notebooks/tsp/tsp-doodles.ipynb

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    "# Traveling Salesman Problem\n",
    "\n",
    "\n",
    "Peter Norvig has an excellent [notebook](https://github.com/norvig/pytudes/blob/master/ipynb/TSP.ipynb) on the Traveling Salesman Problem (TSP), which inspired me to play around some as well.\n",
    "\n",
    "The first part of the notebook duplicates his work, and thereafter I dabble with some other techniques to solve the TSP. None of these are really aimed at getting an efficient solvers, but to better understand aspects of the problems and play around with some libraries as well.\n",
    "\n",
    "For the first part, follow Norvig's notebook for explanations."
   ]
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  {
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   "source": [
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "%config InlineBackend.figure_format = 'svg'"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import itertools\n",
    "import random\n",
    "import time"
   ]
  },
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   "execution_count": 3,
   "metadata": {
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   "source": [
    "Tour = list\n",
    "\n",
    "def all_tours(cities):\n",
    "    start = first(cities)\n",
    "    return [[start] + Tour(rest) for rest in itertools.permutations(cities - {start})]\n",
    "\n",
    "def first(collection):\n",
    "    return next(iter(collection))\n",
    "\n",
    "def tour_length(tour):\n",
    "    return sum(distance(tour[i-1], tour[i]) for i in range(len(tour)))\n",
    "\n",
    "def shortest_tour(tours):\n",
    "    return min(tours, key=tour_length)"
   ]
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   "source": [
    "class Point(complex):\n",
    "    x = property(lambda p: p.real)\n",
    "    y = property(lambda p: p.imag)\n",
    "    \n",
    "    def __repr__(self):\n",
    "        return \"({}, {})\".format(self.x, self.y)\n",
    "    \n",
    "    def __lt__(self, other):\n",
    "        \"\"\" needed for heapq \"\"\"\n",
    "        return self.x < other.x\n",
    "    \n",
    "City = Point\n",
    "\n",
    "def distance(A, B):\n",
    "    return abs(A - B)\n",
    "\n",
    "def cities(n, max_x=900, max_y=600, seed=1):\n",
    "    \"\"\" Randomly generated cities\"\"\"\n",
    "    random.seed(n * seed)\n",
    "    return frozenset(City(random.randrange(max_x), random.randrange(max_y)) for _ in range(n))"
   ]
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   "source": [
    "def tsp_alltours(cities):\n",
    "    \"\"\" Find shortest tour by considering all permutations \"\"\"\n",
    "    return shortest_tour(all_tours(cities))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
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   "source": [
    "def plot_tour(tour):\n",
    "    f, ax = plt.subplots()\n",
    "    plot_lines(ax, list(tour) + [tour[0]])\n",
    "    plot_lines(ax, [first(tour)], style=\"rs\")\n",
    "    \n",
    "    ax.axis('scaled')\n",
    "    ax.axis('off')\n",
    "    \n",
    "def plot_lines(ax, points, style=\"bo-\"):\n",
    "    ax.plot([p.x for p in points], [p.y for p in points], style)   "
   ]
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   "source": [
    "def plot_tsp(algorithm, cities):\n",
    "    t_start = time.clock()\n",
    "    shortest_path = algorithm(cities)\n",
    "    t_end = time.clock()\n",
    "    \n",
    "    plot_tour(shortest_path)\n",
    "    \n",
    "    print(\"Tour of {} cities has total length {}\".format(len(cities), tour_length(shortest_path)))\n",
    "    print(\"Took {:.2f} seconds\".format(t_end - t_start))"
   ]
  },
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     "text": [
      "Tour of 9 cities has total length 2023.2435500436966\n",
      "Took 0.32 seconds\n"
     ]
    },
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    "plot_tsp(tsp_alltours, cities(9))"
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   "source": [
    "# Approximations\n",
    "\n",
    "Considering all tours is only feasible for a very limited number of cities. As Norvig points out, simple heuristics can be surprisingly good and are able to handle extremely large problem instances."
