On approximation algorithms for hierarchical MAX-SAT

We prove upper and lower bounds on performance guarantees of approximation algorithms for the hierarchical MAX-SAT (H-MAX-SAT) problem. This problem is representative of an important class of PSPACE-hard problems involving graphs, Boolean formulas and other structures that are defined “succinctly”. Our first result is that for some constant ε<1, it is PSPACE-hard to approximate the function H-MAX-SAT to within ratio ε. We obtain our result using a known characterization of PSPACE in terms of probabilistically checkable debate systems. As an immediate application, we obtain non-approximability results for functions on hierarchical graphs by combining our result with previously known approximation-preserving reductions to other problems. For example, it is PSPACE-hard to approximate H-MAX-CUT and H-MAX-INDEPENDENT-SET to within some constant factor. Our second result is that there is an efficient approximation algorithm for H-MAX-SAT with performance guarantee 2/3. The previous best bound claimed for this problem was 1/2. One new technique of our algorithm can be used to obtain approximation algorithms for other problems, such as hierarchical MAX-CUT, which are simpler than previously known algorithms and which have performance guarantees that match the previous best bounds