5-Applications.ipynb

{
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Applications\n",
    "## Optimization on graphs / operations research"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "using LightGraphs\n",
    "import LightGraphsFlows\n",
    "using GraphPlot\n",
    "import Random\n",
    "using Colors\n",
    "\n",
    "using JuMP\n",
    "import Cbc\n",
    "import Clp"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Optimization on graphs\n",
    "\n",
    "Find a subset of the vertices / of the edges:\n",
    "- Minimizing a quantity (sum of costs, shortest path)\n",
    "- Respecting some criteria\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "\\min_{x, y} & f(x, y)\\\\\n",
    "& x_{i} \\in \\{0,1\\} \\,\\, \\forall i \\in V\\\\\n",
    "& y_{i,j} \\in \\left[0,1\\right] \\,\\, \\forall (i,j) \\in E\\\\\n",
    "& (x, y) \\in S\n",
    "\\end{align}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## JuMP\n",
    "\n",
    "Julia package and Embedded Domain-Specific Language (DSL)\n",
    "\n",
    "Macro-based syntax to define models, variables and constraints."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
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      "text/latex": [
       "$$ \\begin{alignat*}{1}\\max\\quad & x_{1} + x_{2} + x_{3} - y_{1,1} - y_{2,1} - y_{3,1} - y_{1,2} - y_{2,2} - y_{3,2} - y_{1,3} - y_{2,3} - y_{3,3}\\\\\n",
       "\\text{Subject to} \\quad & x_{1} - 4 y_{1,1} - 4 y_{1,2} - 4 y_{1,3} \\leq 0.0\\\\\n",
       " & x_{2} - 4 y_{2,1} - 4 y_{2,2} - 4 y_{2,3} \\leq 0.0\\\\\n",
       " & x_{3} - 4 y_{3,1} - 4 y_{3,2} - 4 y_{3,3} \\leq 0.0\\\\\n",
       " & x_{1} \\geq 0.0\\\\\n",
       " & x_{2} \\geq 0.0\\\\\n",
       " & x_{3} \\geq 0.0\\\\\n",
       " & y_{1,1} binary\\\\\n",
       " & y_{2,1} binary\\\\\n",
       " & y_{3,1} binary\\\\\n",
       " & y_{1,2} binary\\\\\n",
       " & y_{2,2} binary\\\\\n",
       " & y_{3,2} binary\\\\\n",
       " & y_{1,3} binary\\\\\n",
       " & y_{2,3} binary\\\\\n",
       " & y_{3,3} binary\\\\\n",
       "\\end{alignat*}\n",
       " $$"
      ],
      "text/plain": [
       "A JuMP Model\n",
       "Maximization problem with:\n",
       "Variables: 12\n",
       "Objective function type: GenericAffExpr{Float64,VariableRef}\n",
       "`GenericAffExpr{Float64,VariableRef}`-in-`MathOptInterface.LessThan{Float64}`: 3 constraints\n",
       "`VariableRef`-in-`MathOptInterface.GreaterThan{Float64}`: 3 constraints\n",
       "`VariableRef`-in-`MathOptInterface.ZeroOne`: 9 constraints\n",
       "Model mode: AUTOMATIC\n",
       "CachingOptimizer state: NO_OPTIMIZER\n",
       "Solver name: No optimizer attached.\n",
       "Names registered in the model: my_constraint, x, y"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m = Model()\n",
    "@variable(m, x[1:3] >= 0)\n",
    "@variable(m, y[1:3, 1:3], Bin)\n",
    "\n",
    "@constraint(m, my_constraint[i=1:3], x[i] <= 4 * sum(y[i,j] for j in 1:3))\n",
    "@objective(m, Max, sum(x) - sum(y))\n",
    "m"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "set_optimizer(m, Cbc.Optimizer)\n",
    "MOI.set(m, MOI.Silent(), true)\n",