   ]
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    "def tsp_nn(cities, tour=None):\n",
    "    if tour == None: tour = [first(cities)]\n",
    "    unvisited = set(cities - set(tour))\n",
    "    \n",
    "    while unvisited:\n",
    "        nearest_city = nearest_neighbor(tour[-1], unvisited)\n",
    "        tour.append(nearest_city)\n",
    "        unvisited.remove(nearest_city)\n",
    "    \n",
    "    return tour\n",
    "\n",
    "def nearest_neighbor(current, collection):\n",
    "    return min(collection, key=lambda x: distance(current, x))\n",
    "\n",
    "def tsp_repeated_nn(cities, repetitions=100):\n",
    "    return min((tsp_nn(cities, [start]) \n",
    "                for start in sample(cities, repetitions)), \n",
    "               key=tour_length)\n",
    "               \n",
    "def sample(population, k, seed=42):\n",
    "    if k == None or k > len(population):\n",
    "        return population\n",
    "    \n",
    "    random.seed(len(population) * k * seed)\n",
    "    return random.sample(population, k)\n"
   ]
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     "text": [
      "Tour of 9 cities has total length 2261.5084843274626\n",
      "Took 0.00 seconds\n"
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    "plot_tsp(tsp_nn, cities(9))"
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      "Tour of 9 cities has total length 2023.2435500436964\n",
      "Took 0.00 seconds\n"
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   "source": [
    "plot_tsp(tsp_repeated_nn, cities(9))"
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  {
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   "metadata": {},
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     "text": [
      "Tour of 200 cities has total length 10363.1597481998\n",
      "Took 0.02 seconds\n"
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   "source": [
    "plot_tsp(tsp_nn, cities(200))"
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  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
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     "text": [
      "Tour of 500 cities has total length 14649.921933409672\n",
      "Took 5.88 seconds\n"
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    "# A* search\n",
    "\n",
    "The naive algorithm that considers all tours clearly does not scale well.\n",
    "Maybe we can do (slightly) better by trying segments that look promising first.\n",
    "\n",
    "What are promising segments? One way to quantify this is to find a lower bound on the tour length\n",
    "for a segment, and then expand the segment with the lowest bound.\n",
    "Obviously the segment length itself provides a simple but naive lower bound,\n",
    "but we can do better. \n",
    "\n",
    "Before diving into details, let's look at the big picture.\n",
    "We keep track of a heap (ordered by the lower bound) of segments,\n",
    "which we initialize with a trivial segment of a single node.\n",
    "In fact, we could even start with the empty set and add every single node,\n",
    "but it is not obvious to me whether that's better or worse.\n",
    "\n",
    "As long as we haven't found an optimal tour,\n",
    "we pop the best segment from the heap, and create new segments\n",
    "by adding an unvisited node to the end.\n",
    "We can then recompute the lower bound and add this segment to the heap,\n",
    "and we do this for every unvisited node.\n",
    "Once all new segments are added, we pop the next best segment from the heap.\n",
    "\n",
    "At some point we will find a complete tour, which gives us an upperbound on the length of the optimal tour.\n",
    "Now we only have to consider segments whose lower bound is less than the upperbound, and once we run out of those,\n",
    "we have found an optimal solution.\n",
    "\n",
    "We start with a simple PriorityQueue which is based on code provided in Stanford's CS 221 class."
