ " 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(column_generation(W))\n",
+ "display(column_generation(W2))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Held and Karp algorithm - attempt 1\n",
+ "\n",
+ "This is the part where we implement the Held and Karp algorithm as described in J.E. Mitchell's slides using Kruskal's algorithm to solve the minimum spanning tree problem.\n",
+ "\n",
+ "However this approach seems to be flawed as the example is easily solved, however the exercise seems to alternate between two identical states. \n",
+ "This may be due to the update rule of $\\lambda$, results may vary by modifying it."
+ ]
+ },
{
"cell_type": "code",
"execution_count": 88,
"metadata": {
- "collapsed": true,
"jupyter": {
- "outputs_hidden": true,
"source_hidden": true
},
"tags": []
@@ -320,7 +605,7 @@
},
{
"cell_type": "code",
- "execution_count": 6,
+ "execution_count": 9,
"metadata": {
"collapsed": true,
"jupyter": {
@@ -330,883 +615,273 @@
},
"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"
+ "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": [
- "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",
- "]"
+ "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": 117,
+ "execution_count": null,
"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"
- }
- ],
+ "outputs": [],
"source": [
- "columnGeneration(W)"
+ "function held_and_karp_algo()\n",
+ " \n",
+ "end"
]
},
{
"cell_type": "code",
- "execution_count": 118,
+ "execution_count": null,
"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"
- }
- ],
+ "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 graph, 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": [
- "columnGeneration(W2)"
+ "function bellman_held_karp()\n",
+ " \n",
+ "end"
]
},
{
"cell_type": "code",
- "execution_count": 99,
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "display(bellman_held_karp(W))\n",
+ "display(bellman_held_karp(W2))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
"metadata": {
- "collapsed": true,
"jupyter": {
- "outputs_hidden": true
+ "source_hidden": true
},
"tags": []
},
- "outputs": [
- {
- "name": "stdout",
- "output_type": "stream",
- "text": [
- "[2.0, 1.0, 1.0, 2.0, 5.0, 1.0]\n",
- "[2.0, 1.0, 1.0, 2.0, 5.0, 1.0]\n",
- "[2.0, 1.0, 1.0, 2.0, 5.0, 1.0]\n",
- "[2.0, 1.0, 1.0, 3.0, 4.0, 1.0]\n",
- "[2.0, 1.0, 1.0, 5.0, 2.0, 1.0]\n",
- "[2.0, 1.0, 2.0, 5.0, 1.0, 1.0]\n",
- "[2.0, 1.0, 3.0, 4.0, 1.0, 1.0]\n",
- "[2.0, 1.0, 5.0, 2.0, 1.0, 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, 2.0, 1.0, 1.0, 1.0, 5.0]\n",
- "[2.0, 3.0, 1.0, 1.0, 1.0, 4.0]\n",
- "[2.0, 5.0, 1.0, 1.0, 1.0, 2.0]\n",
- "[2.0, 5.0, 1.0, 1.0, 1.0, 2.0]\n",
- "[2.0, 5.0, 1.0, 1.0, 1.0, 2.0]\n",
- "[2.0, 5.0, 1.0, 1.0, 1.0, 2.0]\n",
- "[2.0, 5.0, 1.0, 1.0, 1.0, 2.0]\n",
- "[2.0, 5.0, 1.0, 1.0, 1.0, 2.0]\n",
- "[2.0, 5.0, 1.0, 1.0, 1.0, 2.0]\n",
- "[2.0, 3.0, 2.0, 1.0, 1.0, 3.0]\n",
- "[2.0, 1.0, 5.0, 1.0, 1.0, 2.0]\n",
- "[2.0, 1.0, 5.0, 1.0, 1.0, 2.0]\n",
- "[2.0, 1.0, 5.0, 1.0, 1.0, 2.0]\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, 1.0, 2.0, 3.0, 3.0]\n",
- "[2.0, 1.0, 1.0, 3.0, 4.0, 1.0]\n",
- "[2.0, 1.0, 1.0, 3.0, 4.0, 1.0]\n",
- "[2.0, 1.0, 1.0, 4.0, 3.0, 1.0]\n",
- "[2.0, 1.0, 1.0, 4.0, 3.0, 1.0]\n",
- "[2.0, 1.0, 1.0, 4.0, 3.0, 1.0]\n",
- "[2.0, 1.0, 1.0, 4.0, 3.0, 1.0]\n",
- "[2.0, 1.0, 1.0, 3.0, 4.0, 1.0]\n",
- "[2.0, 1.0, 1.0, 3.0, 4.0, 1.0]\n",
- "[2.0, 1.0, 1.0, 3.0, 4.0, 1.0]\n",
- "[2.0, 1.0, 1.0, 3.0, 4.0, 1.0]\n",
- "[2.0, 1.0, 1.0, 4.0, 3.0, 1.0]\n",
- "[2.0, 1.0, 1.0, 2.0, 2.0, 4.0]\n",
- "[2.0, 1.0, 1.0, 2.0, 1.0, 5.0]\n",
- "[2.0, 1.0, 1.0, 2.0, 1.0, 5.0]\n",
- "[2.0, 1.0, 1.0, 2.0, 1.0, 5.0]\n",
- "[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",
- "[2.0, 1.0, 3.0, 1.0, 1.0, 4.0]\n",
- "[2.0, 4.0, 3.0, 1.0, 1.0, 1.0]\n",
- "[2.0, 5.0, 2.0, 1.0, 1.0, 1.0]\n",
- "[2.0, 5.0, 2.0, 1.0, 1.0, 1.0]\n",
