Tight Double Exponential Lower Bounds

Ivan Bliznets, Markus Hecher

Abstract: The majority of established algorithms have polynomial, exponential, or factorial runtime complexities. Examples of problems that admit tight double or even triple exponential bounds on computational complexity are relatively rare. The Choosability problem is one such example. For this problem, Marx and Mitsou [ICALP 2016] presented a quite sophisticated proof that shows there is no \(\mathcal {O}(2^{2^{o(tw)}})n^{O(1)}\) time algorithm parameterized by treewidth tw, assuming ETH. In our paper, we show how we almost immediately come to the same conclusion knowing the reduction from \(\forall \exists \)-TQBF to the Choosability problem. Besides, in some sense, we provide a factory that produces problems with tight double exponential lower bounds not only in terms of treewidth but also pathwidth, cutwidth, and bandwidth. It was suspected that the \(\Pi ^\textrm{P}_2\)-complete or \(\Sigma ^\textrm{P}_2\)-complete problems require a double exponential time algorithm in terms of treewidth. However, in our paper, we provide a counterexample to this statement.