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alpar@9
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1 /* MAXFLOW, Maximum Flow Problem */
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alpar@9
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2
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alpar@9
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3 /* Written in GNU MathProg by Andrew Makhorin <mao@gnu.org> */
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alpar@9
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4
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alpar@9
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5 /* The Maximum Flow Problem in a network G = (V, E), where V is a set
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alpar@9
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6 of nodes, E within V x V is a set of arcs, is to maximize the flow
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alpar@9
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7 from one given node s (source) to another given node t (sink) subject
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8 to conservation of flow constraints at each node and flow capacities
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9 on each arc. */
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alpar@9
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10
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alpar@9
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11 param n, integer, >= 2;
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alpar@9
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12 /* number of nodes */
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alpar@9
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13
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alpar@9
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14 set V, default {1..n};
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alpar@9
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15 /* set of nodes */
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alpar@9
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16
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alpar@9
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17 set E, within V cross V;
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alpar@9
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18 /* set of arcs */
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alpar@9
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19
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alpar@9
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20 param a{(i,j) in E}, > 0;
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21 /* a[i,j] is capacity of arc (i,j) */
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alpar@9
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22
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alpar@9
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23 param s, symbolic, in V, default 1;
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alpar@9
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24 /* source node */
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alpar@9
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25
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alpar@9
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26 param t, symbolic, in V, != s, default n;
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alpar@9
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27 /* sink node */
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alpar@9
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28
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alpar@9
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29 var x{(i,j) in E}, >= 0, <= a[i,j];
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alpar@9
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30 /* x[i,j] is elementary flow through arc (i,j) to be found */
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alpar@9
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31
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32 var flow, >= 0;
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33 /* total flow from s to t */
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34
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alpar@9
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35 s.t. node{i in V}:
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36 /* node[i] is conservation constraint for node i */
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alpar@9
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37
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alpar@9
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38 sum{(j,i) in E} x[j,i] + (if i = s then flow)
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alpar@9
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39 /* summary flow into node i through all ingoing arcs */
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alpar@9
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40
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alpar@9
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41 = /* must be equal to */
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alpar@9
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42
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alpar@9
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43 sum{(i,j) in E} x[i,j] + (if i = t then flow);
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alpar@9
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44 /* summary flow from node i through all outgoing arcs */
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45
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alpar@9
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46 maximize obj: flow;
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alpar@9
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47 /* objective is to maximize the total flow through the network */
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48
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alpar@9
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49 solve;
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50
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alpar@9
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51 printf{1..56} "="; printf "\n";
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52 printf "Maximum flow from node %s to node %s is %g\n\n", s, t, flow;
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alpar@9
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53 printf "Starting node Ending node Arc capacity Flow in arc\n";
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alpar@9
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54 printf "------------- ----------- ------------ -----------\n";
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55 printf{(i,j) in E: x[i,j] != 0}: "%13s %11s %12g %11g\n", i, j,
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56 a[i,j], x[i,j];
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alpar@9
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57 printf{1..56} "="; printf "\n";
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58
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alpar@9
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59 data;
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60
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alpar@9
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61 /* These data correspond to an example from [Christofides]. */
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62
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63 /* Optimal solution is 29 */
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64
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65 param n := 9;
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66
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67 param : E : a :=
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68 1 2 14
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69 1 4 23
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70 2 3 10
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71 2 4 9
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72 3 5 12
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73 3 8 18
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74 4 5 26
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75 5 2 11
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76 5 6 25
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77 5 7 4
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78 6 7 7
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79 6 8 8
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80 7 9 15
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81 8 9 20;
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82
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alpar@9
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83 end;
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