- "[2.0, 5.0, 2.0, 1.0, 1.0, 1.0]\n",
- "[2.0, 5.0, 2.0, 1.0, 1.0, 1.0]\n",
- "[2.0, 3.0, 4.0, 1.0, 1.0, 1.0]\n",
- "[2.0, 2.0, 5.0, 1.0, 1.0, 1.0]\n",
- "[2.0, 2.0, 5.0, 1.0, 1.0, 1.0]\n",
- "[2.0, 2.0, 5.0, 1.0, 1.0, 1.0]\n",
- "[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",
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- "[2.0, 1.0, 1.0, 2.0, 3.0, 3.0]\n",
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- "[2.0, 2.0, 1.0, 5.0, 1.0, 1.0]\n",
- "["
- ]
- },
- {
- "ename": "InterruptException",
- "evalue": "InterruptException:",
- "output_type": "error",
- "traceback": [
- "InterruptException:",
- "",
- "Stacktrace:",
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- " [2] wait at .\\task.jl:710 [inlined]",
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- " [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)"
+ "## 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": 90,
- "metadata": {},
+ "execution_count": 3,
+ "metadata": {
+ "collapsed": true,
+ "jupyter": {
+ "outputs_hidden": true,
+ "source_hidden": true
+ },
+ "tags": []
+ },
"outputs": [
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List([Compose.Property{Compose.LineWidthPrimitive}(Compose.LineWidthPrimitive[Compose.LineWidthPrimitive(1.2247448713915892mm)]), Compose.Property{Compose.FillPrimitive}(Compose.FillPrimitive[Compose.FillPrimitive(RGBA{Float64}(0.0,0.0,0.0,0.0))]), Compose.Property{Compose.StrokePrimitive}(Compose.StrokePrimitive[Compose.StrokePrimitive(RGBA{Float64}(0.8274509803921568,0.8274509803921568,0.8274509803921568,1.0))])]), 0, false, false, false, false, nothing, nothing, 0.0, Symbol(\"\")), Compose.Context(Measures.BoundingBox{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}},Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}((0.0w, 0.0h), (1.0w, 1.0h)), nothing, nothing, nothing, nothing, List([]), List([]), List([Compose.Property{Compose.LineWidthPrimitive}(Compose.LineWidthPrimitive[Compose.LineWidthPrimitive(1.2247448713915892mm)]), Compose.Property{Compose.StrokePrimitive}(Compose.StrokePrimitive[Compose.StrokePrimitive(RGBA{Float64}(0.8274509803921568,0.8274509803921568,0.8274509803921568,1.0))])]), 0, false, false, false, false, nothing, nothing, 0.0, Symbol(\"\")), Compose.Context(Measures.BoundingBox{Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}},Tuple{Measures.Length{:w,Float64},Measures.Length{:h,Float64}}}((0.0w, 0.0h), (1.0w, 1.0h)), nothing, nothing, nothing, nothing, List([]), List([]), List([Compose.Property{Compose.FontSizePrimitive}(Compose.FontSizePrimitive[Compose.FontSizePrimitive(4.0mm)]), Compose.Property{Compose.StrokePrimitive}(Compose.StrokePrimitive[Compose.StrokePrimitive(RGBA{Float64}(0.0,0.0,0.0,0.0))]), Compose.Property{Compose.FillPrimitive}(Compose.FillPrimitive[Compose.FillPrimitive(RGBA{Float64}(0.0,0.0,0.0,1.0))])]), 0, false, false, false, false, nothing, nothing, 0.0, Symbol(\"\")), 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), 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+ "columnGeneration (generic function with 1 method)"
]
},
- "execution_count": 90,
+ "execution_count": 3,
"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",
+ "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, Bin)\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",
- "S2 = hk(W2)\n",
+ " st = Matrix(adjacency_matrix(Graph(kruskal_mst(g, W_updated[2:n,2:n]))))\n",
+ " T = zeros(n,n)\n",
"\n",
- "gplot(Graph(S2), nodelabel=1:size(W2,1))"
+ " 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"
]
}
],
diff --git a/heldAndKarp_fluff.ipynb b/heldAndKarp_fluff.ipynb
index c7255e9..3b811f1 100644
--- a/heldAndKarp_fluff.ipynb
+++ b/heldAndKarp_fluff.ipynb
@@ -1536,15 +1536,15 @@
],
"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,
diff --git a/wheel_like_1-trees.png b/wheel_like_1-trees.png
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--
GitLab