Scilab function

min_qcost_flow - minimum quadratic cost flow

### Calling Sequence

[c,phi,flag] = min_qcost_flow(eps,g)

### Parameters

• eps : scalar, precision
• g : graph list
• c : value of cost
• phi : row vector of the value of flow on the arcs
• flag : feasible problem flag (0 or 1)

### Description

min_qcost_flow computes the minimum quadratic cost flow in the network g. It returns the total cost of the flows on the arcs c and the row vector of the flows on the arcs phi. eps is the precision of the iterative algorithm. If the problem is not feasible (impossible to find a compatible flow for instance), flag is equal to 0, otherwise it is equal to 1.

The bounds of the flow are given by the elements edge_min_cap and edge_max_cap of the graph list. The value of the maximum capacity must be greater than or equal to the value of the minimum capacity. If the value of edge_min_cap or edge_max_cap is not given (empty row vector []), it is assumed to be equal to 0 on each edge.

The costs on the edges are given by the elements edge_q_orig and edge_q_weight of the graph list. The cost on arc u is given by:

(1/2)*edge_q_weight[u](phi[u]-edge_q_orig[u])^2

The costs must be non negative. If the value of edge_q_orig or edge_q_weight is not given (empty row vector []), it is assumed to be equal to 0 on each edge.

This function uses an algorithm due to M. Minoux.

### Examples

```ta=[1 1 2 2 2 3 4 4 5 6 6 6 7 7 7 8 9 10 12 12 13 13 13 14 15 14 9 11 10 1 8];
he=[2 6 3 4 5 1 3 5 1 7 10 11 5 8 9 5 8 11 10 11 9 11 15 13 14 4 6 9 1 12 14];
g=make_graph('foo',1,15,ta,he);
g('node_x')=[194 191 106 194 296 305 305 418 422 432 552 550 549 416 548];
g('node_y')=[56 221 316 318 316 143 214 321 217 126 215 80 330 437 439];
show_graph(g);
g1=g; ma=arc_number(g1);
rand('uniform');
while %T then
g1('edge_min_cap')=round(5*rand(1,ma));
g1('edge_max_cap')=round(20*rand(1,ma))+30*ones(1,ma);
g1('edge_q_orig')=0*ones(1,ma);
g1('edge_q_weight')=ones(1,ma);
[c,phi,flag]=min_qcost_flow(0.001,g1);
if flag==1 then break; end;
end;
x_message(['The cost is: '+string(c);
'Showing the flow on the arcs']);
ii=find(phi<>0); edgecolor=phi; edgecolor(ii)=11*ones(ii);
g1('edge_color')=edgecolor;
edgefontsize=8*ones(1,ma); edgefontsize(ii)=18*ones(ii);
g1('edge_font_size')=edgefontsize;
g1('edge_label')=string(phi);
show_graph(g1);
```