* The model is based on a graph G = (V,E). * We have one binary variable x[e] for each edge e in E. That variable * is set to 1 if edge e is selected in the optimal tour and 0 otherwise. *
** The model contains only one explicit constraint: *
for each v in V: sum(u in V : u != v) x[uv] == 2* This states that from all the edges incident to a node u, exactly two * must be selected in the optimal tour. * *
* The above constraints ensures that the selected edges form tours. However, * it allows multiple tours, also known as subtours. So we need a constraint * that requires that there is only one tour (which then necessarily hits * all the nodes). This constraint is known as subtour elimination constraint * and is *
sum(e in S) x[e] <= |S|-1 for each subtour S
* Since there are exponentially many subtours in a graph, this constraint
* is not stated explicitly. Instead we check for any solution that the
* optimizer finds, whether it satisfies the subtour elimination constraint.
* If it does then we accept the solution. Otherwise we reject the solution
* and augment the model by the violated subtour eliminiation constraint.
*
* * This lazy addition of constraints is implemented using * a preintsol callback that rejects any solution that violates a * subtour elimination constraint and injects a violated subtour elimination * constraint in case the solution candidate came from an integral node. *
** An important thing to note about this strategy is that dual reductions * have to be disabled. Since the optimizer does not see the whole model * (subtour elimination constraints are only generated on the fly), dual * reductions may cut off the optimal solution. *
*/ public final class TSP { /** Number of nodes in the instance. */ private final int nodes; /** Number of edges in the instance. */ private final int edges; /** X coordinate of nodes. */ private final double[] nodeX; /** Y coordinate of nodes. */ private final double[] nodeY; /** Variable indices for the edges. */ private final int[][] x; /** Construct a new random instance with random seed 0. * @param nodes The number of nodes in the instance. */ public TSP(int nodes) { this(nodes, 0); } /** Construct a new random instance. * @param nodes The number of nodes in the instance. * @param seed Random number seed. */ public TSP(int nodes, int seed) { this.nodes = nodes; edges = (nodes * (nodes - 1)) / 2; nodeX = new double[nodes]; nodeY = new double[nodes]; Random rand = new Random(seed); for (int i = 0; i < nodes; ++i) { nodeX[i] = 4.0 * rand.nextDouble(); nodeY[i] = 4.0 * rand.nextDouble(); } x = new int[nodes][]; for (int i = 0; i < nodes; ++i) x[i] = new int[nodes]; } /** Get the distance between two nodes. * @param u First node. * @param v Second node. * @return The distance betweenu and v.
* The distance is symmetric.
*/
public double distance(int u, int v) {
return Math.sqrt((nodeX[u] - nodeX[v]) * (nodeX[u] - nodeX[v]) +
(nodeY[u] - nodeY[v]) * (nodeY[u] - nodeY[v]));
}
/** Find the tour rooted at 0 in a solution.
* As a side effect, the tour is printed to the console.
* @param sol The current solution.
* @param from Stores the tour. from[u] yields the
* predecessor of u in the tour. If
* from[u] is negative then u
* is not in the tour.
* This parameter can be null.
* @return The length of the tour.
*/
private int findTour(double[] sol, int[] from) {
if (from == null)
from = new int[nodes];
Arrays.fill(from, -1);
boolean[] inTour = new boolean[edges]; // Marks edges on the subtour
int u = 0;
int used = 0;
System.out.print("0");
do {
for (int v = 0; v < nodes; ++v) {
if (u == v) // no self-loops
continue;
else if (from[v] != -1) // node already on tour
continue;
else if (sol[x[u][v]] < 0.5) // edge not selected in solution
continue;
else if (inTour[x[u][v]]) // edge already on tour
continue;
else {
System.out.print(" -> " + v);
inTour[x[u][v]] = true;
from[v] = u;
used += 1;
u = v;
break;
}
}
} while (u != 0);
System.out.println();
return used;
}
/** Integer solution check callback.
*/
private final class PreIntsolListener extends AbstractPreIntsolListener {
@Override
public Double preIntsolEvent(XPRSprob prob, int soltype, double cutoff) {
System.out.println("Checking candidate solution ...");
// Get current solution and check whether it is feasible
double[] sol = prob.getCallbackSolution();
int[] from = new int[nodes];
int used = findTour(sol, from);
System.out.print("Solution is ");
if (used < nodes) {
// The tour given by the current solution does not pass through
// all the nodes and is thus infeasible.
