void kMST_ILP::solve()
{
try {
// initialize CPLEX
env = IloEnv();
model = IloModel( env );
addTreeConstraints(); // call first, initialises edges
// add model-specific constraints
if( model_type == "scf" ) modelSCF();
else if( model_type == "mcf" ) modelMCF();
else if( model_type == "mtz" ) modelMTZ();
else {
cerr << "No existing model chosen\n";
exit( -1 );
}
addObjectiveFunction();
// build model
cplex = IloCplex( model );
// export model to a text file
//cplex.exportModel( "model.lp" );
// set parameters
setCPLEXParameters();
// solve model
cout << "Calling CPLEX solve ...\n";
cplex.solve();
cout << "CPLEX finished.\n\n";
cout << "CPLEX status: " << cplex.getStatus() << "\n";
cout << "Branch-and-Bound nodes: " << cplex.getNnodes() << "\n";
cout << "Objective value: " << cplex.getObjValue() << "\n";
cout << "CPU time: " << Tools::CPUtime() << "\n\n";
// show result
IloNumArray edgesSelected(env, edges.getSize()), flowRes(env, edges.getSize()), uRes(env, instance.n_nodes);
cplex.getValues(edgesSelected, edges);
if (model_type == "scf") {
cplex.getValues(flowRes, flow_scf);
} else if (model_type == "mtz") {
try {
cplex.getValues(uRes, u);
} catch ( IloException& e ) {
cerr << "Exception while extracting u: " << e << endl;
uRes = IloNumArray(env, 0);
}
}
cout << "Edges:\n";
for (unsigned int i=0; i<edges.getSize(); i++) {
// skip unused ones
if (((int)edgesSelected[i]) == 0 ) {
continue;
}
if (i == instance.n_edges) {
cout << endl;
}
bool direction = ( i >= instance.n_edges);
cout << " " << setw(4) << i << ": " << ((int)edgesSelected[i]) << " ";
// flow
if (model_type == "scf") {
cout << "f: " << setw(2) << (flowRes[i]);
} else if (model_type == "mtz") {
if (uRes.getSize() != 0) {
cout << "u: " ;
if (i < instance.n_edges) {
//cout << setw(2) << instance.edges[i % instance.n_edges].v1 << ": ";
cout << setw(2) <<((int)uRes[ instance.edges[i % instance.n_edges].v1]) << " ";
//cout << setw(2) << instance.edges[i % instance.n_edges].v2 << ": ";
cout << setw(2) << ((int)uRes[ instance.edges[i % instance.n_edges].v2]) ;
} else {
//cout << setw(2) << instance.edges[i % instance.n_edges].v2 << ": ";
cout << setw(2) <<((int)uRes[ instance.edges[i % instance.n_edges].v2]) << " ";
//cout << setw(2) << instance.edges[i % instance.n_edges].v1 << ": ";
cout << setw(2) <<((int)uRes[ instance.edges[i % instance.n_edges].v1]) ;
}
}
}
cout << " " << Tools::edgeToString(instance.edges[i % instance.n_edges], direction) ;
cout << endl;
}
} catch (IloAlgorithm::CannotExtractException& e) {
cerr << "CannotExtractException: " << e << endl ;
IloExtractableArray failed = e.getExtractables();
for (IloInt i = 0; i < failed.getSize(); ++i) {
cerr << "\t" << failed[i] << std::endl;
}
} catch( IloException& e ) {
cerr << "kMST_ILP: exception " << e << "\n";
exit( -1 );
}
catch( ... ) {
cerr << "kMST_ILP: unknown exception.\n";
exit( -1 );
}
}
Variables *kMST_ILP::modelMCF()
{
MCFVariables *v = new MCFVariables();
/***** generic part ***/
const vector<Instance::Edge> edges = directed_edges(instance.edges);
const u_int n_edges = edges.size();
/* $x_{ij} \in \{0, 1\}$ variables denote whether edge (i, j) is active. */
v->xs = createVarArrayXs(env, edges, n_edges);
/* $v_i \in \{0, 1\}$ variables denote whether node i is active. */
v->vs = createVarArrayVs(env, instance.n_nodes);
/* add objective function */
addObjectiveFunction(env, model, v->xs, edges, n_edges);
/* There are exactly k - 1 edges not counting edges from the artificial root node 0. */
addConstraint_k_minus_one_active_edges(env,model,v->xs,edges,n_edges,this->k);
/* Exactly one node is chosen as the tree root. */
addConstraint_one_active_outgoing_arc_for_node_zero(env,model,v->xs,edges,n_edges);