    "optimize!(m)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3-element Array{Float64,1}:\n",
       " 12.0\n",
       " 12.0\n",
       " 12.0"
      ]
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     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "JuMP.value.(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Flow problems"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g = SimpleDiGraph(6)\n",
    "add_edge!(g, 1, 2)\n",
    "add_edge!(g, 1, 3)\n",
    "add_edge!(g, 3, 5)\n",
    "add_edge!(g, 5, 6)\n",
    "add_edge!(g, 2, 4)\n",
    "add_edge!(g, 2, 5)\n",
    "add_edge!(g, 4, 6)\n",
    "\n",
    "capacity = [\n",
    "    0 2 1 0 0 0\n",
    "    0 0 0 1 1 0\n",
    "    0 0 0 0 1 0\n",
    "    0 0 0 0 0 1\n",
    "    0 0 0 0 0 3\n",
    "    0 0 0 0 0 0\n",
    "]\n",
    "Random.seed!(5)\n",
    "edge_labels = [capacity[src(e),dst(e)] for e in edges(g)]\n",
    "gplot(g, nodelabel=1:6, edgelabel=edge_labels)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Flow problem\n",
    "\n",
    "Basic version: max-flow problem\n",
    "\n",
    "Objective: send maximum units of flow from **source** (s) to **sink** (t)  \n",
    "Constraints:\n",
    "- Edge capacity (max flow)\n",
    "- Flow conservation: at each non-terminal vertex, flow in = flow out\n",
    "\n",
    "Applications in logistics, physical flows, machine learning\n",
    "\n",
    "Formulation:\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "\\max_{f} & \\sum_{i\\in V, (i,t)\\in E} f_{it} \\\\\n",
    "& \\sum_{i,(i,v) \\in E} f_{iv} = \\sum_{j,(v,j) \\in E} f_{vj} & \\forall v \\in V \\backslash \\{s,t\\} \\\\\n",
    "& 0 \\leq f_{ij} \\leq c_{ij} & \\forall (i,j) \\in E\n",
    "\\end{align}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Version 1\n",
    "\n",
    "JuMP from scratch"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
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       "$$ \\begin{alignat*}{1}\\max\\quad & f_{Edge 4 => 6} + f_{Edge 5 => 6}\\\\\n",
       "\\text{Subject to} \\quad & f_{Edge 1 => 2} - f_{Edge 2 => 4} - f_{Edge 2 => 5} = 0.0\\\\\n",
       " & f_{Edge 1 => 3} - f_{Edge 3 => 5} = 0.0\\\\\n",
       " & f_{Edge 2 => 4} - f_{Edge 4 => 6} = 0.0\\\\\n",
       " & f_{Edge 2 => 5} + f_{Edge 3 => 5} - f_{Edge 5 => 6} = 0.0\\\\\n",
       " & f_{Edge 1 => 2} \\geq 0.0\\\\\n",
       " & f_{Edge 1 => 3} \\geq 0.0\\\\\n",
       " & f_{Edge 2 => 4} \\geq 0.0\\\\\n",
       " & f_{Edge 2 => 5} \\geq 0.0\\\\\n",
       " & f_{Edge 3 => 5} \\geq 0.0\\\\\n",
       " & f_{Edge 4 => 6} \\geq 0.0\\\\\n",
       " & f_{Edge 5 => 6} \\geq 0.0\\\\\n",
       " & f_{Edge 1 => 2} \\leq 2.0\\\\\n",
       " & f_{Edge 1 => 3} \\leq 1.0\\\\\n",
       " & f_{Edge 2 => 4} \\leq 1.0\\\\\n",
       " & f_{Edge 2 => 5} \\leq 1.0\\\\\n",
       " & f_{Edge 3 => 5} \\leq 1.0\\\\\n",
       " & f_{Edge 4 => 6} \\leq 1.0\\\\\n",
       " & f_{Edge 5 => 6} \\leq 3.0\\\\\n",
       "\\end{alignat*}\n",
       " $$"
      ],
      "text/plain": [
       "A JuMP Model\n",
       "Maximization problem with:\n",
       "Variables: 7\n",
       "Objective function type: GenericAffExpr{Float64,VariableRef}\n",
       "`GenericAffExpr{Float64,VariableRef}`-in-`MathOptInterface.EqualTo{Float64}`: 4 constraints\n",