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    "import heapq\n",
    "\n",
    "class PriorityQueue:\n",
    "    \"\"\" \n",
    "    Heap based on code from Stanford CS 221\n",
    "    \n",
    "    cs221.stanford.edu\n",
    "    \"\"\"\n",
    "    def  __init__(self):\n",
    "        self.DONE = -100000\n",
    "        self.heap = []\n",
    "        self.priorities = {}  # Map from state to priority\n",
    "\n",
    "    # Insert |state| into the heap with priority |newPriority| if\n",
    "    # |state| isn't in the heap or |newPriority| is smaller than the existing\n",
    "    # priority.\n",
    "    # Return whether the priority queue was updated.\n",
    "    def update(self, state, new_priority):\n",
    "        old_priority = self.priorities.get(state)\n",
    "        if old_priority == None or new_priority < old_priority:\n",
    "            self.priorities[state] = new_priority\n",
    "            heapq.heappush(self.heap, (new_priority, state))\n",
    "            return True\n",
    "        return False\n",
    "\n",
    "    # Returns (state with minimum priority, priority)\n",
    "    # or (None, None) if the priority queue is empty.\n",
    "    def pop(self):\n",
    "        while len(self.heap) > 0:\n",
    "            priority, state = heapq.heappop(self.heap)\n",
    "            if self.priorities[state] == self.DONE: continue  # Outdated priority, skip\n",
    "            self.priorities[state] = self.DONE\n",
    "            return (state, priority)\n",
    "        return (None, None) # Nothing left...\n",
    "    \n",
    "    def __repr__(self):\n",
    "        return \"PQ {}\".format(self.heap)"
   ]
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     "text": [
      "Tour of 7 cities has total length 1818.6687444623317\n",
      "Took 0.01 seconds\n"
     ]
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    "Now let's find a lower bound. \n",
    "Clearly the segment length provides a lower bound, but we can do better by noting that for each unvisited city,\n",
    "we have to add an edge to our current segment that is at least as far away as the closest other city.\n",
    "\n",
    "There are some refinements possible, for example by noting that for a tour, each city has 2 edges,\n",
    "so the minimum tour length $T$ is at least as large as\n",
    "$$\n",
    "T \\ge 1/2 * \\sum_{c \\in C} d_c(1) + d_c(2)\n",
    "$$\n",
    "where $C$ is the set of cities, and $d_c(1)$ is the distance to the closest city from $c$ and $d_c(2)$ is the distance to the second closest city from $c$."
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    "def compute_shortest_distances(nodes):\n",
    "    return {city: min(distance(city, other) \n",
    "                      for other in nodes if other != city) \n",
    "            for city in nodes}\n",
    "    \n",
    "\n",
    "def segment_length(segment):\n",
    "    return sum(distance(segment[i-1], segment[i]) for i in range(1, len(segment)))\n",
    "\n",
    "\n",
    "def lower_bound(nodes):\n",
    "    shortest_distances = compute_shortest_distances(nodes)\n",
    "    lb_empty = sum(shortest_distances.values())\n",
    "    \n",
    "    def ls(segment, unvisited):\n",
    "        if len(unvisited) == 0:\n",
    "            return tour_length(segment)\n",
    "        return segment_length(segment) + sum(shortest_distances[node] for node in unvisited)\n",
    "    \n",
    "    return ls"
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   "metadata": {},
   "source": [
    "For debugging, it's useful to be able to plot segments as well."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def plot_segment(cities, segment):\n",
    "    f, ax = plt.subplots()\n",
    "    plot_lines(ax, list(segment), style=\"r.-\")\n",
    "    plot_lines(ax, list(cities), style=\"bo\")\n",
    "    \n",
    "    ax.axis('scaled')\n",
    "    ax.axis('off')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "ename": "NameError",
     "evalue": "name 'example' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mNameError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-18-435eb39c0671>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mplot_segment\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexample\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mlist\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexample\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;36m3\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;31mNameError\u001b[0m: name 'example' is not defined"
     ]
    }
   ],
   "source": [
    "plot_segment(example, list(example)[:3])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now the meat of the algorithm.\n",
    "Our algorithm can be seen as a search over a search graph, which is different from the TSP graph.\n",
