Answer: (i) 2

Explanation:

• Classic 2-approximation for Minimum Vertex Cover (greedy matching-based):

  C = ∅
  while E ≠ ∅:
    pick any edge (u,v) ∈ E
    add both u and v to C
    remove all edges incident to u or v
  

  Any optimal cover must include at least one endpoint of each picked edge, and picked edges are disjoint. So $|C| ≤ 2·|OPT|$.

• Ratio 2 is tight; a simple maximal-matching algorithm achieves it. No polynomial algorithm can achieve $(2-\epsilon)$ unless $P=NP$ (inapproximability). The naive greedy "pick highest-degree vertex" can be $\Theta(\log n)$ bad, not 2.

Correct choice: (i) 2