On the query complexity of clique size and maximum satisfiability

Abstract

The paper explores the bounded query complexity of approximating the size of the maximum clique in a graph (clique size) and the number of simultaneously satisfiable clauses in a 3CNF formula (MAX3SAT). The results show that for certain approximation factors, approximating clique size and MAX3SAT are complete for corresponding bounded query classes under metric reductions. The completeness result is important because it shows that queries and approximation are interchangeable: NP queries can be used to solve NP-approximation problems and solutions to NP-approximation problems answer queries to NP oracles. Completeness also shows the existence of approximation preserving reductions from many NP-approximation problems to approximating clique size and MAX3SAT (e.g., from approximating chromatic number to approximating clique size). Finally, since query complexity is a quantitative complexity measure, these results also provide a framework for comparing the complexities of approximating clique size and approximating MAX3SAT.<>