Lower Bounds for QBFs of Bounded Treewidth

The problem of deciding the validity (QSat) of quantified Boolean formulas (QBF) is a vivid research area in both theory and practice. In the field of parameterized algorithmics, the well-studied graph measure treewidth turned out to be a successful parameter. A well-known result by Chen [9] is that QSat when parameterized by the treewidth of the primal graph and the quantifier rank of the input formula is fixed-parameter tractable. More precisely, the runtime of such an algorithm is polynomial in the formula size and exponential in the treewidth, where the exponential function in the treewidth is a tower, whose height is the quantifier rank. A natural question is whether one can significantly improve these results and decrease the tower while assuming the Exponential Time Hypothesis (ETH). In the last years, there has been a growing interest in the quest of establishing lower bounds under ETH, showing mostly problem-specific lower bounds up to the third level of the polynomial hierarchy. Still, an important question is to settle this as general as possible and to cover the whole polynomial hierarchy. In this work, we show lower bounds based on the ETH for arbitrary QBFs parameterized by treewidth and quantifier rank. More formally, we establish lower bounds for QSat and treewidth, namely, that under ETH there cannot be an algorithm that solves QSat of quantifier rank i in runtime significantly better than i-fold exponential in the treewidth and polynomial in the input size. In doing so, we provide a reduction technique to compress treewidth that encodes dynamic programming on arbitrary tree decompositions. Further, we describe a general methodology for a more finegrained analysis of problems parameterized by treewidth that are at higher levels of the polynomial hierarchy. Finally, we illustrate the usefulness of our results by discussing various applications of our results to problems that are located higher on the polynomial hierarchy, in particular, various problems from the literature such as projected model counting problems.