Treewidth (tw) is an important parameter that, when bounded, yields tractability for many problems. For example, graph problems expressible in Monadic Second Order (MSO) logic and QUANTIFIED SAT or, more generally, QUANTIFIED CSP, are FPT parameterized by the tw of the input's (primal) graph plus the length of the MSO-formula [Courcelle, Information & Computation 1990] and the quantifier rank [Chen, ECAI 2004], resp. The algorithms from these (meta-)results have running times whose dependence on tw is a tower of exponents. A conditional lower bound by Fichte et al. [LICS 2020] shows that, for QUANTIFIED SAT, the height of this tower is equal to the number of quantifier alternations. Lower bounds showing that at least double-exponential factors in the running time are necessary are rare: there are very few (for tw and vertex cover vc parameterizations) and they are for problems that are complete for #NP, $\Sigma_2^p$, $\Pi_2^p$, or higher levels of the polynomial hierarchy. We show, for the first time, that it is not necessary to go higher up in the polynomial hierarchy to obtain such lower bounds. We design a novel, yet simple versatile technique based on Sperner families to obtain such lower bounds and apply it to 3 problems: METRIC DIMENSION, STRONG METRIC DIMENSION, and GEODETIC SET. We prove that they do not admit $2^{2^{o(tw)}} \cdot n^{O(1)}$-time algorithms, even on bounded diameter graphs, unless the ETH fails. For STRONG METRIC DIMENSION, the lower bound holds even for vc. We complement our lower bounds with matching upper bounds.

翻译：树宽（tw）是一个重要参数，当其有界时可使许多问题变得可解。例如，可用一元二阶（MSO）逻辑表达的图问题以及量化SAT（更一般地，量化CSP）问题，在输入（原始）图的tw加上MSO公式长度[Courcelle, Information & Computation 1990]和量词秩[Chen, ECAI 2004]为参数的情况下，属于FPT。这些（元）结果的算法运行时间中关于tw的依赖关系是指数塔形式。Fichte等人[LICS 2020]的条件性下界表明，对于量化SAT，该指数塔的高度等于量词交替次数。证明运行时间中至少需要双指数因子的下界很少：针对tw和顶点覆盖（vc）参数化的问题仅有极少数，且这些问题是#NP、$\Sigma_2^p$、$\Pi_2^p$或多项式层级中更高层级的完全问题。我们首次证明，要获得此类下界无需深入多项式层级的高层。我们基于Sperner族设计了一种新颖而简单的通用技术来获得此类下界，并将其应用于三个问题：度量维度、强度量维度和测地集。我们证明这些问题即使在有界直径图上也不存在$2^{2^{o(tw)}} \cdot n^{O(1)}$时间的算法（除非ETH失效）。对于强度量维度，该下界对vc参数化同样成立。我们还用匹配的上界补充了下界结果。