A vertex cover is a set S of vertices such that every edge has at least one endpoint in S. The minimum vertex cover problem is NP-hard, but admits a famous 2-approximation: repeatedly pick any uncovered edge and add both endpoints. The connection to König's theorem is exact for bipartite graphs: min vertex cover = max matching. By LP relaxation and half-integrality, the integrality gap is exactly 2. Improving beyond a 1.36-approximation would violate the Unique Games Conjecture (Khot-Regev 2008), placing vertex cover at a complexity frontier.