Minimum vertex cover algorithm with linear programming
Consider the graph
  2-4---7
 /  |\ /|
1   | × |
 \  |/ \|
  3-5---6

Setting $x_1,x_2,x_3,x_6$ or $x_7$ to 0 gives your LP a value of 4, while setting $x_4$ or $x_5$ to 0 gives your LP a value of 5. However, if you set $x_1=0$, you will finally get a vertex cover of size 5, while the optimal vertex cover is of size 4.