/* No edge leads back to the artificial root node 0. */
addConstraint_no_active_incoming_arc_for_node_zero(env,model,v->xs,edges,n_edges);
IloExprArray e_in_degree = createExprArray_in_degree(env, edges, n_edges, v->xs, instance);
IloExprArray e_out_degree = createExprArray_out_degree(env, edges, n_edges, v->xs, instance);
/* Inactive nodes have no outgoing active edges, active ones at most k - 1. TODO: A tighter bound is to take the sum of incoming goods - 1.*/
addConstraint_bound_on_outgoing_arcs(model,v->vs,e_out_degree,instance,this->k);
/* Active nodes have at least one active arc.*/
addConstraint_active_node_at_least_one_active_arc(model,v->vs,e_in_degree, e_out_degree,instance);
/* Exactly one incoming edge for an active node and none for an inactive node (omitting artificial root). */
addConstraint_in_degree_one_for_active_node_zero_for_inactive(model,v->vs,e_in_degree,instance);
//note: position matters. Tried worse positions than this one
/* $\sum_{i > 0} v_i = k$. Ensure that exactly k nodes are active. */
addConstraint_k_nodes_active(env, model, v->vs, instance, this->k);
e_in_degree.endElements();
e_out_degree.endElements();
/***** MCF specific part ***/
/* $f^k_{ij} \in \{0, 1\}$ variables denote the flow on edge (i, j) for commodity k. */
for (u_int i = 0; i < instance.n_nodes; i++) {
v->fss.push_back(IloBoolVarArray(env, n_edges));
}
for (u_int k = 0; k < n_edges; k++) {
const u_int i = edges[k].v1;
const u_int j = edges[k].v2;
for (u_int l = 0; l < (u_int) instance.n_nodes; l++) {
v->fss[l][k] = IloBoolVar(env, Tools::indicesToString("f", l, i, j).c_str());
}
}
/*
* Each commodity l is generated once by the artificial root node if node l is active, not at all otherwise:
* $\forall l \in \{1, \ldots, n\}: \sum_{j:j>0,(0,j) \in A} f^l_{0j} == v_l$
*/
for (u_int c = 1; c < instance.n_nodes; c++){
IloExpr e_one_commodity(env);
for (u_int m = 0; m < n_edges; m++) {
const u_int i = edges[m].v1;
const u_int j = edges[m].v2;
if (i == 0 && j > 0){
e_one_commodity += v->fss[c][m];
}
}
model.add(e_one_commodity == v->vs[c]);
e_one_commodity.end();
}
/*
* The artifical root generates k commodities:
* $\forall l \in \{0,\ldots,n\}\sum_{j:j>0,(0,j) \in A} f^l_{0j} = k$.
*/
IloExpr e_root_generates_k(env);
for (u_int c = 0; c < instance.n_nodes; c++){
for (u_int m = 0; m < n_edges; m++) {
const u_int i = edges[m].v1;
const u_int j = edges[m].v2;
if (i == 0 && j > 0){
e_root_generates_k += v->fss[c][m];
}
}
}
model.add(e_root_generates_k == this->k);
e_root_generates_k.end();
/*
* No commodity is generated for the artificial root:
* $\forall i, j: f^0_{ij} = 0$.
*/
for (u_int m = 0; m < n_edges; m++) {
model.add(v->fss[0][m] == 0);
}
/*
* Transmitted commodities end up at the target node:
* $\forall l>0: \sum_i f^l_{il} = \sum_j f^l_{0j}$. (here: = v_l)
*/
for (u_int c = 1; c < (u_int) instance.n_nodes; c++){
IloExpr e_commodity_reaches_target(env);
for (u_int m = 0; m < n_edges; m++) {
const u_int i = edges[m].v1;
const u_int j = edges[m].v2;
if (i != c && j == c){
e_commodity_reaches_target += v->fss[c][m];
}
}
model.add(e_commodity_reaches_target == v->vs[c]);
e_commodity_reaches_target.end();
}
/*
* Once reached, the commodity never leaves the target node:
* $\forall l>0: \sum_j f^l_{lj} = 0$.
*/
for (u_int c = 1; c < (u_int) instance.n_nodes; c++){
IloExpr e_commodity_stays_at_target(env);
for (u_int m = 0; m < n_edges; m++) {
const u_int i = edges[m].v1;
const u_int j = edges[m].v2;
if (i == c && j != c){
e_commodity_stays_at_target += v->fss[c][m];
}
}
model.add(e_commodity_stays_at_target == 0);
e_commodity_stays_at_target.end();
}
/*
* Flow is conserved when not at target node.
