The paper by Sartaj Sahni and Teofilo Gonzalez explores the P-completeness of approximation problems for several combinatorial optimization problems, including the traveling salesperson problem, cycle covers, 0-1 integer programming, multicommodity network flows, quadratic assignment, and general partitions. The authors show that the approximation problem for these P-complete problems is also P-complete, indicating that if P ≠ NP, then no polynomial-time approximation algorithm can guarantee solutions that are close to optimal. In contrast, they present a linear-time approximation algorithm for the clustering problem, demonstrating that some problems can be efficiently approximated. The paper provides detailed reductions and proofs for various problems, showing that the approximation problem for each is P-complete. The authors also discuss the implications of these results and highlight the difficulty of finding good approximate solutions for P-complete problems, especially in practical applications.The paper by Sartaj Sahni and Teofilo Gonzalez explores the P-completeness of approximation problems for several combinatorial optimization problems, including the traveling salesperson problem, cycle covers, 0-1 integer programming, multicommodity network flows, quadratic assignment, and general partitions. The authors show that the approximation problem for these P-complete problems is also P-complete, indicating that if P ≠ NP, then no polynomial-time approximation algorithm can guarantee solutions that are close to optimal. In contrast, they present a linear-time approximation algorithm for the clustering problem, demonstrating that some problems can be efficiently approximated. The paper provides detailed reductions and proofs for various problems, showing that the approximation problem for each is P-complete. The authors also discuss the implications of these results and highlight the difficulty of finding good approximate solutions for P-complete problems, especially in practical applications.