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/* 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;