Bounded queries, approximations, and the Boolean hierarchy
Information and Computation
STACS '03 Proceedings of the 20th Annual Symposium on Theoretical Aspects of Computer Science
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This paper investigates the computational complexity of approximating several \NP-optimization problems using the number of queries to an \NP\ oracle as a complexity measure. The results show a tradeoff between the closeness of the approximation and the number of queries required. For an approximation factor $k(n)$, $\log \log_{k(n)} n$ 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 $\log \log_{k(n)} n - c$ queries to any oracle unless $\Pe = \NP$, where $c$ is a constant that does not depend on $k$. These results hold for approximation factors $k(n) \geq 2$ that belong to a class of functions which includes any integer constant function, $\log n$, $\log^{a} n$, and $n^{1/a}$. Similar results are obtained for Graph Coloring, Set Cover, and other \NP-optimization problems.