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/* MAXFLOW, Maximum Flow Problem */
/* Written in GNU MathProg by Andrew Makhorin <mao@gnu.org> */
/* The Maximum Flow Problem in a network G = (V, E), where V is a set of nodes, E within V x V is a set of arcs, is to maximize the flow from one given node s (source) to another given node t (sink) subject to conservation of flow constraints at each node and flow capacities on each arc. */
param n, integer, >= 2;/* number of nodes */
set V, default {1..n};/* set of nodes */
set E, within V cross V;/* set of arcs */
param a{(i,j) in E}, > 0;/* a[i,j] is capacity of arc (i,j) */
param s, symbolic, in V, default 1;/* source node */
param t, symbolic, in V, != s, default n;/* sink node */
var x{(i,j) in E}, >= 0, <= a[i,j];/* x[i,j] is elementary flow through arc (i,j) to be found */
var flow, >= 0;/* total flow from s to t */
s.t. node{i in V}:/* node[i] is conservation constraint for node i */
sum{(j,i) in E} x[j,i] + (if i = s then flow) /* summary flow into node i through all ingoing arcs */
= /* must be equal to */
sum{(i,j) in E} x[i,j] + (if i = t then flow); /* summary flow from node i through all outgoing arcs */
maximize obj: flow;/* objective is to maximize the total flow through the network */
solve;
printf{1..56} "="; printf "\n";printf "Maximum flow from node %s to node %s is %g\n\n", s, t, flow;printf "Starting node Ending node Arc capacity Flow in arc\n";printf "------------- ----------- ------------ -----------\n";printf{(i,j) in E: x[i,j] != 0}: "%13s %11s %12g %11g\n", i, j, a[i,j], x[i,j];printf{1..56} "="; printf "\n";
data;
/* These data correspond to an example from [Christofides]. */
/* Optimal solution is 29 */
param n := 9;
param : E : a := 1 2 14 1 4 23 2 3 10 2 4 9 3 5 12 3 8 18 4 5 26 5 2 11 5 6 25 5 7 4 6 7 7 6 8 8 7 9 15 8 9 20;
end;
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