simplex-glpk with modified glpk for fpgaHEADmaster

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+/* MAXCUT, Maximum Cut Problem */
+
+/* Written in GNU MathProg by Andrew Makhorin <mao@gnu.org> */
+
+/* The Maximum Cut Problem in a network G = (V, E), where V is a set
+   of nodes, E is a set of edges, is to find the partition of V into
+   disjoint sets V1 and V2, which maximizes the sum of edge weights
+   w(e), where edge e has one endpoint in V1 and other endpoint in V2.
+
+   Reference:
+   Garey, M.R., and Johnson, D.S. (1979), Computers and Intractability:
+   A guide to the theory of NP-completeness [Network design, Cuts and
+   Connectivity, Maximum Cut, ND16]. */
+
+set E, dimen 2;
+/* set of edges */
+
+param w{(i,j) in E}, >= 0, default 1;
+/* w[i,j] is weight of edge (i,j) */
+
+set V := (setof{(i,j) in E} i) union (setof{(i,j) in E} j);
+/* set of nodes */
+
+var x{i in V}, binary;
+/* x[i] = 0 means that node i is in set V1
+   x[i] = 1 means that node i is in set V2 */
+
+/* We need to include in the objective function only that edges (i,j)
+   from E, for which x[i] != x[j]. This can be modeled through binary
+   variables s[i,j] as follows:
+
+      s[i,j] = x[i] xor x[j] = (x[i] + x[j]) mod 2,                  (1)
+
+   where s[i,j] = 1 iff x[i] != x[j], that leads to the following
+   objective function:
+
+      z = sum{(i,j) in E} w[i,j] * s[i,j].                           (2)
+
+   To describe "exclusive or" (1) we could think that s[i,j] is a minor
+   bit of the sum x[i] + x[j]. Then introducing binary variables t[i,j],
+   which represent a major bit of the sum x[i] + x[j], we can write:
+
+      x[i] + x[j] = s[i,j] + 2 * t[i,j].                             (3)
+
+   An easy check shows that conditions (1) and (3) are equivalent.
+
+   Note that condition (3) can be simplified by eliminating variables
+   s[i,j]. Indeed, from (3) it follows that:
+
+      s[i,j] = x[i] + x[j] - 2 * t[i,j].                             (4)
+
+   Since the expression in the right-hand side of (4) is integral, this
+   condition can be rewritten in the equivalent form:
+
+      0 <= x[i] + x[j] - 2 * t[i,j] <= 1.                            (5)
+
+   (One might note that (5) means t[i,j] = x[i] and x[j].)
+
+   Substituting s[i,j] from (4) to (2) leads to the following objective
+   function:
+
+      z = sum{(i,j) in E} w[i,j] * (x[i] + x[j] - 2 * t[i,j]),       (6)
+
+   which does not include variables s[i,j]. */
+
+var t{(i,j) in E}, binary;
+/* t[i,j] = x[i] and x[j] = (x[i] + x[j]) div 2 */
+
+s.t. xor{(i,j) in E}: 0 <= x[i] + x[j] - 2 * t[i,j] <= 1;
+/* see (4) */
+
+maximize z: sum{(i,j) in E} w[i,j] * (x[i] + x[j] - 2 * t[i,j]);
+/* see (6) */
+
+data;
+
+/* In this example the network has 15 nodes and 22 edges. */
+
+/* Optimal solution is 20 */
+
+set E :=
+   1 2, 1 5, 2 3, 2 6, 3 4, 3 8, 4 9, 5 6, 5 7, 6 8, 7 8, 7 12, 8 9,
+   8 12, 9 10, 9 14, 10 11, 10 14, 11 15, 12 13, 13 14, 14 15;
+
+end;