QBF as an Alternative to Courcelle's Theorem
This provides a new approach for algorithm designers in artificial intelligence and theoretical computer science to efficiently solve complex problems on bounded-treewidth graphs, though it is incremental as it builds on existing methods like Courcelle's Theorem and dynamic programming.
The paper tackles the problem of designing fixed-parameter linear algorithms for AI problems parameterized by treewidth by proposing reductions to quantified Boolean formulas (QBF), resulting in algorithms with concrete and tight runtime guarantees that are essentially optimal in treewidth dependence.
We propose reductions to quantified Boolean formulas (QBF) as a new approach to showing fixed-parameter linear algorithms for problems parameterized by treewidth. We demonstrate the feasibility of this approach by giving new algorithms for several well-known problems from artificial intelligence that are in general complete for the second level of the polynomial hierarchy. By reduction from QBF we show that all resulting algorithms are essentially optimal in their dependence on the treewidth. Most of the problems that we consider were already known to be fixed-parameter linear by using Courcelle's Theorem or dynamic programming, but we argue that our approach has clear advantages over these techniques: on the one hand, in contrast to Courcelle's Theorem, we get concrete and tight guarantees for the runtime dependence on the treewidth. On the other hand, we avoid tedious dynamic programming and, after showing some normalization results for CNF-formulas, our upper bounds often boil down to a few lines.