       "`VariableRef`-in-`MathOptInterface.GreaterThan{Float64}`: 7 constraints\n",
       "`VariableRef`-in-`MathOptInterface.LessThan{Float64}`: 7 constraints\n",
       "Model mode: AUTOMATIC\n",
       "CachingOptimizer state: EMPTY_OPTIMIZER\n",
       "Solver name: Clp\n",
       "Names registered in the model: f, flow_conservation"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m = Model(Clp.Optimizer)\n",
    "\n",
    "@variable(m, 0 <= f[e in edges(g)] <= capacity[src(e), dst(e)])\n",
    "\n",
    "@constraint(m, flow_conservation[v in vertices(g); v ∉ (1,6)],\n",
    "    sum(f[Edge(i,v)] for i in inneighbors(g, v)) == sum(f[Edge(v,j)] for j in outneighbors(g, v))\n",
    ")\n",
    "\n",
    "@objective(m, Max, sum(f[e] for e in edges(g) if dst(e) == 6))\n",
    "m"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "MOI.set(m, MOI.Silent(), true)\n",
    "optimize!(m)\n",
    "fopt = round.(Int, JuMP.value.(f));"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Random.seed!(5)\n",
    "edge_labels = [\"$(fopt[e]) / $(capacity[src(e),dst(e)])\" for e in edges(g)]\n",
    "gplot(g, nodelabel=1:6, edgelabel=edge_labels)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Version 2\n",
    "\n",
    "Using LightGraphsFlows.jl"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(3, [0 2 … 0 0; -2 0 … 1 0; … ; 0 -1 … 0 2; 0 0 … -2 0])"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(max_flow_value, flow_var) = LightGraphsFlows.maximum_flow(\n",
    "    g, # graph\n",
    "    1, # source\n",
    "    6, # target\n",
    "    capacity\n",
    ")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
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     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
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   "source": [
    "Random.seed!(5)\n",
    "edge_labels = [\"$(flow_var[src(e),dst(e)]) / $(capacity[src(e),dst(e)])\" for e in edges(g)]\n",
    "gplot(g, nodelabel=1:6, edgelabel=edge_labels)"
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Minimum spanning tree\n",
    "\n",
    "Given a (weighted) connected graph,  \n",
    "how to connect all vertices at least cost?\n",
    "\n",
    "Application example: how to build a connected road system, a communication network...\n",
    "\n",
    "Custom algorithms: Kruskal and Prim $\\rightarrow$ `kruskal_mst` & `prim_mst`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
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       "this._drag.y=w;this._drag.id=r.identifier;!q.length&&a.mousemove(e).mouseup(l);q.push({el:this,move_scope:h,start_scope:n,end_scope:p});d&&k.on(\"snap.drag.start.\"+this.id,d);b&&k.on(\"snap.drag.move.\"+this.id,b);f&&k.on(\"snap.drag.end.\"+this.id,f);k(\"snap.drag.start.\"+this.id,n||h||this,v,w,r)}if(!arguments.length){var v;return this.drag(function(a,b){this.attr({transform:v+(v?\"T\":\"t\")+[a,b]})},function(){v=this.transform().local})}this._drag={};r.push({el:this,start:u});this.mousedown(u);return this};\n",
