/* MFVSP, Minimum Feedback Vertex Set Problem */

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

 /* The Minimum Feedback Vertex 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 vertices, 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 Vertex Set, GT8]. */

 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 in V}, binary;

 /* x[i] = 1 means that i is a feedback vertex */

 /* 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] + x[j]);

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

 minimize obj: sum{i in V} x[i];

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

    vertices */

 solve;

 printf "Minimum feedback vertex set:\n";

 printf{i in V: x[i]} "%d\n", i;

 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;