This paper presents strong negative results on the approximability of two important minimization problems: Graph Coloring and Set Covering. For Graph Coloring, it is shown that there exists a constant ε > 0 such that no polynomial-time approximation algorithm can achieve a ratio of n^ε unless P = NP. Similarly, for Set Covering, it is shown that the problem cannot be approximated within a ratio of c log N for any constant c < 1/4 unless NP is contained in DTIME(n^{poly log n}). The results are derived using recent advances in interactive proof systems and probabilistically checkable proofs. These techniques are used to establish lower bounds on the approximation ratios for both problems. The paper also discusses related minimization problems, such as Clique Cover, Fractional Chromatic Number, and Dominating Set, showing that they share similar approximability properties. For Graph Coloring, the paper presents a construction that reduces the maximum independent set problem to the chromatic number problem, showing that if the maximum independent set problem cannot be approximated within a certain factor, then the chromatic number problem also cannot be approximated within that factor. This construction uses a linear algebraic approach and involves transforming the graph into a new structure that preserves the approximability properties. For Set Covering, the paper uses a reduction from two-prover one-round interactive proof systems. This reduction shows that if the Set Covering problem cannot be approximated within a certain factor, then the corresponding interactive proof system cannot be simulated efficiently. The paper also discusses the implications of these results for related problems, including Hypergraph Transversal, Minimum Hitting Set, and Minimum Dominating Set. The paper concludes by showing that the results have significant implications for the approximability of these problems, demonstrating that they are hard to approximate within certain factors unless P = NP or NP is contained in a certain complexity class.This paper presents strong negative results on the approximability of two important minimization problems: Graph Coloring and Set Covering. For Graph Coloring, it is shown that there exists a constant ε > 0 such that no polynomial-time approximation algorithm can achieve a ratio of n^ε unless P = NP. Similarly, for Set Covering, it is shown that the problem cannot be approximated within a ratio of c log N for any constant c < 1/4 unless NP is contained in DTIME(n^{poly log n}). The results are derived using recent advances in interactive proof systems and probabilistically checkable proofs. These techniques are used to establish lower bounds on the approximation ratios for both problems. The paper also discusses related minimization problems, such as Clique Cover, Fractional Chromatic Number, and Dominating Set, showing that they share similar approximability properties. For Graph Coloring, the paper presents a construction that reduces the maximum independent set problem to the chromatic number problem, showing that if the maximum independent set problem cannot be approximated within a certain factor, then the chromatic number problem also cannot be approximated within that factor. This construction uses a linear algebraic approach and involves transforming the graph into a new structure that preserves the approximability properties. For Set Covering, the paper uses a reduction from two-prover one-round interactive proof systems. This reduction shows that if the Set Covering problem cannot be approximated within a certain factor, then the corresponding interactive proof system cannot be simulated efficiently. The paper also discusses the implications of these results for related problems, including Hypergraph Transversal, Minimum Hitting Set, and Minimum Dominating Set. The paper concludes by showing that the results have significant implications for the approximability of these problems, demonstrating that they are hard to approximate within certain factors unless P = NP or NP is contained in a certain complexity class.