       "A.undrag=function(){for(var b=r.length;b--;)r[b].el==this&&(this.unmousedown(r[b].start),r.splice(b,1),k.unbind(\"snap.drag.*.\"+this.id));!r.length&&a.unmousemove(e).unmouseup(l);return this}});C.plugin(function(a,v,y,C){y=y.prototype;var A=/^\\s*url\\((.+)\\)/,w=String,z=a._.$;a.filter={};y.filter=function(d){var f=this;\"svg\"!=f.type&&(f=f.paper);d=a.parse(w(d));var k=a._.id(),u=z(\"filter\");z(u,{id:k,filterUnits:\"userSpaceOnUse\"});u.appendChild(d.node);f.defs.appendChild(u);return new v(u)};k.on(\"snap.util.getattr.filter\",\n",
       "function(){k.stop();var d=z(this.node,\"filter\");if(d)return(d=w(d).match(A))&&a.select(d[1])});k.on(\"snap.util.attr.filter\",function(d){if(d instanceof v&&\"filter\"==d.type){k.stop();var f=d.node.id;f||(z(d.node,{id:d.id}),f=d.id);z(this.node,{filter:a.url(f)})}d&&\"none\"!=d||(k.stop(),this.node.removeAttribute(\"filter\"))});a.filter.blur=function(d,f){null==d&&(d=2);return a.format('<feGaussianBlur stdDeviation=\"{def}\"/>',{def:null==f?d:[d,f]})};a.filter.blur.toString=function(){return this()};a.filter.shadow=\n",
       "function(d,f,k,u,p){\"string\"==typeof k&&(p=u=k,k=4);\"string\"!=typeof u&&(p=u,u=\"#000\");null==k&&(k=4);null==p&&(p=1);null==d&&(d=0,f=2);null==f&&(f=d);u=a.color(u||\"#000\");return a.format('<feGaussianBlur in=\"SourceAlpha\" stdDeviation=\"{blur}\"/><feOffset dx=\"{dx}\" dy=\"{dy}\" result=\"offsetblur\"/><feFlood flood-color=\"{color}\"/><feComposite in2=\"offsetblur\" operator=\"in\"/><feComponentTransfer><feFuncA type=\"linear\" slope=\"{opacity}\"/></feComponentTransfer><feMerge><feMergeNode/><feMergeNode in=\"SourceGraphic\"/></feMerge>',\n",
       "{color:u,dx:d,dy:f,blur:k,opacity:p})};a.filter.shadow.toString=function(){return this()};a.filter.grayscale=function(d){null==d&&(d=1);return a.format('<feColorMatrix type=\"matrix\" values=\"{a} {b} {c} 0 0 {d} {e} {f} 0 0 {g} {b} {h} 0 0 0 0 0 1 0\"/>',{a:0.2126+0.7874*(1-d),b:0.7152-0.7152*(1-d),c:0.0722-0.0722*(1-d),d:0.2126-0.2126*(1-d),e:0.7152+0.2848*(1-d),f:0.0722-0.0722*(1-d),g:0.2126-0.2126*(1-d),h:0.0722+0.9278*(1-d)})};a.filter.grayscale.toString=function(){return this()};a.filter.sepia=\n",
       "function(d){null==d&&(d=1);return a.format('<feColorMatrix type=\"matrix\" values=\"{a} {b} {c} 0 0 {d} {e} {f} 0 0 {g} {h} {i} 0 0 0 0 0 1 0\"/>',{a:0.393+0.607*(1-d),b:0.769-0.769*(1-d),c:0.189-0.189*(1-d),d:0.349-0.349*(1-d),e:0.686+0.314*(1-d),f:0.168-0.168*(1-d),g:0.272-0.272*(1-d),h:0.534-0.534*(1-d),i:0.131+0.869*(1-d)})};a.filter.sepia.toString=function(){return this()};a.filter.saturate=function(d){null==d&&(d=1);return a.format('<feColorMatrix type=\"saturate\" values=\"{amount}\"/>',{amount:1-\n",
       "d})};a.filter.saturate.toString=function(){return this()};a.filter.hueRotate=function(d){return a.format('<feColorMatrix type=\"hueRotate\" values=\"{angle}\"/>',{angle:d||0})};a.filter.hueRotate.toString=function(){return this()};a.filter.invert=function(d){null==d&&(d=1);return a.format('<feComponentTransfer><feFuncR type=\"table\" tableValues=\"{amount} {amount2}\"/><feFuncG type=\"table\" tableValues=\"{amount} {amount2}\"/><feFuncB type=\"table\" tableValues=\"{amount} {amount2}\"/></feComponentTransfer>',{amount:d,\n",