* $\forall j, l s.t. j \neq l: \sum_i f^l_{ij} = \sum_i f^l_{ji}$.
*/
for (u_int c = 0; c < (u_int) instance.n_nodes; c++){
IloExprArray e_in_flow(env, instance.n_nodes);
IloExprArray e_out_flow(env, instance.n_nodes);
for (u_int m = 0; m < instance.n_nodes; m++){
e_in_flow[m] = IloExpr(env);
e_out_flow[m] = IloExpr(env);
}
for (u_int m = 0; m < n_edges; m++) {
const u_int i = edges[m].v1;
const u_int j = edges[m].v2;
e_out_flow[i] += v->fss[c][m];
e_in_flow[j] += v->fss[c][m];
}
for (u_int m = 1; m < instance.n_nodes; m++){
if (m != c) {
model.add(e_in_flow[m] == e_out_flow[m]);
}
}
e_in_flow.endElements();
e_out_flow.endElements();
}
/*
* Commodities may only be transmitted on active edges:
* $\forall l, i, j: f^l_{ij} \leq x_{ij}$.
*/
for (u_int c = 0; c < (u_int) instance.n_nodes; c++){
for (u_int m = 0; m < n_edges; m++) {
model.add(v->fss[c][m] <= v->xs[m]);
}
}
/*
* For each commodity l , the total flow is <= k if node l is active, 0 otherwise
* (works well for all before g05, k=n/2 which is a bit slower with this)
*/
for (u_int c = 1; c < (u_int) instance.n_nodes; c++){
IloExpr e_total_flow(env);
for (u_int m = 0; m < n_edges; m++) {
e_total_flow += v->fss[c][m];
}
model.add(e_total_flow <= this->k * v->vs[c]);
e_total_flow.end();
}
return v;
}
Variables *kMST_ILP::modelMTZ()
{
MTZVariables *v = new MTZVariables();
/***** generic part ***/
const vector<Instance::Edge> edges = directed_edges(instance.edges);
const u_int n_edges = edges.size();
/* $x_{ij} \in \{0, 1\}$ variables denote whether edge (i, j) is active. */
v->xs = createVarArrayXs(env, edges, n_edges);
/* $v_i \in \{0, 1\}$ variables denote whether node i is active. */
v->vs = createVarArrayVs(env, instance.n_nodes);
/* add objective function */
addObjectiveFunction(env, model, v->xs, edges, n_edges);
/* There are exactly k - 1 edges not counting edges from the artificial root node 0. */
addConstraint_k_minus_one_active_edges(env,model,v->xs,edges,n_edges,this->k);
/* Exactly one node is chosen as the tree root. */
addConstraint_one_active_outgoing_arc_for_node_zero(env,model,v->xs,edges,n_edges);
/* No edge leads back to the artificial root node 0. */
addConstraint_no_active_incoming_arc_for_node_zero(env,model,v->xs,edges,n_edges);
IloExprArray e_in_degree = createExprArray_in_degree(env, edges, n_edges, v->xs, instance);
IloExprArray e_out_degree = createExprArray_out_degree(env, edges, n_edges, v->xs, instance);
/* Inactive nodes have no outgoing active edges, active ones at most k - 1. TODO: A tighter bound is to take the sum of incoming goods - 1.*/
addConstraint_bound_on_outgoing_arcs(model,v->vs,e_out_degree,instance,this->k);
/* Active nodes have at least one active arc.*/
addConstraint_active_node_at_least_one_active_arc(model,v->vs,e_in_degree, e_out_degree,instance);
/* Exactly one incoming edge for an active node and none for an inactive node (omitting artificial root). */
addConstraint_in_degree_one_for_active_node_zero_for_inactive(model,v->vs,e_in_degree,instance);
//note: position matters. Tried worse positions than this one
/* $\sum_{i > 0} v_i = k$. Ensure that exactly k nodes are active. */
addConstraint_k_nodes_active(env, model, v->vs, instance, this->k);
e_in_degree.endElements();
e_out_degree.endElements();
/***** MTZ specific part ***/
/* $u_i \in [0, k]$ variables are used to impose an order on nodes. */
v->us = IloIntVarArray(env, instance.n_nodes);
for (u_int i = 0; i < instance.n_nodes; i++) {
v->us[i] = IloIntVar(env, 0, k, Tools::indicesToString("u", i).c_str());
}
IloExpr e3(env);
e3 += v->us[0];
/* $u_0 = 0$. Set level of artificial root 0 to 0. */
model.add(e3 == 0);
e3.end();
for (u_int k = 0; k < n_edges; k++) {
const u_int i = edges[k].v1;
const u_int j = edges[k].v2;
IloExpr e4(env);
e4 = v->us[i] + v->xs[k] - v->us[j] - (-v->xs[k] + 1) * this->k;
/* $\forall i, j: u_i + x_{ij} \leq u_j + (1 - x_{ij})k$.
