On bounded queries and approximation

This paper investigates the computational complexity of approximating NP-optimization problems using the number of queries to an NP oracle as a complexity measure. The results show a trade-off between the closeness of the approximation and the number of queries required. For an approximation factor k(n), loglog_k(n) n queries to an NP oracle can be used to approximate the maximum clique size of a graph within a factor of k(n). However, this approximation cannot be achieved using fewer than loglog_k(n) n-c queries to any oracle unless P=NP, where c is a constant that does not depend on k. These results hold when k(n) belongs to a class of functions which include any integer constant function, log n, log^a n and n¹a/. Similar results are obtained for graph coloring, set cover and other NP-optimization problems