Problems Parameterized by Treewidth Tractable in Single Exponential Time: A Logical Approach

摘要

We introduce a variant of modal logic, named Existential Counting Modal Logic (ECML), which captures a good number of problems known to be tractable in single exponential time when parameterized by treewidth. It appears that all these results can be subsumed by the theorem that model checking of ECML admits an algorithm with such complexity. We extend ECML by adding connectivity requirements and, using the Cut&Count technique introduced by Cygan et al. [4], prove that problems expressible in the extension are also tractable in single exponential time when parameterized by treewidth; however, using randomization. The need for navigational character of the introduced logic is informally justified by a negative result that two expository problems involving non-acyclic local conditions, C_1 -Vertex Deletion and Girth > l Vertex Deletion for l ≥ 5, do not admit such a robust algorithm unless Exponential Time Hypothesis fails.