* Enforce order hierarchy on nodes. */
model.add(e4 <= 0);
e4.end();
}
for (u_int i = 0; i < instance.n_nodes; i++) {
/* $\forall i: u_i <= nv_i$ force order of inactive nodes to 0 */
/* helps with big instances 6,7,8 */
model.add(v->us[i] <= v->vs[i] * (int) instance.n_nodes);
}
return v;
}
Variables *kMST_ILP::modelSCF()
{
SCFVariables *v = new SCFVariables();
const vector<Instance::Edge> edges = directed_edges(instance.edges);
const u_int n_edges = edges.size();
/* $x_{ij} \in \{0, 1\}$ variables denote whether edge (i, j) is active. */
v->xs = createVarArrayXs(env, edges, n_edges);
/* $v_i \in \{0, 1\}$ variables denote whether node i is active. */
v->vs = createVarArrayVs(env, instance.n_nodes);
/* add objective function */
addObjectiveFunction(env, model, v->xs, edges, n_edges);
/* There are exactly k - 1 edges not counting edges from the artificial root node 0. */
addConstraint_k_minus_one_active_edges(env,model,v->xs,edges,n_edges,this->k);
/* Exactly one node is chosen as the tree root. */
addConstraint_one_active_outgoing_arc_for_node_zero(env,model,v->xs,edges,n_edges);
/* No edge leads back to the artificial root node 0. */
addConstraint_no_active_incoming_arc_for_node_zero(env,model,v->xs,edges,n_edges);
IloExprArray e_in_degree = createExprArray_in_degree(env, edges, n_edges, v->xs, instance);
IloExprArray e_out_degree = createExprArray_out_degree(env, edges, n_edges, v->xs, instance);
/* Inactive nodes have no outgoing active edges, active ones at most k - 1. TODO: A tighter bound is to take the sum of incoming goods - 1.*/
addConstraint_bound_on_outgoing_arcs(model,v->vs,e_out_degree,instance,this->k);
/* Active nodes have at least one active arc.*/
addConstraint_active_node_at_least_one_active_arc(model,v->vs,e_in_degree, e_out_degree,instance);
/* Exactly one incoming edge for an active node and none for an inactive node (omitting artificial root). */
addConstraint_in_degree_one_for_active_node_zero_for_inactive(model,v->vs,e_in_degree,instance);
//note: position matters. Tried worse positions than this one
/* $\sum_{i > 0} v_i = k$. Ensure that exactly k nodes are active. */
addConstraint_k_nodes_active(env, model, v->vs, instance, this->k);
e_in_degree.endElements();
e_out_degree.endElements();
/* $f_{ij} \in [0, k - 1]$ variables denote the number of goods on edge (i, j). */
v->fs = createVarArrayFs(env, edges, n_edges);
IloExprArray e_in_flow = createExprArray_in_flow(env, edges, n_edges, v->fs, instance);
IloExprArray e_out_flow = createExprArray_out_flow(env, edges, n_edges, v->fs, instance);
/*
* Active nodes consume exactly 1 commodity, inactive nodes conserve flow.
* $\forall i \neq 0: \sum_j (f_{ji} - f_{ij}) == v_i$
*/
for (u_int i = 0; i < instance.n_nodes; i++) {
if (i == 0){
/* Don't add a constraint for the artificial root. */
} else if (i > 0) {
/* outflow = inflow -1 for active nodes, same for inactive nodes. */
model.add(v->vs[i] == e_in_flow[i] - e_out_flow[i]);
}
}
e_in_flow.endElements();
e_out_flow.endElements();
/* $\forall i, j \neq 0: f_{ij} \leq kx_{ij}$. Only active edges transport goods.
* TODO: bound sum of incoming goods per node.
* $\forall i, j s.t. i or j is 0: f_{ij} = kx_{ij}$. Only a single edge
* incident on the artificial root transports goods. */
for (u_int k = 0; k < n_edges; k++) {
const u_int i = edges[k].v1;
const u_int j = edges[k].v2;
if (i == 0 || j == 0) {
model.add(v->fs[k] == this->k * v->xs[k]);
} else {
model.add(v->fs[k] <= (this->k) * v->xs[k]);
}
}
return v;
}