// If soltype is non-zero then we reject by returning null.
// If instead soltype is zero then the solution came from an
// integral node. In this case we can reject by adding a cut
// that cuts off that solution. Note that we must NOT return
// null in that case because that would result in just dropping
// the node, no matter whether we add cuts or not.
System.out.println("infeasible (" + used + " edges)");
if (soltype != 0) {
return null;
}
else {
// The tour is too short. Get the edges on the tour and
// add a subtour elimination constraint
int[] ind = new int[used];
double[] val = new double[used];
for (int u = 0, next = 0; u < nodes; ++u) {
if (from[u] >= 0)
ind[next++] = x[u][from[u]];
}
Arrays.fill(val, 1);
// Since we created the constraint in the original space,
// we must crush it to the presolved space before we can
// add it.
RowInfo r = prob.presolveRow(ind, val, 'L', used - 1);
if (r != null)
prob.addCut(1, r);
else
throw new RuntimeException("failed to presolve subtour elimination constraint");
}
return cutoff; // Do not return null as that would drop the node
}
else {
System.out.println("feasible");
return cutoff;
}
}
}
/** Create a feasible tour and add this as initial MIP solution. */
private void createInitialTour(XPRSprob prob) {
int[] ind = new int[nodes];
double[] val = new double[nodes];
// Create a tour that just visits each node in order.
for (int i = 0; i < nodes; ++i) {
ind[i] = x[i][(i + 1) % nodes];
val[i] = 1.0;
}
prob.addMipSol(val, ind);
}
/** Solve the TSP represented by this instance.
*/
public void solve() {
try (XPRSprob prob = new XPRSprob(null)) {
// Create variables. We create one variable for each edge in
// the (complete) graph. x[u][v] gives the index of the variable
// that represents edge uv. Since edges are undirected, x[v][u]
// gives the same variable.
// All variables are binary.
for (int i = 0; i < nodes; ++i) {
for (int j = i + 1; j < nodes; ++j) {
x[j][i] = x[i][j] = prob.binVar("x_" + i + "_" + j);
}
}
// Objective. All variables are in the objective and their
// respective coefficient is the distance between the two nodes.
int[] objInd = new int[edges];
double[] objVal = new double[edges];
for (int i = 0, nz = 0; i < nodes; ++i) {
for (int j = i + 1; j < nodes; ++j) {
objInd[nz] = x[i][j];
objVal[nz] = distance(i, j);
++nz;
}
}
prob.setObjective(objInd, objVal, ObjSense.MINIMIZE);
// Constraint: In the graph that is induced by the selected
// edges, each node should have degree 2.
// This is the only constraint we add explicitly.
// Subtour elimination constraints are added
// dynamically via a callback.
for (int u = 0; u < nodes; ++u) {
int[] ind = new int[nodes - 1];
double[] val = new double[nodes - 1];
for (int v = 0, next = 0; v < nodes; ++v) {
if (u != v)
ind[next++] = x[u][v];
}
Arrays.fill(val, 1.0);
prob.addRow(ind, val, 'E', 2);
}
/* Note: with Java 1.8 this could also be written differently:
IntStream.range(0, nodes).forEach(
u -> prob.addRow(IntStream.range(0, nodes).filter(v -> v != u).map(v -> x[u][v]).toArray(),
DoubleStream.generate(() -> 1).limit(nodes - 1).toArray(),
'E', 2));
*/
// Create a starting solution.
// This is optional but having a feasible solution available right
// from the beginning can improve optimizer performance.
createInitialTour(prob);
// Write out the model in case we want to look at it.
prob.writeProb("tsp.lp", "l");
// We don't have all constraints explicitly in the matrix, hence
// we must disable dual reductions. Otherwise MIP presolve may
// cut off the optimal solution.
prob.controls().setMIPDualReductions(0);
// Add a callback that rejects solutions that do not satisfy
// the subtour constraints.
prob.addPreIntsolListener(new PreIntsolListener());
// Add a message listener to display log information.
prob.addMessageListener(DefaultMessageListener::console);
prob.mipOptimize();
double[] sol = prob.getSolution();
// Print the optimal tour.
System.out.println("Tour with length " + prob.attributes().getMIPBestObjVal());
findTour(sol, null);
}
}
public static void main(String[] args) {
new TSP(10).solve();
}
}