       "amount2:1-d})};a.filter.invert.toString=function(){return this()};a.filter.brightness=function(d){null==d&&(d=1);return a.format('<feComponentTransfer><feFuncR type=\"linear\" slope=\"{amount}\"/><feFuncG type=\"linear\" slope=\"{amount}\"/><feFuncB type=\"linear\" slope=\"{amount}\"/></feComponentTransfer>',{amount:d})};a.filter.brightness.toString=function(){return this()};a.filter.contrast=function(d){null==d&&(d=1);return a.format('<feComponentTransfer><feFuncR type=\"linear\" slope=\"{amount}\" intercept=\"{amount2}\"/><feFuncG type=\"linear\" slope=\"{amount}\" intercept=\"{amount2}\"/><feFuncB type=\"linear\" slope=\"{amount}\" intercept=\"{amount2}\"/></feComponentTransfer>',\n",
       "{amount:d,amount2:0.5-d/2})};a.filter.contrast.toString=function(){return this()}});return C});\n",
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     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Random.seed!(33)\n",
    "ba = barabasi_albert(8, 3, 3)\n",
    "\n",
    "st = kruskal_mst(ba)\n",
    "edge_width = [e ∈ st ? 2.5 : 1.0 for e in edges(ba)]\n",
    "edge_color = [e ∈ st ? colorant\"red\" : colorant\"lightgray\" for e in edges(ba)]\n",
    "\n",
    "gplot(ba, nodelabel = vertices(ba), edgelinewidth = edge_width, edgestrokec = edge_color)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
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Compose.LinePrimitive{Tuple{Measures.Measure,Measures.Measure}}(Tuple{Measures.Measure,Measures.Measure}[(0.43189254145891887cx, -0.11042668460002412cy), (0.9524264020282822cx, -0.9255066930790062cy)]), Compose.LinePrimitive{Tuple{Measures.Measure,Measures.Measure}}(Tuple{Measures.Measure,Measures.Measure}[(0.0958742579106337cx, -0.7884050216178183cy), (0.7571834202873071cx, -0.4301356935977119cy)]), Compose.LinePrimitive{Tuple{Measures.Measure,Measures.Measure}}(Tuple{Measures.Measure,Measures.Measure}[(0.10525814879142927cx, -0.8455441282005967cy), (0.9128999080068618cx, -0.9849642434581141cy)]), Compose.LinePrimitive{Tuple{Measures.Measure,Measures.Measure}}(Tuple{Measures.Measure,Measures.Measure}[(0.8579224330071861cx, -0.4733696164549142cy), (0.9769771883924637cx, -0.9146627271019052cy)])], Symbol(\"\"))]), List([Compose.Property{Compose.LineWidthPrimitive}(Compose.LineWidthPrimitive[Compose.LineWidthPrimitive(0.42426406871192845mm), 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      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Random.seed!(33)\n",
    "ba = barabasi_albert(8, 3, 3)\n",
    "\n",
    "weights = rand(8, 8)\n",
    "weights += weights'\n",
    "\n",
    "st = kruskal_mst(ba, weights)\n",
    "\n",
    "edge_width = [e ∈ st ? 2.5 : 1.0 for e in edges(ba)]\n",
    "edge_color = [e ∈ st ? colorant\"red\" : colorant\"lightgray\" for e in edges(ba)]\n",
    "\n",
    "lin_weights = [round(weights[src(e), dst(e)], digits=2) for e in edges(ba)]\n",
    "gplot(ba, nodelabel = vertices(ba), edgelinewidth = edge_width, edgestrokec = edge_color, edgelabel=lin_weights)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# What goes into LightGraphs?\n",
    "\n",
    "- Established problems $\\neq$ ongoing research\n",
    "- Fast algorithms $\\neq$ no NP-hard problems\n",
    "- Exact methods (guaranteed optimal solution)"
   ]
  }
 ],
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