    "We define the current state by the segment (a tuple), and unvisited nodes (a frozenset) (which could be computed from the segment as well).\n",
    "We make sure both of these are immutable so we can hash them.\n",
    "Furthermore, this helps to avoid bugs."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# state = (segment, unvisited)\n",
    "\n",
    "\n",
    "# two helper functions to \n",
    "def extend_segment(segment, node):\n",
    "    return segment + (node, )\n",
    "\n",
    "def remove_fset(collection, element):\n",
    "    return frozenset(set(collection) - {element})\n",
    "\n",
    "\n",
    "class TSP_Problem:\n",
    "    def __init__(self, nodes):\n",
    "        self.nodes = nodes\n",
    "        self.lb = lower_bound(nodes)\n",
    "    \n",
    "    # use NN to get a crude upperbound\n",
    "    # nn_tour = tsp_nn(nodes)\n",
    "    # upper_bound = tour_length(nn_tour)\n",
    "    \n",
    "    def initial_state(self):\n",
    "        start = first(self.nodes)\n",
    "        return (start, ), remove_fset(self.nodes, start)\n",
    "\n",
    "    def new_states_and_cost(self, state):\n",
    "        segment, unvisited = state\n",
    "        current = segment[-1]\n",
    "        \n",
    "        for node in unvisited:\n",
    "            new_segment = extend_segment(segment, node)\n",
    "            new_unvisited = remove_fset(unvisited, node)\n",
    "            yield (new_segment, new_unvisited), self.lb(new_segment, new_unvisited)\n",
    "            \n",
    "    def is_goal(self, state):\n",
    "        _, unvisited = state\n",
    "        return len(unvisited) == 0\n",
    "    \n",
    "def solve_tsp_problem(tsp_problem, max_time=30):    \n",
    "    queue = PriorityQueue()\n",
    "    queue.update(tsp_problem.initial_state(), 0)\n",
    "        \n",
    "    t_start = time.clock()\n",
    "    while time.clock() - t_start < max_time:\n",
    "        state, cost = queue.pop()\n",
    "        segment, unvisited = state\n",
    "                \n",
    "        # if goal achieved, this must be shortest tour\n",
    "        if len(unvisited) == 0:\n",
    "            tour, _ = state\n",
    "            assert abs(cost - tour_length(tour)) < 1e-5, \"Lower bound {} does not correspond to tour length {}\".format(cost, tour_length(tour))\n",
    "            return tour            \n",
    "        \n",
    "        for new_state, new_cost in tsp_problem.new_states_and_cost(state):\n",
    "            queue.update(new_state, new_cost)\n",
    "            \n",
    "    print(\"Warning: no optimal solution found\")\n",
    "    return tsp_repeated_nn(tsp_problem.nodes)\n",
    "            \n",
    "def tsp_pq(cities):\n",
    "    tsp_problem = TSP_Problem(cities)\n",
    "    return solve_tsp_problem(tsp_problem)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true,
    "scrolled": false
   },
   "outputs": [],
   "source": [
    "for n in range(7, 13):\n",
    "    plot_tsp(tsp_pq, cities(n))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We note that we can handle slightly bigger problems now, \n",
    "but the gains are small, and we quickly hit the exponential wall.\n",
    "\n",
    "Potentially, we can improve performance with a tighter lower bound,\n",
    "but there are simply too many potential tours that need to be checked."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "# Branch and bound using LPs\n",
    "\n",
    "A very successful approach is the branch and bound / branch and cut methods.\n",
    "\n",
    "One general approach to solving hard problems is to relax! \n",
    "In this particular case, we can formulate the TSP as an Integer Linear Program:\n",
    "For each edge, we create a decision variable $x_e \\in \\{0, 1\\}$ that denotes whether the edge is included in the tour.\n",
    "Now we have to ensure that each node has exactly 2 adjacent edges.\n",
    "\n",
    "Unfortunately, there are two problems:\n",
    "The obvious one is that integer problems are hard.\n",
    "Furthermore, this does not ensure that we find 2 (or more) disconnected tours (or subtours).\n",
    "To avoid these, we have to include subtour elimination constraints that make such solutions infeasible.\n",
    "The problem is that there are exponentially many such constraints.\n",
    "\n",
    "So how do we relax? First, we let go of the integer condition and solve the LP.\n",
    "Furthermore, we ignore *all* subtour elimination constraints.\n",
    "Solving the relaxed LP is easy, and if it returns an integer solution without subtours \n",
    "we know we found an optimal solution.\n",
    "\n",
    "Furthermore, this approach has the benefit of providing us with another lower bound on the optimal value,\n",
    "which helps quantify how close a given tour is to the optimal tour.\n",
    "\n",
    "However, we often won't be so lucky, and need to deal with (1) non-integer solutions\n",
