Lower Bounds for QBFs of Bounded Treewidth

Abstract

The problem of deciding the validity (QSAT) of quantified Boolean formulas (QBF) is a vivid research area with strong interest in practical as well as theoretical advances. This is witnessed by various solvers competing in the annual QBF evaluation as well as a notable progress in the theoretical understanding of the problem. One reason for the interest in QBF is that it is the prototypical problem for the polynomial hierarchy; another lies in its convenience for encoding problems from artificial intelligence and beyond. 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 [10] in parameterized complexity is that QSAT when parameterized by the treewidth of the primal graph of the input formula together with the quantifier depth of the 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 depth.A natural question is whether one can significantly improve these results and decrease the tower while taking only standard assumptions in computational complexity into account, i.e., assuming the Exponential Time Hypothesis (ETH). Intuitively, under ETH one cannot solve 3-SAT in time 2 o (n) where n is the number of variables. 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 second level of the polynomial hierarchy and a handful for the third level. 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 depth) and thereby cover the full polynomial hierarchy. More formally, we establish lower bounds for QSAT and treewidth, namely, that under ETH there cannot be an algorithm that solves QSAT of quantifier depth i in runtime significantly better than i-fold exponential in the treewidth and polynomial in the input size. In doing so, we provide a versatile reduction technique to compress treewidth that encodes the essence of dynamic programming on arbitrary tree decompositions. By construction of the reduction, our results naturally carry over to the larger parameter pathwidth. As a by-product, our result renders the algorithm by Chen [10] asymptotically worst-case optimal.Further, we describe a general methodology for a more fine-grained analysis of problems parameterized by treewidth that are at higher levels of the polynomial hierarchy to foster research on lower bounds (parameterized by treewidth). 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.