This paper shows that several P-complete problems, such as the traveling salesperson, cycle covers, 0-1 integer programming, multicommodity network flows, quadratic assignment, and others, have their approximation problems also P-complete. This means that if P ≠ NP, then no polynomial-time approximation algorithm can guarantee a good approximation for these problems. The authors demonstrate that for these problems, any polynomial-time approximation algorithm would produce arbitrarily bad results on some inputs. They also present a linear-time approximation algorithm for the clustering problem, which contrasts with the P-complete nature of other problems. The paper provides detailed proofs showing that the approximation problems for these P-complete problems are also P-complete, using reductions from known P-complete problems like the Hamiltonian cycle problem. The results highlight the computational difficulty of approximating these problems, even when exact solutions are not feasible in polynomial time.This paper shows that several P-complete problems, such as the traveling salesperson, cycle covers, 0-1 integer programming, multicommodity network flows, quadratic assignment, and others, have their approximation problems also P-complete. This means that if P ≠ NP, then no polynomial-time approximation algorithm can guarantee a good approximation for these problems. The authors demonstrate that for these problems, any polynomial-time approximation algorithm would produce arbitrarily bad results on some inputs. They also present a linear-time approximation algorithm for the clustering problem, which contrasts with the P-complete nature of other problems. The paper provides detailed proofs showing that the approximation problems for these P-complete problems are also P-complete, using reductions from known P-complete problems like the Hamiltonian cycle problem. The results highlight the computational difficulty of approximating these problems, even when exact solutions are not feasible in polynomial time.