    "and (2) subtours.\n",
    "\n",
    "To handle non-integer solutions, we use the branch and bound method:\n",
    "Suppose edge $e$ is \"fractionally part of the tour\", i.e. $0 < x_e < 1$,\n",
    "then we branch into two problems: one for which we force $x_e = 0$, and one for which $x_e = 1$,\n",
    "and find the best one between the two.\n",
    "Of course we might have to repeat this for multiple such edges.\n",
    "This explains the branching part.\n",
    "The bounding part of the algorithm is about pruning the search tree of suboptimal paths.\n",
    "In particular, if we find a feasible integer solution with value $v$, \n",
    "then we can prune all paths where the current lower bound is larger than $v$.\n",
    "The branching factor is again exponential in the worst case, so there are no guarantees that\n",
    "this method is faster than the all-tours algorithm.\n",
    "\n",
    "Given an integer solution, we have to check whether there are any subtours.\n",
    "If we find a subtour, we add a constraint that prohibits those nodes forming a subtour,\n",
    "and solve the augmented LP using branch-and-bound again.\n",
    "\n",
    "Below I've used [cvxpy](www.cvxpy.org) to implement this strategy.\n",
    "While it is not tailored to LPs, it's intuitive and easy to use."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import cvxpy as cvx\n",
    "import numpy as np\n",
    "import collections"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 117,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def create_lp_relaxation(cities):\n",
    "    edges = {i: (a, b) for i, (a, b) in enumerate((a, b) \n",
    "                                                  for a in cities for b in cities \n",
    "                                                  if a < b)}\n",
    "    city_edges = {city: {i for i, (a, b) in edges.items() \n",
    "                         if city == a or city == b} \n",
    "                  for city in cities}\n",
    "    \n",
    "    m = len(edges)\n",
    "    x = cvx.Variable(m)\n",
    "\n",
    "    edge_cost = np.array([distance(edges[i][0], edges[i][1]) for i in range(m)])\n",
    "    objective = cvx.Minimize(edge_cost * x)\n",
    "\n",
    "    constraints = [x >= 0] + [x <= 1]\n",
    "    for city, indices in city_edges.items():\n",
    "        u = np.zeros(m)\n",
    "        u[list(indices)] = 1\n",
    "        constraints += [(u * x) == 2]\n",
    "\n",
    "    return cvx.Problem(objective, constraints), edges\n",
    "\n",
    "def get_solution(problem):\n",
    "    \"\"\"\n",
    "    Helper function to extract (optimal) solution from a problem\n",
    "    \"\"\"\n",
    "    return np.array(problem.variables()[0].value).squeeze()\n",
    "\n",
    "def find_subtour(solution, edges):\n",
    "    \"\"\"\n",
    "    Function that searches for a (sub)tour in the solution.\n",
    "    \"\"\"\n",
    "    \n",
    "    # find the edges that are used for the tour\n",
    "    active_edges = [i for i, v in enumerate(solution) if v > 1-1e-3]\n",
    "\n",
    "    # find which nodes are connected to create the search tree\n",
    "    active_nodes_edges = collections.defaultdict(set)\n",
    "    for e in active_edges:\n",
    "        a, b = edges[e]\n",
    "        active_nodes_edges[a].add(b)\n",
    "        active_nodes_edges[b].add(a)\n",
    "    \n",
    "    # find the nodes of the subtour\n",
    "    to_explore = [a]\n",
    "    visited = set()\n",
    "    while to_explore:\n",
    "        node = to_explore.pop()\n",
    "        visited.add(node)\n",
    "        for other in active_nodes_edges[node]:\n",
    "            if other not in visited:\n",
    "                to_explore.append(other)\n",
    "    \n",
    "    # find the edges of the subtour\n",
    "    return [e for e in edges if edges[e][0] in visited and edges[e][1] in visited], len(visited)\n",
    "        \n",
    "def find_noninteger(x):\n",
    "    \"\"\"\n",
    "    Find a \"fractional\" edge\n",
    "    \"\"\"\n",
    "    \n",
    "    # find the index of the value closest to 0.5\n",
    "    i = np.argmin(abs(x-0.5))\n",
    "    # make sure the value is not 0 or 1\n",
    "    v = min(x[i], 1-x[i])\n",
    "    if v > 1e-4:\n",
    "        return i\n",
    "    \n",
    "def add_constraints(problem, constraints):\n",
    "    return cvx.Problem(problem.objective, \n",
    "                       problem.constraints[:] + constraints)\n",
    "\n",
    "def add_subtour_elimination_constraints(problem, subtour, length, m):\n",
    "    u = np.zeros(m)\n",
    "    u[subtour] = 1\n",
    "    constraints = [u * problem.variables()[0] < length-0.5]\n",
    "    return cvx.Problem(problem.objective, problem.constraints[:] + constraints)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 131,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def tsp_bb(cities, tmax=60):\n",
