False: exact NP-hardness rules out constant-factor approximation

Exact optimization of minimum vertex cover is NP-hard. Indeed, a polynomial-time algorithm computing ${\mathrm{OPT}}_{\mathrm{V}\mathrm{C}}(G)$ exactly would decide VERTEX COVER by computing ${\mathrm{OPT}}_{\mathrm{V}\mathrm{C}}(G)$ and comparing it with the integer $k$, which answers a problem that [F1] records as NP-complete; hence no polynomial-time exact optimizer exists unless $P=NP$.

Nonetheless minimum vertex cover admits an unconditional constant-factor approximation: by [F2] the maximal-matching endpoints form a deterministic polynomial-time computed vertex cover of value at most $2{\mathrm{OPT}}_{\mathrm{V}\mathrm{C}}$ on every finite simple graph, which by [F3] is precisely a polynomial-time $2$-approximation in the value-inequality sense. No hypothesis $P\mathrm{\ne }NP$ is used.

The four-vertex path $P_4$ with vertices $v_1,v_2,v_3,v_4$ and edges $v_1{v}_2,v_2{v}_3,v_3{v}_4$ illustrates both sides. The matching $\{v_2{v}_3\}$ is maximal with endpoint set $\{v_2,v_3\}$, a vertex cover of size $2$; one vertex meets at most two of the three edges, so no cover of size $1$ exists and ${\mathrm{OPT}}_{\mathrm{V}\mathrm{C}}(P_4)=2$. Scanning the edges in the order $v_1{v}_2,v_2{v}_3,v_3{v}_4$, the greedy procedure instead inserts $v_1{v}_2$ and then $v_3{v}_4$ and returns all four vertices, so $\mathrm{\mid }C\mathrm{\mid }=4=2\cdot 2$ realizes the factor-two upper bound, while the matching of the middle edge has size $1\le {\mathrm{OPT}}_{\mathrm{V}\mathrm{C}}(P_4)=2$.

Minimum vertex cover has NP-hard exact optimization by step 1.1 and an unconditional polynomial-time factor-two approximation by step 2.1, so it refutes the unqualified claim of [F1]. This does not refute a separate claim that approximation is impossible unless $P=NP$; such a conditional lower bound needs additional evidence, such as a gap reduction. The four-vertex path illustrates tightness, while the general theorem establishes the uniform ratio.