Therefore, we will call LP, the following linear relaxation of F1 On the other hand, once the integrality conditions have been removed from F2, symmetry conditions (14) alone do not longer guarantee that all the vertices will be even. However, constraints (15) should also be satisfied by any solution x to F2. In what follows, we show that LPO fg is stronger that LPOr ;. Consider a feasible solution x to LPOpe2 and define again ye = a4; + 2;;, Ve = (i,9) € Er and yj = xj;, V(i,j) € A. If « satisfies the symmetry constraints (28) for every vertex, y satisfies all the system equations and also the balanced-set constraints (20) for each vertex. Then, from any feasible solution to LPOr2 we obtain a feasible solution to LPOr, with the same cost. However, the converse is not true, as the following example shows. The graph in Figure 1, in which all the links are required, represents a feasible solution to LPO; (although the balanced-set conditions are not satisfied for every subset S of vertices). However, there is no way of directing the edges so as, with the same cost, obtaining a symmetric graph. Therefore, the lower bound obtained using LPO; cannot be greater than the lower bound obtained using LPOr2. This fact is reflected in column LBO in the tables in Section 4. Figure 1: Feasible solution to LPOr1 Table 2: MCPP instances from Madrigueras graph Table 1: MCPP instances from Albaida graph Table 4: MRPP instances from Albaida graph Table 3: MCPP instances from Aldaya graph Table 5: MRPP instances from Madrigueras graph Table 6: MRPP instances from Aldaya graph Table 7: MGRP instances from Albaida graph Table 9: MGRP instances from Aldaya graph