    "    n = len(cities)\n",
    "    relaxation, edges = create_lp_relaxation(cities)\n",
    "    \n",
    "    upper_bound = 1e9\n",
    "    best_solution = None\n",
    "    \n",
    "    # add the initial relaxation\n",
    "    unsolved_queue = [(-1e9, relaxation)]\n",
    "    t = time.clock()\n",
    "    while unsolved_queue and time.clock() - t < tmax:\n",
    "        lower_bound, current_lp = unsolved_queue.pop(0)\n",
    "        \n",
    "        # if we found a better solution than the lower bound of the current \n",
    "        # relaxation, we can ignore this problem\n",
    "        if lower_bound > upper_bound:\n",
    "            continue\n",
    "        \n",
    "        value = current_lp.solve()        \n",
    "        if current_lp.status != cvx.OPTIMAL:\n",
    "            continue\n",
    "        \n",
    "        # find fractional edges\n",
    "        x = get_solution(current_lp)\n",
    "        non_integer_x = find_noninteger(x)\n",
    "        \n",
    "        if non_integer_x is not None and value < upper_bound:\n",
    "            # need to branch on variable x\n",
    "            unsolved_queue.append((value, \n",
    "                                   add_constraints(current_lp, \n",
    "                                                   [current_lp.variables()[0][non_integer_x] == 0])))\n",
    "            unsolved_queue.append((value, \n",
    "                                   add_constraints(current_lp, \n",
    "                                                   [current_lp.variables()[0][non_integer_x] == 1])))\n",
    "            continue\n",
    "            \n",
    "        if non_integer_x is None and value < upper_bound:\n",
    "            # found integer solution, need to check for subtours\n",
    "            subtour, length = find_subtour(x, edges)\n",
    "            if length < n:\n",
    "                unsolved_queue.append((value, add_subtour_elimination_constraints(current_lp, \n",
    "                                                                                  subtour, \n",
    "                                                                                  length, \n",
    "                                                                                  len(edges))))\n",
    "            else:\n",
    "                # found optimal solution\n",
    "                upper_bound = value\n",
    "                best_solution = x\n",
    "                \n",
    "    if best_solution is None:\n",
    "        print(\"Warning, no solution found\")\n",
    "        return None\n",
    "    \n",
    "    return [edges[i] for i, v in enumerate(best_solution) if v > 1-1e-3]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 132,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def plot_edges(edges):\n",
    "    f, ax = plt.subplots()\n",
    "\n",
    "    for edge in edges:\n",
    "        plot_lines(ax, edge, style=\"bo-\")\n",
    "\n",
    "    ax.axis('scaled')\n",
    "    ax.axis('off')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's solve an example with 20 cities.\n",
    "I also plot the intermediate integer solutions that have subtours \n",
    "to visualize the subtour elimination process."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 136,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Found tour in 0.58 seconds\n"
     ]
    },
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   "source": [
    "t_start = time.clock()\n",
    "solution = tsp_bb(cities(20))\n",
    "print(\"Found tour in {:.2f} seconds\".format(time.clock() - t_start))\n",
    "plot_edges(solution)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## A better lower bound\n",
    "\n",
    "We can also use the relaxation to obtain a lower bound on the optimal value that is better than our simple heuristic we used before:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 137,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Found lower bound of 7502.572881506507 in 0.83 seconds\n"
     ]
    }
   ],
   "source": [
    "n = 200\n",
    "\n",
    "relaxation, edges = create_lp_relaxation(cities(n))\n",
    "t_start = time.clock()\n",
    "lb = relaxation.solve()\n",
    "\n",
    "print(\"Found lower bound of {} in {:.2f} seconds\".format(lb, time.clock() - t_start))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 138,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
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     "execution_count": 138,
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   "source": [
    "lb = lower_bound(cities(n))\n",
    "lb([], cities(n))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 139,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Tour of 200 cities has total length 9474.624239249088\n",
      "Took 0.96 seconds\n"
     ]
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