/* MFASP, Minimum Feedback Arc Set Problem */

/* Written in GNU MathProg by Andrew Makhorin <mao@gnu.org> */

/* The Minimum Feedback Arc Set Problem for a given directed graph

   G = (V, E), where V is a set of vertices and E is a set of arcs, is

   to find a minimal subset of arcs, which being removed from the graph

   make it acyclic.

   Reference:

   Garey, M.R., and Johnson, D.S. (1979), Computers and Intractability:

   A guide to the theory of NP-completeness [Graph Theory, Covering and

   Partitioning, Minimum Feedback Arc Set, GT9]. */

param n, integer, >= 0;

/* number of vertices */

set V, default 1..n;

/* set of vertices */

set E, within V cross V,

default setof{i in V, j in V: i <> j and Uniform(0,1) <= 0.15} (i,j);

/* set of arcs */

printf "Graph has %d vertices and %d arcs\n", card(V), card(E);

var x{(i,j) in E}, binary;

/* x[i,j] = 1 means that (i->j) is a feedback arc */

/* It is known that a digraph G = (V, E) is acyclic if and only if its

   vertices can be assigned numbers from 1 to |V| in such a way that

   k[i] + 1 <= k[j] for every arc (i->j) in E, where k[i] is a number

   assigned to vertex i. We may use this condition to require that the

   digraph G = (V, E \ E'), where E' is a subset of feedback arcs, is

   acyclic. */

var k{i in V}, >= 1, <= card(V);

/* k[i] is a number assigned to vertex i */

s.t. r{(i,j) in E}: k[j] - k[i] >= 1 - card(V) * x[i,j];

/* note that x[i,j] = 1 leads to a redundant constraint */

minimize obj: sum{(i,j) in E} x[i,j];

/* the objective is to minimize the cardinality of a subset of feedback

   arcs */

solve;

printf "Minimum feedback arc set:\n";

printf{(i,j) in E: x[i,j]} "%d %d\n", i, j;

data;

/* The optimal solution is 3 */

param n := 15;

set E := 1 2, 2 3, 3 4, 3 8, 4 9, 5 1, 6 5, 7 5, 8 6, 8 7, 8 9, 9 10,

         10 11, 10 14, 11 15, 12 7, 12 8, 12 13, 13 8, 13 12, 13 14,

         14 9, 15 14;

end;